Abstract

Recently, we demonstrated an interferometric materials analysis scheme at terahertz frequencies based on the self-mixing effect in terahertz quantum cascade lasers. Here, we examine the impact of variations in laser operating parameters, target characteristics, laser–target system properties, and the quality calibration standards on our scheme. We show that our coherent scheme is intrinsically most sensitive to fluctuations in interferometric phase, arising primarily from variations in external cavity length. Moreover we demonstrate that the smallest experimental uncertainties in the determination of extinction coefficients are expected for lossy materials.

© 2014 Optical Society of America

1. Introduction

In [1], we exploited the self-mixing effect in terahertz (THz) quantum cascade lasers (QCLs) to create an ultimately compact and easy to implement materials analysis and coherent imaging scheme for remote targets. Our scheme enables the complex refractive index of targets to be measured by combining a local oscillator, mixer, and the detector all in a single THz QCL through forming a self-mixing interferometer. This approach is non-destructive, and has been demonstrated in the reflection mode, permitting its use on thick samples. Potential applications areas in the THz frequency range include biomedical, pharmaceutical, security, industrial, and cultural heritage [216]. Moreover, the self-mixing interferometer at the heart of our scheme has been demonstrated over a wide range of devices and wavelengths, including THz QCLs [1,1720], mid-infrared QCLs [2123] and interband cascade lasers [24], and near infrared [25] and visible diode lasers [26]. In this work, we quantify the impact of variations in system parameters on our scheme, evaluating its usefulness for different laser–target conditions. Throughout, we point to the THz QCL as an exemplar device for which we have demonstrated the effectiveness of the scheme. However, the reader should bear in mind that our scheme is not fundamentally linked to the particular emission frequency of the device nor to the particular device type.

Our method for materials analysis involves interrogating an external target by applying a small linear frequency sweep to the laser and measuring the resulting interferometric signal. This signal is imprinted with information about the complex refractive index of the target, which can be retrieved through model fitting and system calibration. There are a number of potential sources of systematic and random error in the scheme, including variations in the laser operating parameters (fluctuations in frequency, relative intensity noise), variations in the target (surface profile and roughness, material homogeneity and granularity), variations in the laser–target system (tilt due to misalignment, vibrations of the target), and variations in system calibration (imperfect calibration standards). In this paper, we first detail how we describe the optical properties of the target and how the interferometric signal resulting from our scheme is modelled. In the process, it becomes clear which characteristics of the interferometric system impact the signal, and the nature by which the optical constants of the target are imprinted on the signal. In addition, we detail the system calibration scheme, and remark on the propagation of calibration errors. Second, we examine how systematic changes to elements of the system manifest as changes in the signal, and identify which of these are dominant. Third, we examine how random fluctuations in the system components affect the interferometric signal, and identify when consideration of these random fluctuations becomes important. Finally, we conclude with a discussion of the implications and applicability of the scheme, drawing on the analysis conducted herein.

2. Background

2.1. Optical properties of the target

We model the optical properties of the target material at a particular spatial point as a (non-magnetic) homogeneous half-space with complex refractive index (using the positive time convention e+jωt) = n − jk (with j1, where n is referred to as the refractive index of the target material, and k is known as its extinction coefficient. Equivalent information is contained in the reflectivity of the material R and the phase-shift incurred by THz radiation on reflection θR. The dependence of (R, θR) on (n, k) is given through the pair of relations [27]

R=(n0n)2+k2(n0+n)2+k2,θR=arctan(2n0kn02n2k2),
where n0 is the refractive index of the incident material (typically air), which we take to be n0 = 1 from this point onward. Conversely, the dependence of (n, k) on (R, θR) is given through the pair of relations [27]
n=1R1+R2Rcos(θR),k=1Rsin(θR)1+R2Rcos(θR).

2.2. Modelling the self-mixing signal

We use a three mirror model to describe the laser system under feedback [28] (see Fig. 1), which is equivalent to the steady-state solution to the well-established model for a laser experiencing optical feedback proposed by Lang and Kobayashi [29]. In this model, only one round-trip in the external cavity is considered (solid arrows in Fig. 1). When the external target is displaced longitudinally, the laser system is swept through a set of compound cavity resonances [19]. The equivalent effect may be obtained by changing the laser frequency, which is accomplished in [1] by applying a linear modulation of the laser driving current. The primary effect of this current sweep is a modulation of both the laser power and the voltage across the laser terminals. The secondary effect, which is of most importance here, is a chirp of the lasing frequency.

 

Fig. 1 Three mirror model for a laser under feedback. Light recirculates in the laser cavity between mirrors M0 and Ms — solid arrows in laser cavity of length L. Light exits through Ms, is reflected from the external target Mext, and is reinjected into the laser cavity — solid arrows in external cavity of length Lext.

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The total phase delay in the external cavity can be decomposed into the transmission phase delay arising from the round-trip through the cavity and the phase change on reflection from the target, which is material dependent. The second order effect of the linear current sweep is a linear chirp of the lasing frequency (600 MHz on 2.59 THz for a change in driving current of 50 mA in [1]), leading to a linear dependence of transmission phase with time. As a consequence of the slow current (and hence frequency) modulation of the laser, this scheme operates the laser as a swept-frequency continuous-wave homodyning transceiver, similar to self-mixing range finders [3032]. Due to the slow rate of the sweep, the laser under feedback transitions from one steady-state to another, effectively stepping continuously through a series of operating frequencies, which results in linear dependence of phase in time. Therefore the time-dependent external phase delay (interferometric phase) φ over one frequency modulation period T as a function of time is of the form

φ(t)=φ0+ΦΔTtθR,
where φ0 is the round-trip transmission phase delay in the external cavity at the start of the frequency sweep, ΦΔ is the interferometric phase deviation caused by the current (frequency) sweep, and θR is the phase change on reflection from the material under test (which is assumed not to change significantly with the small frequency sweep). Clearly, φ is a function of the instantaneous laser frequency, which depends on the level of feedback in the laser system.

According to the Lang and Kobayashi model for a semiconductor laser under optical feedback in a steady state [29], the laser frequency satisfies the phase condition (sometimes called the ‘excess phase equation’)

φSφFB=Csin(φFB+arctanα),
where φFB represents the total external round-trip phase at the perturbed laser frequency, φS represents the total external round-trip phase at the solitary laser frequency, C is the feedback parameter that depends on the amount of light coupled back into the laser cavity, and α is the linewidth enhancement factor [3337]. Note that both the solitary phase φS and the resulting feedback phase φFB will be of the form in (3) and that we have suppressed the explicit time dependence in (4) for notational simplicity. However, for concreteness, it is useful to think of φS as the phase stimulus varying linearly in time according to (3) and of φFB as solved from (4) being the resultant feedback phase. Solutions to (4) are not possible in closed form and therefore require numerical solution [38, 39]. The interferometric phase change is directly observable through the change in emitted optical power, or equivalently through the change in voltage across the laser terminals [1, 40]. In our scheme, the linear current sweep leads to a quasi-linear voltage change near threshold. As such, the self-mixing signal embedded in the modulated voltage signal is related to the phase change through
V=V0+βcos(φFB),
where V is the voltage waveform obtained after the removal of a common reference slope to remove the primary effect of the linear current sweep, V0 is a dc component of this signal (corresponding to a material-dependent voltage offset from the reference slope), and β is the modulation index [1]. For this modulation scheme, note that V is a function of time through its dependence on the interferometric phase φFB (whose explicit time dependence has been suppressed for notational simplicity).

The refractive index n and the extinction coefficient k directly affect the self-mixing voltage in our model through the phase-shift on reflection θR in (3), which can be obtained from (1). Moreover, the reflectivity of the target R is directly linked to the model parameters C and α through the definition of the feedback parameter C, given by [36]

CC(α)=τextτLκext1+α2,
where τext is the round-trip propagation time in the external cavity, τL is the round-trip delay of the solitary laser, and κext is the coupling coefficient that depends on the reflectivity of the emitting mirror of the laser Rs and the reflectivity of the target R (associated with mirrors Ms and Mext in Fig. 1, respectively), as
κext=εRRs(1Rs).
Here ε represents the fraction of emitted power that is re-injected into the laser cavity, accounting for power loss on transmission in the external cavity and partial re-injection into the laser cavity. The reflectivity of the external target is usually not known — however we shall later make use of the form of (7) to recover unknown reflectivities using a pair of laser–target system calibration standards.

2.3. Measured reflectivity and phase-shift on reflection proxies

We introduce two observable variables (RM and θRM) which are related to the physical properties of the target (R and θR) in a known way. We will term them “proxies” for the actual reflectivity RAR (sometimes termed reflectance) and phase-shift on reflection θRAθR, where we have introduced the superscript A to make the distinction of actual target properties clear from measured target properties (which have been designated with the superscript M). From (6) and (7), we take our proxy for RA as RM, (theoretically) given by

RM=C1+α2=ε(1Rs)ninLinRsLextRA,
so that RM is a multiple of RA in the ideal case. For the reflection phase proxy θRM we conveniently choose the actual reflection phase with the reference plane translated to coincide with the plane of the laser exit mirror. Therefore, θRM contains not only the actual reflection phase θRA, but also the phase delay due to the transmission to the target and back. This is in accordance with (3), evaluated at a particular (unperturbed) frequency ν0 during the frequency sweep:
θRM=θRA4πcν0Lext,
so that, ideally, θRM is θRA together with an additive term corresponding to the total phase-shift on transmission. Notice that, ideally, both RM and θRM are linear in RA and θRA, respectively. Of course in practice the ideal case is not achieved, and so we must first calibrate the linear dependence of the reflectivity and phase proxies on their actual counterparts. Our approach to calibration is detailed next in Section 2.4. Following calibration, we will examine precisely how systematic and random changes in laser–target system parameters impact RM and θRM in Sections 3 and 4.

2.4. System calibration

Due to the linear dependence of RM on RA in practice, which may result from external reflections other than that from the target (including reflections from the cryostat shield and the window), we write

RM=aR+bRRA
where RA is the actual reflectivity of the material under test, aR and bR are unknown parameters to be determined, and RM = C2/(1 + α2), as defined earlier [see (8)], is representative of the material’s measured, but uncalibrated reflectivity. Along similar lines, to account for systematic phase uncertainties, we express θR [see (9)] as
θRM=aθ+bθθRA,
where θRA is the actual phase shift on reflection, aθ and bθ are unknown parameters to be determined, and θRM (which we call the phase proxy) is representative of the uncalibrated phase shift on reflection with the reference plane translated to the laser exit mirror. Equations (10) and (11) contain four unknown parameters, aR, bR, aθ, and bθ, which can be determined by model fitting (see Appendix C) from two measurements on materials with known ( RA, θRA) values, which can be viewed as a set of four linear equations with four unknowns. Denoting the calibration pairs of measured and actual reflectivities and phase-shifts for our two standards as ( R1M, R1A) ( θR,1M, θR,1A), and ( R2M, R2A) ( θR,2M, θR,2A) respectively, the set of four linear equations are
R1A=aR+bRR1M,R2A=aR+bRR2M,θR,1A=aθ+bθθR,1M,θR,2A=aθ+bθθR,2M.
The solution of this system of equations is straightforward and provides values for aR, bR, aθ, and bθ. In particular,
aR=R1AR2MR1MR2AR1MR2M,bR=R1AR2MR1MR2M,
aθ=θR,1AθR,2MθR,1MθR,2AθR,1MθR,2M,bθ=θR,1AθR,2AθR,1MθR,2M.
Once these values have been obtained we can readily calculate actual values for θRA and RA for the material under test using (10) and (11).

It is clear that an error in calibration standard ΔRA and ΔθRA will propagate as both an additive error (through the constant calibration terms) and as a multiplicative error (through the linear terms) to each measurement ( RMor θRM). Therefore, for this scheme, it is crucial that the optical constants of calibration standards be known accurately in order to minimise the introduction of calibration error, which would compound other errors present in the system.

3. Impact of systematic variations

In practice, we combine several measurements from the same (homogeneous) target material, either from repeated measurements at the same location on the target, from multiple spatial locations of the same target material, or both. In an ideal situation, all of these measurements would be identical. However, in practice there will be some error leading to differences in recorded results which will need to be accounted for.

There are a number of potential sources of systematic errors which contribute to changes in the self-mixing signal. One of the potential sources in the laser operating parameters is a systematic change in lasing frequency ν (arising for example from phase-instability or large linewidth) from the unperturbed frequency ν0 (other than the intended change caused by the frequency sweep). Potential sources in target characteristics are systematic local variation in external cavity length due to the target surface profile, and systematic spatial variation in optical constants of the target material (unintentional material inhomogeneity). Another potential source of error in laser–target system characteristics is the tilt of the target, leading to systematic change in external cavity length for different lateral positions of the probing beam on the target.

Changes in external cavity length (ΔL), lasing frequency (Δν), phase-shift on reflection ( ΔθRA), and amplitude reflection coefficient of the external target ( ΔRA) — all of which can take positive or negative values — enter the excess phase equation as changes to (3) through

φ+Δφ=4πc(ν0+Δν)(Lext+ΔL)(θRA+θRA),
as well as interacting nonlinearly with changes in the feedback parameter (6) through the solution of (4) via
C+ΔC=ε1+α2(1Rs)ninLinRs(Lext+ΔL)(RA+ΔRA).
Note that the approach taken in this analysis is similar to the concept of noise-equivalent phase [25, 41].

From (9) and (15), changes in external cavity length and changes in lasing frequency impact the “measured” phase-shift on reflection (9) through

θRM=θRA+ΔθRA4πc(ν0Lext+ν0ΔL+ΔνLext+ΔνΔL).
From (17), we see that changes in external cavity length and changes in lasing frequency affect the proxy for the phase-shift on reflection θRM in the same way. Directly from (8) and (16), these changes also impact the “measured” reflectivity through
RM=ε(1Rs)ninLinRs(Lext+ΔL)(RA+ΔRA).
From (18), we see that changes in external cavity length and changes in reflectivity affect the proxy for the reflectivity RM in the same way.

In practice, several measurements of the target material are acquired across its spatial extent. Clearly, if systematic variations in lasing frequency (the known frequency sweep) and external cavity length (such as tilt of the target surface) can be accounted for (see Appendix A), then the remaining variations in the self-mixing signal can be viewed as a consequence of variations in the reflectivity and phase-shift on reflection of the material (provided the shape of the target surface is known, e.g. it is flat).

Equations (17) and (18) can be rewritten in terms relative to wavelength and frequency. Writing the change in external cavity length in terms of the unperturbed laser wavelength λ0L = γλ0, so that the fractional change in external cavity length relative to λ0 is γ = ΔL/λ0), the external cavity length in terms of the unperturbed laser wavelength (Lext = Γλ0), and the change in lasing frequency in terms of the unperturbed lasing frequency ν0ν = η ν0, so that the fractional change in frequency is η = Δ ν/ν0), (17) can be cast as

θRM=θRA+ΔθRA4π{Γ+γ+η(Γ+γ)}.
Therefore, with reference to a particular external cavity length and unperturbed laser wavelength (resulting in a particular phase-shift on transmission 4πΓ), and based on the condition πθRM+4πΓθRAπ, an absolute limit on the ability to uniquely recover the phase-shift on reflection (with respect to a particular phase-shift on transmission 4πΓ) from (19) is
ΔθRA4π=14γ+η(Γ+γ)ΔθRA4π+14.
Similarly, (18) can be cast as
RM=ε(1Rs)ninLinRs(Γ+γ)λ0(RA+ΔRA).
When γ ≪ Γ (that is ΔLLext), RM does not change significantly with small changes in the surface profile. Hence, θRM will usually be comparatively more sensitive to such changes, and thereby contribute most to changes in n and k.

Figure 2 shows quantitatively how the systematic changes discussed here are manifested as equivalent changes in amplitude reflection coefficient ΔRA and phase-shift on reflection ΔθRA. For example, consider an optically-flat target with uncertainty in external cavity length on the order of λ0/10 (γ = 0.1). For all semiconductor laser wavelengths, it is possible to polish a target sample to λ0/10; for terahertz frequencies, this is particularly easy to accomplish. From the diagonal blue γ = 0.1 line and with λ0 = 100 μm, one obtains uncertainty equivalent to ΔRA=104. If one wishes to determine the corresponding uncertainty in θRA, one examines the horizontal red γ = 0.1 line to find uncertainty equivalent to ΔθRA=4π·101 rad. As such we can confirm that θRA is more sensitive than RA to changes in cavity length.

 

Fig. 2 Systematic sensitivity of the system when η ≈ 0 and ηΓ 0, in terms of equivalent change in amplitude reflection coefficient ΔRA and phase-shift on reflection ΔθRA, for a range of laser wavelengths (frequencies). Blue contours are lines of constant λ0γ. Red contours are lines of constant γ.

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4. Impact of random variations

There are a number of noise sources that contribute to noise in the self-mixing signal. Potential sources related to operation of the laser are shot-noise associated with the current source, temperature fluctuations which lead to instantaneous frequency deviations, laser intensity and phase noise, and laser sensitivity to fluctuations in the optical feedback as the frequency is swept. Potential noise sources in target characteristics are random local variation in the external cavity length arising from surface profile variation, and random spatial variation in optical constants. Potential noise sources in laser–target system characteristics are fluctuations in external cavity length due to vibrations, and spatial variations in the material properties of the external cavity.

Here, we examine how the distribution of random fluctuations in the measurement variables ( RM, θRM) propagates through to (n, k) parameter space. We assume a bivariate Gaussian error distribution in ( RM, θRM) space, denoted here by

(RMθRM)~N(μ,Σ),
where
μ=(𝔼RM𝔼θRM),andΣ=(Var(RM)Cov(RM,θRM)Cov(RM,θRM)Var(θRM)).
Note that μ and Σ can be estimated directly from measured data — once the systematic error has been removed. Moreover, we can quantify how random temporal or spatial fluctuations in lasing frequency (such as relative intensity noise), external cavity length (for example due to surface roughness), and material properties (for example due to sample inhomogeneity) impact the covariance of the measurement errors Σ.

We assume that there are fluctuations in lasing frequency ν, surface deviation from known systematic tilt (manifested through Lext), reflectivity RA, and phase-shift on reflection θRA, but all other quantities are fixed. Under the mild assumption of mutual independence of the random variations in ( θRA,ν, Lext) and in ( RA,Lext), meaning in particular that all covariances between the random variations in these quantities are zero, we have expected “measurements” [see also (8) and (9)]

𝔼RM=ε(1Rs)ninLinRs𝔼RA𝔼Lext,𝔼θRM=𝔼θRA4πc𝔼ν𝔼Lext.

The covariance and variances of the “measurements” are

Var(RM)=(ε(1Rs)ninLinRs)2(Var(Lext)Var(RA)+(𝔼Lext)2Var(RA)+(𝔼RA)2Var(Lext)),
Var(θRM)=Var(θMA)+(4πc)2(Var(Lext)Var(ν)+(𝔼Lext)2Var(ν)+(𝔼ν)2Var(Lext)),
Cov(RM,θRM)=ε(1Rs)ninLinRs4πc𝔼RA𝔼νVar(Lext).
We note particularly that variations in Lext contribute most to spurious variations in θRM. This can be seen by examining the second and third terms in the parenthesis of Var( θRM), and noting that the coefficient in front of the variance in (𝔼Lext)2Var(ν) is much smaller in magnitude than that in front of (𝔼ν)2Var(Lext) for most cases, reinforcing the earlier point that target smoothness is usually much more important than a pure spectrum. Moreover, as Lext and RA are typically of the similar order, variations in Lext are as important as variations in RA, once again reinforcing our earlier analysis.

Assuming a calibrated linear laser–target system relationship between a “measured” pair ( RM, θRM) and the actual pair ( RA, θRA) (see Section 2.4), we may write

(RAθRA)=A(RMθRM)+b~N(Aμ+b,AΣAT)
where A contains the calibration terms bR and bθ on the diagonal, and b contains the calibration terms aR and aθ. Then, via the delta method (see e.g. [42]), we may write the (local) distribution in (n, k) space as
(nk)=g(RM,θRM)~approxN(g(Aμ+b),JgAΣATJgT),
where Jg is the Jacobian of the transformation from g(n,k)(RA,θRA) evaluated at +b (see Appendix B).

In practice, an average of measurements is used rather than a single data point, so that

(n^k^)=g(RM¯,θRM¯),RM¯=1Ni=1NRMi,θRM¯=1Ni=1NθR,iM,
where, by assuming measurement errors are uncorrelated and therefore independent due to the Gaussian model here,
(RM¯θRM¯)~N(μ,Σ/N).
Therefore, the error-reduction factor of 1/N will propagate through the linear transformation (and Jacobian) to reduce the corresponding (local) covariance matrices in ( RA, θRA) and (n, k) space by a factor of 1/N.

The impact of random variations in optical constants is illustrated in Fig. 3, where lines of constant n (black) and k (red) are plotted for a range of (RA, θRA) pairs. Superimposed on Fig. 3 are the optical constants of several plastics at 2.59 THz, including those used in our study [1]. Particularly evident from Fig. 3 is that, as k decreases, the same variation in θRAtranslates to greater uncertainty in k. This means that the technique performs better for lossy materials. On the other hand, over the range of plastics considered here, the same variation in RA translates to similar levels of uncertainty in n.

 

Fig. 3 Sensitivity of the system to random variations in ( RA, θRA) for a range of plastics. 95% confidence intervals are drawn for each plastic, with Var(RA)0.00316 (Var( RA) = 10−5) and Var(θRA)0.00316 rad (Var( θRA) = 10−5 rad2), at ν = 2.59 THz with Var(ν)=10 kHz (Var(ν) = 108 Hz2), and an external cavity of Lext = 500 mm. Black contours are lines of constant n. Red contours are lines of constant k. Plastics at 2.59 THz are POM (dot), PA6 (circle), PVC (×), HDPE (cross), PTFE (star), PMMA (square), HDPE Black (diamond), and PC (triangle).

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Figure 4 shows the sensitivity of swept-frequency self-mixing signals as the external cavity or material properties change, synthesised for a 2.59 THz laser with a periodic sawtooth frequency chirp of 600 MHz and an external cavity length of 500 mm. Panels on the left are error-free self-mixing signals, and panels on the right are instances of the same signals with random error at every time-instant in Lext, RA, and θRA. The waveforms with random error show what one might expect to see in practice using a digital oscilloscope with simulated trace persistence.

 

Fig. 4 Sensitivity of swept-frequency self-mixing signals to systematic and random changes. (a) Systematic change in external cavity length (ΔL) of 0, 250, and 500 wavelengths on Lext = 500 mm, for fixed RA = 0.01 and θRA= 0.05 rad. (b) Systematic change in RA from 0.001, 0.01, and 0.1, for fixed Lext = 500 mm and θRA= 0.05 rad. (c) Systematic change in θRAfrom 0.01 rad, 0.05 rad, and 0.1 rad, for fixed Lext = 500 mm and RA = 0.01. (d)–(f) As (a)–(c), with additive zero-mean Gaussian variation in Lext with standard deviation 10−4 λ0 m; RAwith standard deviation 10−2; and θRAwith standard deviation 10−2 rad.

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5. Discussion

In [1], we demonstrated an imaging and material analysis scheme based on self-mixing with a THz QCL. There are many possible sources of systematic and random variation which may influence the scheme, arising from the laser operating parameters, the target characteristics, and the laser–target system characteristics. Here, we analysed the impact of such systematic and random errors on our material analysis scheme. As this scheme is coherent, it is not surprising that it is intrinsically most sensitive to fluctuations in interferometric phase. The main contributor to phase fluctuations was seen to be both systematic and random variations in the external cavity length, for example due to variation in surface profile of the remote target, tilt of the target, or mechanical vibration. Nevertheless, effects arising from sample tilt can be corrected for as described in Appendix A. Analysis of random noise sources also indicates that lossy materials lead to smaller experimental uncertainty in k. Moreover, the quality of calibration standards was seen to be crucial for the scheme. Characteristics of lasers ill-suited to our scheme include phase-instability under feedback, high phase and intensity noise, susceptibility to polarisation switching, and shorter wavelengths. As such, the THz QCL is particularly well suited to our scheme due to its narrow linewidth, long wavelength, and remarkable stability under feedback [22].

Appendix A: accounting for systematic variation in the external cavity

If systematic variation in the external cavity (due to alignment or surface profile) is known, then it can clearly be accounted for directly in (17) and (18). On the other hand, if systematic variation in the external cavity is not known, but can be assumed to be of a particular form, then it can be removed from RM and θRM by fitting the known (spatial) error description to data in the least-squares sense and accounting for it as above. This approach will be particularly effective if the target (or parts of the target) are of the same (homogeneous) material.

For example, if there is known to be systematic planar tilt over a particular (homogeneous) material patch on the target, then we may fit a plane to that region and remove, by writing

RMi=β0+βx(xix0)+βy(yiy0),
for each observation, where (xi, yi) represents the spatial coordinates on the target, (x0, y0) represents a particular relative coordinate (such as the center or a corner of the target), and (β0, βx, βy) represents the (unknown) parameters of the plane.

If we arrange N (one-dimensional) measured values into a column vector v — such as “measured” reflectivity —, then we may write the planar model as a (general) linear model v = +ε, where ε is a zero-mean vector of random errors, and

A=(1x1x0y1y01x2x0y2y01xNx0yNy0),β=(β0βxβy),
where A is the design matrix for the linear model, and β is the vector of unknown parameters. The least-squares solution β̂ can be found by solving the linear system of equations (the normal equations)
(ATA)β^=ATv.
The tilt-removed values (keeping the constant term) would then be
vi˜=viβx^(xix0)βy^(yiy0),i=1,,N.

Appendix B: Jacobian of the transformation from (n, k) to ( R,θR)

The Jacobian of the transformation in (2) from g(n,k)(R,θR)is given by

Jg=(nRnθRkRkθR,)
where each element of Jg can be calculated as
nR=2(2R+(1+R)cos(θR))(1+R2Rcos(θR))2,nθR=2R(1+R)sin(θR)(1+R2Rcos(θR))2,
kR=2(1+R)sin(θR)(1+R2Rcos(θR))2,kθR=2R(2R+(1+R)cos(θR))(1+R2Rcos(θR))2.

Appendix C: materials analysis model fitting

Our scheme contains a parametric model, based directly on the steady state solution to the Lang and Kobayashi model, that describes well the set of experimentally acquired time domain traces. Equations (3)(5) form a model with six key parameters, namely C, α, θR, ΦΔ, V0, and β. The information about the complex refractive index to be extracted is encoded mainly in C, α, and θR.

To extract self-mixing model parameters, we fit the model to data in the least-squares sense, for each spatial pixel of the target. Given an experimental signal (V1,V2,...,Vn) at times (t1,t2,...,tn), the goal is to obtain a parameter vector θ* that explains the systematic variation inherent in the observed self-mixing signal. This is achieved by finding a set of model parameters θ that minimises the (weighted) sum of squared differences between the (discrete) self-mixing signal and the corresponding discretely evaluated model. In other words,

θ*argminθΘS(θ),withS(θ)=k=1nwk(VkV(tk;θ))2,
where Θ is the permissible parameter space, and {wk} are (optional) weights. To proceed, for each candidate parameter vector θ ∈ Θ, we need to solve the excess-phase equation (4), and compute our modelled waveform V in (5) at the collection of time-points {t1,t2,...,tn}. Note that solving (4) requires care [38, 39]. This (weighted) least-squares approach is particularly useful when the acquired self-mixing signal is extremely noisy, as is typically the case with low-reflectivity targets.

Acknowledgments

This research was supported under Australian Research Council’s Discovery Projects funding scheme (DP 120 103703) and the European Cooperation in Science and Technology (COST) Action BM1205. Y.L.L. acknowledges support under the Queensland Government’s Smart Futures Fellowships programme. P.D. acknowledges support from the EPSRC (UK).

References and links

1. A. D. Rakić, T. Taimre, K. Bertling, Y. L. Lim, P. Dean, D. Indjin, Z. Ikonić, P. Harrison, A. Valavanis, S. P. Khanna, M. Lachab, S. J. Wilson, E. H. Linfield, and A. G. Davies, “Swept-frequency feedback interferometry using terahertz frequency QCLs: a method for imaging and materials analysis,” Opt. Express 21, 22194–22205 (2013). [CrossRef]  

2. T. Bardon, R. K. May, P. F. Taday, and M. Strlič, “Systematic study of terahertz time-domain spectra of historically informed black inks,” Analyst 138, 4859–4869 (2013). [CrossRef]   [PubMed]  

3. M. Bessou, H. Duday, J.-P. Caumes, S. Salort, B. Chassagne, A. Dautant, A. Ziéglé, and E. Abraham, “Advantage of terahertz radiation versus X-ray to detect hidden organic materials in sealed vessels,” Opt. Commun. 285, 4175–4179 (2012). [CrossRef]  

4. J.-P. Caumes, A. Younus, S. Salort, B. Chassagne, B. Recur, A. Ziéglé, A. Dautant, and E. Abraham, “Terahertz tomographic imaging of XVIIIth Dynasty Egyptian sealed pottery,” Appl. Opt. 50, 3604–3608 (2011). [CrossRef]   [PubMed]  

5. A. G. Davies, A. D. Burnett, W. Fan, E. H. Linfield, and J. E. Cunningham, “Terahertz spectroscopy of explosives and drugs,” Mater. Today 11, 18–26 (2008). [CrossRef]  

6. W. H. Fan, A. Burnett, P. C. Upadhya, J. Cunningham, E. H. Linfield, and A. G. Davies, “Far-infrared spectroscopic characterization of explosives for security applications using broadband terahertz time-domain spectroscopy,” Appl. Spectrosc. 61, 638–643 (2007). [CrossRef]   [PubMed]  

7. K. Fukunaga and M. Picollo, “Characterisation of Works of Art,” in Terahertz Spectroscopy and Imaging, K.-E. Peiponen, A. Zeitler, and M. Kuwata-Gonokami, (eds.), pp. 521–538, (Springer, Berlin, 2013).

8. D. Giovannacci, D. Martos-Levif, G. C. Walker, M. Menu, and V. Detalle, “Terahertz applications in cultural heritage: case studies,” in Fundamentals of Laser-Assisted Micro- and Nanotechnologies 2013, Proc. SPIE9065, 906510 (2013). [CrossRef]  

9. B. Hu and M. Nuss, “Imaging with terahertz waves,” Opt. Lett. 20, 1716–1718 (1995). [CrossRef]   [PubMed]  

10. S. Y. Huang, Y. X. J. Wang, D. K. W. Yeung, A. T. Ahuja, Y.-T. Zhang, and E. Pickwell-MacPherson, “Tissue characterization using terahertz pulsed imaging in reflection geometry,” Phys. Med. Biol. 54, 149–160 (2009). [CrossRef]  

11. P. U. Jepsen, J. K. Jensen, and U. Møller, “Characterization of aqueous alcohol solutions in bottles with THz reflection spectroscopy,” Opt. Express 16, 9318–9331 (2008). [CrossRef]   [PubMed]  

12. P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging — Modern techniques and applications,” Laser Photon.Rev. 5, 124–166 (2011). [CrossRef]  

13. N. Laman, S. S. Harsha, D. Grischkowsky, and J. S. Melinger, “7 GHz resolution waveguide THz spectroscopy of explosives related solids showing new features,” Opt. Express 16, 4094–4105 (2008). [CrossRef]   [PubMed]  

14. E. Pickwell and V. P. Wallace, “Biomedical applications of terahertz technology,” J. Phys. D: Appl. Phys. 39, R301–R310 (2006). [CrossRef]  

15. A. S. Skryl, J. B. Jackson, M. I. Bakunov, M. Menu, and G. A. Mourou, “Terahertz time-domain imaging of hidden defects in wooden artworks: application to a Russian icon painting,” Appl. Opt. 53, 1033–1038 (2014). [CrossRef]   [PubMed]  

16. G. C. Walker, J. W. Bowen, J. Labaune, J. B. Jackson, S. Hadjiloucas, J. Roberts, G. Mourou, and M. Menu, “Terahertz deconvolution,” Opt. Express 20, 27230 (2012). [CrossRef]   [PubMed]  

17. P. Dean, Y. L. Lim, A. Valavanis, R. Kliese, M. Nikolić, S. P. Khanna, M. Lachab, D. Indjin, Z. Ikonić, P. Harrison, A. D. Rakić, E. H. Linfield, and A. G. Davies, “Terahertz imaging through self-mixing in a quantum cascade laser,” Opt. Lett. 36, 2587–2589 (2011). [CrossRef]   [PubMed]  

18. R. P. Green, J.-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008). [CrossRef]  

19. Y. L. Lim, P. Dean, M. Nikolić, R. Kliese, S. P. Khanna, M. Lachab, A. Valavanis, D. Indjin, Z. Ikonić, P. Harrison, E. H. Linfield, A. G. Davies, S. J. Wilson, and A. D. Rakić, “Demonstration of a self-mixing displacement sensor based on terahertz quantum cascade lasers,” Appl. Phys. Lett. 99, 081108 (2011). [CrossRef]  

20. F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, M. S. Vitiello, and G. Scamarcio, “Imaging of free carriers in semiconductors via optical feedback in terahertz quantum cascade lasers,” Appl. Phys. Lett. 104, 041112 (2014). [CrossRef]  

21. J. von Staden, T. Gensty, W. Elsäßer, G. Giuliani, and C. Mann, “Measurements of the α factor of a distributed-feedback quantum cascade laser by an optical feedback self-mixing technique,” Opt. Lett. 31, 2574–2576 (2006). [CrossRef]   [PubMed]  

22. F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, S. Borri, M. S. Vitiello, H. E. Beere, D. A. Ritchie, and G. Scamarcio, “Intrinsic stability of quantum cascade lasers against optical feedback,” Opt. Express 21, 13748–13757 (2013). [CrossRef]   [PubMed]  

23. F. P. Mezzapesa, L. L. Columbo, A. Ancona, M. Dabbicco, V. Spagnolo, M. Brambilla, P. M. Lugarà, and G. Scamarcio, “On Line Sensing of Ultrafast Laser Microdrilling Processes by Optical Feedback Interferometry,” Phys. Procedia 41, 670–676 (2013). [CrossRef]  

24. K. Bertling, Y. L. Lim, T. Taimre, D. Indjin, P. Dean, R. Weih, S. Höfling, M. Kamp, M. von Edlinger, J. Koeth, and A. D. Rakić, “Demonstration of the self-mixing effect in interband cascade lasers,” Appl. Phys. Lett. 103, 231107 (2013). [CrossRef]  

25. G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A: Pure Appl. Opt. 4, S283–S294 (2002). [CrossRef]  

26. S. Donati, “Laser interferometry by induced modulation of cavity field,” J. Appl. Phys. 49, 495–497 (1978). [CrossRef]  

27. A. D. Rakić, “Algorithm for the determination of intrinsic optical constants of metal films: application to aluminum,” Appl. Opt. 34, 4755–4767 (1995). [CrossRef]  

28. K. Petermann, Laser Diode Modulation and Noise (Kluwer, Dordrecht, 1991).

29. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980). [CrossRef]  

30. G. Beheim and K. Fritsch, “Range finding using frequency-modulated laser diode,” Appl. Opt. 25, 1439–1442 (1986). [CrossRef]   [PubMed]  

31. F. Gouaux, N. Servagent, and T. Bosch, “Absolute distance measurement with an optical feedback interferometer,” Appl. Opt. 37, 6684–6689 (1998). [CrossRef]  

32. S. Shinohara, H. Yoshida, H. Ikeda, K. Nishide, and M. Sumi, “Compact and high-precision range finder with wide dynamic range and its application,” IEEE Trans. Instrum. Meas. 41, 40–44 (1992). [CrossRef]  

33. G. A. Acket, D. Lenstra, A. J. den Boef, and B. H. Verbeek, “The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers,” IEEE J. Quantum Electron. 20, 1163– 1169 (1984). [CrossRef]  

34. C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982). [CrossRef]  

35. M. Osiński and J. Buus, “Linewidth broadening factor in semiconductor lasers — An overview,” IEEE J. Quantum Electron. 23, 9–29 (1987). [CrossRef]  

36. P. Spencer, P. Rees, and I. Pierce, “Theoretical analysis,” in Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor LasersD. M. Kane and K. A. Shore, eds. (John Wiley & Sons, 2005), pp. 23–54. [CrossRef]  

37. T. Taimre and A. D. Rakić, “On the nature of Acket’s characteristic parameter C in semiconductor lasers,” Appl. Opt. 53, 1001–1006 (2014). [CrossRef]   [PubMed]  

38. R. Kliese, T. Taimre, A. A. A. Bakar, Y. L. Lim, K. Bertling, M. Nikolić, J. Perchoux, T. Bosch, and A. D. Rakić, “Solving self-mixing equations for arbitrary feedback levels: a concise algorithm,” Appl. Opt. 53, 3723–3736 (2014). [CrossRef]   [PubMed]  

39. G. Plantier, C. Bès, and T. M. Bosch, “Behavioral model of a self-mixing laser diode sensor,” IEEE J. Quantum Electron. 41, 1157–1167 (2005). [CrossRef]  

40. Y. L. Lim, K. Bertling, P. Rio, J. Tucker, and A. D. Rakić, “Displacement and distance measurement using the change in junction voltage across a laser diode due to the self-mixing effect,” in Photonics: Design, Technology, and Packaging II, D. Abbott, Y. S. Kivshar, H. H. Rubinsztein-Dunlop, and S. Fan, eds., Proc. SPIE6038, 60381O-1 (2006).

41. S. Donati, Electro-Optical Instrumentation: Sensing and Measuring with Lasers (Prentice Hall, Upper Saddle River, 2004).

42. D. P. Kroese, T. Taimre, and Z. I. Botev, Handbook of Monte Carlo Methods (Wiley, New York, 2011). [CrossRef]  

References

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  1. A. D. Rakić, T. Taimre, K. Bertling, Y. L. Lim, P. Dean, D. Indjin, Z. Ikonić, P. Harrison, A. Valavanis, S. P. Khanna, M. Lachab, S. J. Wilson, E. H. Linfield, and A. G. Davies, “Swept-frequency feedback interferometry using terahertz frequency QCLs: a method for imaging and materials analysis,” Opt. Express 21, 22194–22205 (2013).
    [CrossRef]
  2. T. Bardon, R. K. May, P. F. Taday, and M. Strlič, “Systematic study of terahertz time-domain spectra of historically informed black inks,” Analyst 138, 4859–4869 (2013).
    [CrossRef] [PubMed]
  3. M. Bessou, H. Duday, J.-P. Caumes, S. Salort, B. Chassagne, A. Dautant, A. Ziéglé, and E. Abraham, “Advantage of terahertz radiation versus X-ray to detect hidden organic materials in sealed vessels,” Opt. Commun. 285, 4175–4179 (2012).
    [CrossRef]
  4. J.-P. Caumes, A. Younus, S. Salort, B. Chassagne, B. Recur, A. Ziéglé, A. Dautant, and E. Abraham, “Terahertz tomographic imaging of XVIIIth Dynasty Egyptian sealed pottery,” Appl. Opt. 50, 3604–3608 (2011).
    [CrossRef] [PubMed]
  5. A. G. Davies, A. D. Burnett, W. Fan, E. H. Linfield, and J. E. Cunningham, “Terahertz spectroscopy of explosives and drugs,” Mater. Today 11, 18–26 (2008).
    [CrossRef]
  6. W. H. Fan, A. Burnett, P. C. Upadhya, J. Cunningham, E. H. Linfield, and A. G. Davies, “Far-infrared spectroscopic characterization of explosives for security applications using broadband terahertz time-domain spectroscopy,” Appl. Spectrosc. 61, 638–643 (2007).
    [CrossRef] [PubMed]
  7. K. Fukunaga and M. Picollo, “Characterisation of Works of Art,” in Terahertz Spectroscopy and Imaging, K.-E. Peiponen, A. Zeitler, and M. Kuwata-Gonokami, (eds.), pp. 521–538, (Springer, Berlin, 2013).
  8. D. Giovannacci, D. Martos-Levif, G. C. Walker, M. Menu, and V. Detalle, “Terahertz applications in cultural heritage: case studies,” in Fundamentals of Laser-Assisted Micro- and Nanotechnologies 2013, Proc. SPIE9065, 906510 (2013).
    [CrossRef]
  9. B. Hu and M. Nuss, “Imaging with terahertz waves,” Opt. Lett. 20, 1716–1718 (1995).
    [CrossRef] [PubMed]
  10. S. Y. Huang, Y. X. J. Wang, D. K. W. Yeung, A. T. Ahuja, Y.-T. Zhang, and E. Pickwell-MacPherson, “Tissue characterization using terahertz pulsed imaging in reflection geometry,” Phys. Med. Biol. 54, 149–160 (2009).
    [CrossRef]
  11. P. U. Jepsen, J. K. Jensen, and U. Møller, “Characterization of aqueous alcohol solutions in bottles with THz reflection spectroscopy,” Opt. Express 16, 9318–9331 (2008).
    [CrossRef] [PubMed]
  12. P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging — Modern techniques and applications,” Laser Photon.Rev. 5, 124–166 (2011).
    [CrossRef]
  13. N. Laman, S. S. Harsha, D. Grischkowsky, and J. S. Melinger, “7 GHz resolution waveguide THz spectroscopy of explosives related solids showing new features,” Opt. Express 16, 4094–4105 (2008).
    [CrossRef] [PubMed]
  14. E. Pickwell and V. P. Wallace, “Biomedical applications of terahertz technology,” J. Phys. D: Appl. Phys. 39, R301–R310 (2006).
    [CrossRef]
  15. A. S. Skryl, J. B. Jackson, M. I. Bakunov, M. Menu, and G. A. Mourou, “Terahertz time-domain imaging of hidden defects in wooden artworks: application to a Russian icon painting,” Appl. Opt. 53, 1033–1038 (2014).
    [CrossRef] [PubMed]
  16. G. C. Walker, J. W. Bowen, J. Labaune, J. B. Jackson, S. Hadjiloucas, J. Roberts, G. Mourou, and M. Menu, “Terahertz deconvolution,” Opt. Express 20, 27230 (2012).
    [CrossRef] [PubMed]
  17. P. Dean, Y. L. Lim, A. Valavanis, R. Kliese, M. Nikolić, S. P. Khanna, M. Lachab, D. Indjin, Z. Ikonić, P. Harrison, A. D. Rakić, E. H. Linfield, and A. G. Davies, “Terahertz imaging through self-mixing in a quantum cascade laser,” Opt. Lett. 36, 2587–2589 (2011).
    [CrossRef] [PubMed]
  18. R. P. Green, J.-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
    [CrossRef]
  19. Y. L. Lim, P. Dean, M. Nikolić, R. Kliese, S. P. Khanna, M. Lachab, A. Valavanis, D. Indjin, Z. Ikonić, P. Harrison, E. H. Linfield, A. G. Davies, S. J. Wilson, and A. D. Rakić, “Demonstration of a self-mixing displacement sensor based on terahertz quantum cascade lasers,” Appl. Phys. Lett. 99, 081108 (2011).
    [CrossRef]
  20. F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, M. S. Vitiello, and G. Scamarcio, “Imaging of free carriers in semiconductors via optical feedback in terahertz quantum cascade lasers,” Appl. Phys. Lett. 104, 041112 (2014).
    [CrossRef]
  21. J. von Staden, T. Gensty, W. Elsäßer, G. Giuliani, and C. Mann, “Measurements of the α factor of a distributed-feedback quantum cascade laser by an optical feedback self-mixing technique,” Opt. Lett. 31, 2574–2576 (2006).
    [CrossRef] [PubMed]
  22. F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, S. Borri, M. S. Vitiello, H. E. Beere, D. A. Ritchie, and G. Scamarcio, “Intrinsic stability of quantum cascade lasers against optical feedback,” Opt. Express 21, 13748–13757 (2013).
    [CrossRef] [PubMed]
  23. F. P. Mezzapesa, L. L. Columbo, A. Ancona, M. Dabbicco, V. Spagnolo, M. Brambilla, P. M. Lugarà, and G. Scamarcio, “On Line Sensing of Ultrafast Laser Microdrilling Processes by Optical Feedback Interferometry,” Phys. Procedia 41, 670–676 (2013).
    [CrossRef]
  24. K. Bertling, Y. L. Lim, T. Taimre, D. Indjin, P. Dean, R. Weih, S. Höfling, M. Kamp, M. von Edlinger, J. Koeth, and A. D. Rakić, “Demonstration of the self-mixing effect in interband cascade lasers,” Appl. Phys. Lett. 103, 231107 (2013).
    [CrossRef]
  25. G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A: Pure Appl. Opt. 4, S283–S294 (2002).
    [CrossRef]
  26. S. Donati, “Laser interferometry by induced modulation of cavity field,” J. Appl. Phys. 49, 495–497 (1978).
    [CrossRef]
  27. A. D. Rakić, “Algorithm for the determination of intrinsic optical constants of metal films: application to aluminum,” Appl. Opt. 34, 4755–4767 (1995).
    [CrossRef]
  28. K. Petermann, Laser Diode Modulation and Noise (Kluwer, Dordrecht, 1991).
  29. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
    [CrossRef]
  30. G. Beheim and K. Fritsch, “Range finding using frequency-modulated laser diode,” Appl. Opt. 25, 1439–1442 (1986).
    [CrossRef] [PubMed]
  31. F. Gouaux, N. Servagent, and T. Bosch, “Absolute distance measurement with an optical feedback interferometer,” Appl. Opt. 37, 6684–6689 (1998).
    [CrossRef]
  32. S. Shinohara, H. Yoshida, H. Ikeda, K. Nishide, and M. Sumi, “Compact and high-precision range finder with wide dynamic range and its application,” IEEE Trans. Instrum. Meas. 41, 40–44 (1992).
    [CrossRef]
  33. G. A. Acket, D. Lenstra, A. J. den Boef, and B. H. Verbeek, “The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers,” IEEE J. Quantum Electron. 20, 1163– 1169 (1984).
    [CrossRef]
  34. C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
    [CrossRef]
  35. M. Osiński and J. Buus, “Linewidth broadening factor in semiconductor lasers — An overview,” IEEE J. Quantum Electron. 23, 9–29 (1987).
    [CrossRef]
  36. P. Spencer, P. Rees, and I. Pierce, “Theoretical analysis,” in Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor LasersD. M. Kane and K. A. Shore, eds. (John Wiley & Sons, 2005), pp. 23–54.
    [CrossRef]
  37. T. Taimre and A. D. Rakić, “On the nature of Acket’s characteristic parameter C in semiconductor lasers,” Appl. Opt. 53, 1001–1006 (2014).
    [CrossRef] [PubMed]
  38. R. Kliese, T. Taimre, A. A. A. Bakar, Y. L. Lim, K. Bertling, M. Nikolić, J. Perchoux, T. Bosch, and A. D. Rakić, “Solving self-mixing equations for arbitrary feedback levels: a concise algorithm,” Appl. Opt. 53, 3723–3736 (2014).
    [CrossRef] [PubMed]
  39. G. Plantier, C. Bès, and T. M. Bosch, “Behavioral model of a self-mixing laser diode sensor,” IEEE J. Quantum Electron. 41, 1157–1167 (2005).
    [CrossRef]
  40. Y. L. Lim, K. Bertling, P. Rio, J. Tucker, and A. D. Rakić, “Displacement and distance measurement using the change in junction voltage across a laser diode due to the self-mixing effect,” in Photonics: Design, Technology, and Packaging II, D. Abbott, Y. S. Kivshar, H. H. Rubinsztein-Dunlop, and S. Fan, eds., Proc. SPIE6038, 60381O-1 (2006).
  41. S. Donati, Electro-Optical Instrumentation: Sensing and Measuring with Lasers (Prentice Hall, Upper Saddle River, 2004).
  42. D. P. Kroese, T. Taimre, and Z. I. Botev, Handbook of Monte Carlo Methods (Wiley, New York, 2011).
    [CrossRef]

2014 (4)

2013 (5)

F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, S. Borri, M. S. Vitiello, H. E. Beere, D. A. Ritchie, and G. Scamarcio, “Intrinsic stability of quantum cascade lasers against optical feedback,” Opt. Express 21, 13748–13757 (2013).
[CrossRef] [PubMed]

F. P. Mezzapesa, L. L. Columbo, A. Ancona, M. Dabbicco, V. Spagnolo, M. Brambilla, P. M. Lugarà, and G. Scamarcio, “On Line Sensing of Ultrafast Laser Microdrilling Processes by Optical Feedback Interferometry,” Phys. Procedia 41, 670–676 (2013).
[CrossRef]

K. Bertling, Y. L. Lim, T. Taimre, D. Indjin, P. Dean, R. Weih, S. Höfling, M. Kamp, M. von Edlinger, J. Koeth, and A. D. Rakić, “Demonstration of the self-mixing effect in interband cascade lasers,” Appl. Phys. Lett. 103, 231107 (2013).
[CrossRef]

A. D. Rakić, T. Taimre, K. Bertling, Y. L. Lim, P. Dean, D. Indjin, Z. Ikonić, P. Harrison, A. Valavanis, S. P. Khanna, M. Lachab, S. J. Wilson, E. H. Linfield, and A. G. Davies, “Swept-frequency feedback interferometry using terahertz frequency QCLs: a method for imaging and materials analysis,” Opt. Express 21, 22194–22205 (2013).
[CrossRef]

T. Bardon, R. K. May, P. F. Taday, and M. Strlič, “Systematic study of terahertz time-domain spectra of historically informed black inks,” Analyst 138, 4859–4869 (2013).
[CrossRef] [PubMed]

2012 (2)

M. Bessou, H. Duday, J.-P. Caumes, S. Salort, B. Chassagne, A. Dautant, A. Ziéglé, and E. Abraham, “Advantage of terahertz radiation versus X-ray to detect hidden organic materials in sealed vessels,” Opt. Commun. 285, 4175–4179 (2012).
[CrossRef]

G. C. Walker, J. W. Bowen, J. Labaune, J. B. Jackson, S. Hadjiloucas, J. Roberts, G. Mourou, and M. Menu, “Terahertz deconvolution,” Opt. Express 20, 27230 (2012).
[CrossRef] [PubMed]

2011 (4)

P. Dean, Y. L. Lim, A. Valavanis, R. Kliese, M. Nikolić, S. P. Khanna, M. Lachab, D. Indjin, Z. Ikonić, P. Harrison, A. D. Rakić, E. H. Linfield, and A. G. Davies, “Terahertz imaging through self-mixing in a quantum cascade laser,” Opt. Lett. 36, 2587–2589 (2011).
[CrossRef] [PubMed]

J.-P. Caumes, A. Younus, S. Salort, B. Chassagne, B. Recur, A. Ziéglé, A. Dautant, and E. Abraham, “Terahertz tomographic imaging of XVIIIth Dynasty Egyptian sealed pottery,” Appl. Opt. 50, 3604–3608 (2011).
[CrossRef] [PubMed]

P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging — Modern techniques and applications,” Laser Photon.Rev. 5, 124–166 (2011).
[CrossRef]

Y. L. Lim, P. Dean, M. Nikolić, R. Kliese, S. P. Khanna, M. Lachab, A. Valavanis, D. Indjin, Z. Ikonić, P. Harrison, E. H. Linfield, A. G. Davies, S. J. Wilson, and A. D. Rakić, “Demonstration of a self-mixing displacement sensor based on terahertz quantum cascade lasers,” Appl. Phys. Lett. 99, 081108 (2011).
[CrossRef]

2009 (1)

S. Y. Huang, Y. X. J. Wang, D. K. W. Yeung, A. T. Ahuja, Y.-T. Zhang, and E. Pickwell-MacPherson, “Tissue characterization using terahertz pulsed imaging in reflection geometry,” Phys. Med. Biol. 54, 149–160 (2009).
[CrossRef]

2008 (4)

P. U. Jepsen, J. K. Jensen, and U. Møller, “Characterization of aqueous alcohol solutions in bottles with THz reflection spectroscopy,” Opt. Express 16, 9318–9331 (2008).
[CrossRef] [PubMed]

N. Laman, S. S. Harsha, D. Grischkowsky, and J. S. Melinger, “7 GHz resolution waveguide THz spectroscopy of explosives related solids showing new features,” Opt. Express 16, 4094–4105 (2008).
[CrossRef] [PubMed]

A. G. Davies, A. D. Burnett, W. Fan, E. H. Linfield, and J. E. Cunningham, “Terahertz spectroscopy of explosives and drugs,” Mater. Today 11, 18–26 (2008).
[CrossRef]

R. P. Green, J.-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

2007 (1)

2006 (2)

2005 (1)

G. Plantier, C. Bès, and T. M. Bosch, “Behavioral model of a self-mixing laser diode sensor,” IEEE J. Quantum Electron. 41, 1157–1167 (2005).
[CrossRef]

2002 (1)

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A: Pure Appl. Opt. 4, S283–S294 (2002).
[CrossRef]

1998 (1)

1995 (2)

1992 (1)

S. Shinohara, H. Yoshida, H. Ikeda, K. Nishide, and M. Sumi, “Compact and high-precision range finder with wide dynamic range and its application,” IEEE Trans. Instrum. Meas. 41, 40–44 (1992).
[CrossRef]

1987 (1)

M. Osiński and J. Buus, “Linewidth broadening factor in semiconductor lasers — An overview,” IEEE J. Quantum Electron. 23, 9–29 (1987).
[CrossRef]

1986 (1)

1984 (1)

G. A. Acket, D. Lenstra, A. J. den Boef, and B. H. Verbeek, “The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers,” IEEE J. Quantum Electron. 20, 1163– 1169 (1984).
[CrossRef]

1982 (1)

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[CrossRef]

1980 (1)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[CrossRef]

1978 (1)

S. Donati, “Laser interferometry by induced modulation of cavity field,” J. Appl. Phys. 49, 495–497 (1978).
[CrossRef]

Abraham, E.

M. Bessou, H. Duday, J.-P. Caumes, S. Salort, B. Chassagne, A. Dautant, A. Ziéglé, and E. Abraham, “Advantage of terahertz radiation versus X-ray to detect hidden organic materials in sealed vessels,” Opt. Commun. 285, 4175–4179 (2012).
[CrossRef]

J.-P. Caumes, A. Younus, S. Salort, B. Chassagne, B. Recur, A. Ziéglé, A. Dautant, and E. Abraham, “Terahertz tomographic imaging of XVIIIth Dynasty Egyptian sealed pottery,” Appl. Opt. 50, 3604–3608 (2011).
[CrossRef] [PubMed]

Acket, G. A.

G. A. Acket, D. Lenstra, A. J. den Boef, and B. H. Verbeek, “The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers,” IEEE J. Quantum Electron. 20, 1163– 1169 (1984).
[CrossRef]

Ahuja, A. T.

S. Y. Huang, Y. X. J. Wang, D. K. W. Yeung, A. T. Ahuja, Y.-T. Zhang, and E. Pickwell-MacPherson, “Tissue characterization using terahertz pulsed imaging in reflection geometry,” Phys. Med. Biol. 54, 149–160 (2009).
[CrossRef]

Ancona, A.

F. P. Mezzapesa, L. L. Columbo, A. Ancona, M. Dabbicco, V. Spagnolo, M. Brambilla, P. M. Lugarà, and G. Scamarcio, “On Line Sensing of Ultrafast Laser Microdrilling Processes by Optical Feedback Interferometry,” Phys. Procedia 41, 670–676 (2013).
[CrossRef]

Bakar, A. A. A.

Bakunov, M. I.

Bardon, T.

T. Bardon, R. K. May, P. F. Taday, and M. Strlič, “Systematic study of terahertz time-domain spectra of historically informed black inks,” Analyst 138, 4859–4869 (2013).
[CrossRef] [PubMed]

Beere, H. E.

F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, S. Borri, M. S. Vitiello, H. E. Beere, D. A. Ritchie, and G. Scamarcio, “Intrinsic stability of quantum cascade lasers against optical feedback,” Opt. Express 21, 13748–13757 (2013).
[CrossRef] [PubMed]

R. P. Green, J.-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Beheim, G.

Beltram, F.

R. P. Green, J.-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Bertling, K.

R. Kliese, T. Taimre, A. A. A. Bakar, Y. L. Lim, K. Bertling, M. Nikolić, J. Perchoux, T. Bosch, and A. D. Rakić, “Solving self-mixing equations for arbitrary feedback levels: a concise algorithm,” Appl. Opt. 53, 3723–3736 (2014).
[CrossRef] [PubMed]

A. D. Rakić, T. Taimre, K. Bertling, Y. L. Lim, P. Dean, D. Indjin, Z. Ikonić, P. Harrison, A. Valavanis, S. P. Khanna, M. Lachab, S. J. Wilson, E. H. Linfield, and A. G. Davies, “Swept-frequency feedback interferometry using terahertz frequency QCLs: a method for imaging and materials analysis,” Opt. Express 21, 22194–22205 (2013).
[CrossRef]

K. Bertling, Y. L. Lim, T. Taimre, D. Indjin, P. Dean, R. Weih, S. Höfling, M. Kamp, M. von Edlinger, J. Koeth, and A. D. Rakić, “Demonstration of the self-mixing effect in interband cascade lasers,” Appl. Phys. Lett. 103, 231107 (2013).
[CrossRef]

Y. L. Lim, K. Bertling, P. Rio, J. Tucker, and A. D. Rakić, “Displacement and distance measurement using the change in junction voltage across a laser diode due to the self-mixing effect,” in Photonics: Design, Technology, and Packaging II, D. Abbott, Y. S. Kivshar, H. H. Rubinsztein-Dunlop, and S. Fan, eds., Proc. SPIE6038, 60381O-1 (2006).

Bès, C.

G. Plantier, C. Bès, and T. M. Bosch, “Behavioral model of a self-mixing laser diode sensor,” IEEE J. Quantum Electron. 41, 1157–1167 (2005).
[CrossRef]

Bessou, M.

M. Bessou, H. Duday, J.-P. Caumes, S. Salort, B. Chassagne, A. Dautant, A. Ziéglé, and E. Abraham, “Advantage of terahertz radiation versus X-ray to detect hidden organic materials in sealed vessels,” Opt. Commun. 285, 4175–4179 (2012).
[CrossRef]

Borri, S.

Bosch, T.

Bosch, T. M.

G. Plantier, C. Bès, and T. M. Bosch, “Behavioral model of a self-mixing laser diode sensor,” IEEE J. Quantum Electron. 41, 1157–1167 (2005).
[CrossRef]

Botev, Z. I.

D. P. Kroese, T. Taimre, and Z. I. Botev, Handbook of Monte Carlo Methods (Wiley, New York, 2011).
[CrossRef]

Bowen, J. W.

Brambilla, M.

F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, M. S. Vitiello, and G. Scamarcio, “Imaging of free carriers in semiconductors via optical feedback in terahertz quantum cascade lasers,” Appl. Phys. Lett. 104, 041112 (2014).
[CrossRef]

F. P. Mezzapesa, L. L. Columbo, A. Ancona, M. Dabbicco, V. Spagnolo, M. Brambilla, P. M. Lugarà, and G. Scamarcio, “On Line Sensing of Ultrafast Laser Microdrilling Processes by Optical Feedback Interferometry,” Phys. Procedia 41, 670–676 (2013).
[CrossRef]

F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, S. Borri, M. S. Vitiello, H. E. Beere, D. A. Ritchie, and G. Scamarcio, “Intrinsic stability of quantum cascade lasers against optical feedback,” Opt. Express 21, 13748–13757 (2013).
[CrossRef] [PubMed]

Burnett, A.

Burnett, A. D.

A. G. Davies, A. D. Burnett, W. Fan, E. H. Linfield, and J. E. Cunningham, “Terahertz spectroscopy of explosives and drugs,” Mater. Today 11, 18–26 (2008).
[CrossRef]

Buus, J.

M. Osiński and J. Buus, “Linewidth broadening factor in semiconductor lasers — An overview,” IEEE J. Quantum Electron. 23, 9–29 (1987).
[CrossRef]

Caumes, J.-P.

M. Bessou, H. Duday, J.-P. Caumes, S. Salort, B. Chassagne, A. Dautant, A. Ziéglé, and E. Abraham, “Advantage of terahertz radiation versus X-ray to detect hidden organic materials in sealed vessels,” Opt. Commun. 285, 4175–4179 (2012).
[CrossRef]

J.-P. Caumes, A. Younus, S. Salort, B. Chassagne, B. Recur, A. Ziéglé, A. Dautant, and E. Abraham, “Terahertz tomographic imaging of XVIIIth Dynasty Egyptian sealed pottery,” Appl. Opt. 50, 3604–3608 (2011).
[CrossRef] [PubMed]

Chassagne, B.

M. Bessou, H. Duday, J.-P. Caumes, S. Salort, B. Chassagne, A. Dautant, A. Ziéglé, and E. Abraham, “Advantage of terahertz radiation versus X-ray to detect hidden organic materials in sealed vessels,” Opt. Commun. 285, 4175–4179 (2012).
[CrossRef]

J.-P. Caumes, A. Younus, S. Salort, B. Chassagne, B. Recur, A. Ziéglé, A. Dautant, and E. Abraham, “Terahertz tomographic imaging of XVIIIth Dynasty Egyptian sealed pottery,” Appl. Opt. 50, 3604–3608 (2011).
[CrossRef] [PubMed]

Columbo, L. L.

F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, M. S. Vitiello, and G. Scamarcio, “Imaging of free carriers in semiconductors via optical feedback in terahertz quantum cascade lasers,” Appl. Phys. Lett. 104, 041112 (2014).
[CrossRef]

F. P. Mezzapesa, L. L. Columbo, A. Ancona, M. Dabbicco, V. Spagnolo, M. Brambilla, P. M. Lugarà, and G. Scamarcio, “On Line Sensing of Ultrafast Laser Microdrilling Processes by Optical Feedback Interferometry,” Phys. Procedia 41, 670–676 (2013).
[CrossRef]

F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, S. Borri, M. S. Vitiello, H. E. Beere, D. A. Ritchie, and G. Scamarcio, “Intrinsic stability of quantum cascade lasers against optical feedback,” Opt. Express 21, 13748–13757 (2013).
[CrossRef] [PubMed]

Cooke, D. G.

P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging — Modern techniques and applications,” Laser Photon.Rev. 5, 124–166 (2011).
[CrossRef]

Cunningham, J.

Cunningham, J. E.

A. G. Davies, A. D. Burnett, W. Fan, E. H. Linfield, and J. E. Cunningham, “Terahertz spectroscopy of explosives and drugs,” Mater. Today 11, 18–26 (2008).
[CrossRef]

Dabbicco, M.

F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, M. S. Vitiello, and G. Scamarcio, “Imaging of free carriers in semiconductors via optical feedback in terahertz quantum cascade lasers,” Appl. Phys. Lett. 104, 041112 (2014).
[CrossRef]

F. P. Mezzapesa, L. L. Columbo, A. Ancona, M. Dabbicco, V. Spagnolo, M. Brambilla, P. M. Lugarà, and G. Scamarcio, “On Line Sensing of Ultrafast Laser Microdrilling Processes by Optical Feedback Interferometry,” Phys. Procedia 41, 670–676 (2013).
[CrossRef]

F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, S. Borri, M. S. Vitiello, H. E. Beere, D. A. Ritchie, and G. Scamarcio, “Intrinsic stability of quantum cascade lasers against optical feedback,” Opt. Express 21, 13748–13757 (2013).
[CrossRef] [PubMed]

Dautant, A.

M. Bessou, H. Duday, J.-P. Caumes, S. Salort, B. Chassagne, A. Dautant, A. Ziéglé, and E. Abraham, “Advantage of terahertz radiation versus X-ray to detect hidden organic materials in sealed vessels,” Opt. Commun. 285, 4175–4179 (2012).
[CrossRef]

J.-P. Caumes, A. Younus, S. Salort, B. Chassagne, B. Recur, A. Ziéglé, A. Dautant, and E. Abraham, “Terahertz tomographic imaging of XVIIIth Dynasty Egyptian sealed pottery,” Appl. Opt. 50, 3604–3608 (2011).
[CrossRef] [PubMed]

Davies, A. G.

Dean, P.

K. Bertling, Y. L. Lim, T. Taimre, D. Indjin, P. Dean, R. Weih, S. Höfling, M. Kamp, M. von Edlinger, J. Koeth, and A. D. Rakić, “Demonstration of the self-mixing effect in interband cascade lasers,” Appl. Phys. Lett. 103, 231107 (2013).
[CrossRef]

A. D. Rakić, T. Taimre, K. Bertling, Y. L. Lim, P. Dean, D. Indjin, Z. Ikonić, P. Harrison, A. Valavanis, S. P. Khanna, M. Lachab, S. J. Wilson, E. H. Linfield, and A. G. Davies, “Swept-frequency feedback interferometry using terahertz frequency QCLs: a method for imaging and materials analysis,” Opt. Express 21, 22194–22205 (2013).
[CrossRef]

P. Dean, Y. L. Lim, A. Valavanis, R. Kliese, M. Nikolić, S. P. Khanna, M. Lachab, D. Indjin, Z. Ikonić, P. Harrison, A. D. Rakić, E. H. Linfield, and A. G. Davies, “Terahertz imaging through self-mixing in a quantum cascade laser,” Opt. Lett. 36, 2587–2589 (2011).
[CrossRef] [PubMed]

Y. L. Lim, P. Dean, M. Nikolić, R. Kliese, S. P. Khanna, M. Lachab, A. Valavanis, D. Indjin, Z. Ikonić, P. Harrison, E. H. Linfield, A. G. Davies, S. J. Wilson, and A. D. Rakić, “Demonstration of a self-mixing displacement sensor based on terahertz quantum cascade lasers,” Appl. Phys. Lett. 99, 081108 (2011).
[CrossRef]

den Boef, A. J.

G. A. Acket, D. Lenstra, A. J. den Boef, and B. H. Verbeek, “The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers,” IEEE J. Quantum Electron. 20, 1163– 1169 (1984).
[CrossRef]

Detalle, V.

D. Giovannacci, D. Martos-Levif, G. C. Walker, M. Menu, and V. Detalle, “Terahertz applications in cultural heritage: case studies,” in Fundamentals of Laser-Assisted Micro- and Nanotechnologies 2013, Proc. SPIE9065, 906510 (2013).
[CrossRef]

Donati, S.

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A: Pure Appl. Opt. 4, S283–S294 (2002).
[CrossRef]

S. Donati, “Laser interferometry by induced modulation of cavity field,” J. Appl. Phys. 49, 495–497 (1978).
[CrossRef]

S. Donati, Electro-Optical Instrumentation: Sensing and Measuring with Lasers (Prentice Hall, Upper Saddle River, 2004).

Duday, H.

M. Bessou, H. Duday, J.-P. Caumes, S. Salort, B. Chassagne, A. Dautant, A. Ziéglé, and E. Abraham, “Advantage of terahertz radiation versus X-ray to detect hidden organic materials in sealed vessels,” Opt. Commun. 285, 4175–4179 (2012).
[CrossRef]

Elsäßer, W.

Fan, W.

A. G. Davies, A. D. Burnett, W. Fan, E. H. Linfield, and J. E. Cunningham, “Terahertz spectroscopy of explosives and drugs,” Mater. Today 11, 18–26 (2008).
[CrossRef]

Fan, W. H.

Fritsch, K.

Fukunaga, K.

K. Fukunaga and M. Picollo, “Characterisation of Works of Art,” in Terahertz Spectroscopy and Imaging, K.-E. Peiponen, A. Zeitler, and M. Kuwata-Gonokami, (eds.), pp. 521–538, (Springer, Berlin, 2013).

Gensty, T.

Giovannacci, D.

D. Giovannacci, D. Martos-Levif, G. C. Walker, M. Menu, and V. Detalle, “Terahertz applications in cultural heritage: case studies,” in Fundamentals of Laser-Assisted Micro- and Nanotechnologies 2013, Proc. SPIE9065, 906510 (2013).
[CrossRef]

Giuliani, G.

R. P. Green, J.-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

J. von Staden, T. Gensty, W. Elsäßer, G. Giuliani, and C. Mann, “Measurements of the α factor of a distributed-feedback quantum cascade laser by an optical feedback self-mixing technique,” Opt. Lett. 31, 2574–2576 (2006).
[CrossRef] [PubMed]

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A: Pure Appl. Opt. 4, S283–S294 (2002).
[CrossRef]

Gouaux, F.

Green, R. P.

R. P. Green, J.-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Grischkowsky, D.

Hadjiloucas, S.

Harrison, P.

Harsha, S. S.

Henry, C. H.

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[CrossRef]

Höfling, S.

K. Bertling, Y. L. Lim, T. Taimre, D. Indjin, P. Dean, R. Weih, S. Höfling, M. Kamp, M. von Edlinger, J. Koeth, and A. D. Rakić, “Demonstration of the self-mixing effect in interband cascade lasers,” Appl. Phys. Lett. 103, 231107 (2013).
[CrossRef]

Hu, B.

Huang, S. Y.

S. Y. Huang, Y. X. J. Wang, D. K. W. Yeung, A. T. Ahuja, Y.-T. Zhang, and E. Pickwell-MacPherson, “Tissue characterization using terahertz pulsed imaging in reflection geometry,” Phys. Med. Biol. 54, 149–160 (2009).
[CrossRef]

Ikeda, H.

S. Shinohara, H. Yoshida, H. Ikeda, K. Nishide, and M. Sumi, “Compact and high-precision range finder with wide dynamic range and its application,” IEEE Trans. Instrum. Meas. 41, 40–44 (1992).
[CrossRef]

Ikonic, Z.

Indjin, D.

K. Bertling, Y. L. Lim, T. Taimre, D. Indjin, P. Dean, R. Weih, S. Höfling, M. Kamp, M. von Edlinger, J. Koeth, and A. D. Rakić, “Demonstration of the self-mixing effect in interband cascade lasers,” Appl. Phys. Lett. 103, 231107 (2013).
[CrossRef]

A. D. Rakić, T. Taimre, K. Bertling, Y. L. Lim, P. Dean, D. Indjin, Z. Ikonić, P. Harrison, A. Valavanis, S. P. Khanna, M. Lachab, S. J. Wilson, E. H. Linfield, and A. G. Davies, “Swept-frequency feedback interferometry using terahertz frequency QCLs: a method for imaging and materials analysis,” Opt. Express 21, 22194–22205 (2013).
[CrossRef]

P. Dean, Y. L. Lim, A. Valavanis, R. Kliese, M. Nikolić, S. P. Khanna, M. Lachab, D. Indjin, Z. Ikonić, P. Harrison, A. D. Rakić, E. H. Linfield, and A. G. Davies, “Terahertz imaging through self-mixing in a quantum cascade laser,” Opt. Lett. 36, 2587–2589 (2011).
[CrossRef] [PubMed]

Y. L. Lim, P. Dean, M. Nikolić, R. Kliese, S. P. Khanna, M. Lachab, A. Valavanis, D. Indjin, Z. Ikonić, P. Harrison, E. H. Linfield, A. G. Davies, S. J. Wilson, and A. D. Rakić, “Demonstration of a self-mixing displacement sensor based on terahertz quantum cascade lasers,” Appl. Phys. Lett. 99, 081108 (2011).
[CrossRef]

Jackson, J. B.

Jensen, J. K.

Jepsen, P. U.

P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging — Modern techniques and applications,” Laser Photon.Rev. 5, 124–166 (2011).
[CrossRef]

P. U. Jepsen, J. K. Jensen, and U. Møller, “Characterization of aqueous alcohol solutions in bottles with THz reflection spectroscopy,” Opt. Express 16, 9318–9331 (2008).
[CrossRef] [PubMed]

Kamp, M.

K. Bertling, Y. L. Lim, T. Taimre, D. Indjin, P. Dean, R. Weih, S. Höfling, M. Kamp, M. von Edlinger, J. Koeth, and A. D. Rakić, “Demonstration of the self-mixing effect in interband cascade lasers,” Appl. Phys. Lett. 103, 231107 (2013).
[CrossRef]

Khanna, S. P.

Kliese, R.

Kobayashi, K.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[CrossRef]

Koch, M.

P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging — Modern techniques and applications,” Laser Photon.Rev. 5, 124–166 (2011).
[CrossRef]

Koeth, J.

K. Bertling, Y. L. Lim, T. Taimre, D. Indjin, P. Dean, R. Weih, S. Höfling, M. Kamp, M. von Edlinger, J. Koeth, and A. D. Rakić, “Demonstration of the self-mixing effect in interband cascade lasers,” Appl. Phys. Lett. 103, 231107 (2013).
[CrossRef]

Kroese, D. P.

D. P. Kroese, T. Taimre, and Z. I. Botev, Handbook of Monte Carlo Methods (Wiley, New York, 2011).
[CrossRef]

Labaune, J.

Lachab, M.

Laman, N.

Lang, R.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[CrossRef]

Lenstra, D.

G. A. Acket, D. Lenstra, A. J. den Boef, and B. H. Verbeek, “The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers,” IEEE J. Quantum Electron. 20, 1163– 1169 (1984).
[CrossRef]

Lim, Y. L.

R. Kliese, T. Taimre, A. A. A. Bakar, Y. L. Lim, K. Bertling, M. Nikolić, J. Perchoux, T. Bosch, and A. D. Rakić, “Solving self-mixing equations for arbitrary feedback levels: a concise algorithm,” Appl. Opt. 53, 3723–3736 (2014).
[CrossRef] [PubMed]

A. D. Rakić, T. Taimre, K. Bertling, Y. L. Lim, P. Dean, D. Indjin, Z. Ikonić, P. Harrison, A. Valavanis, S. P. Khanna, M. Lachab, S. J. Wilson, E. H. Linfield, and A. G. Davies, “Swept-frequency feedback interferometry using terahertz frequency QCLs: a method for imaging and materials analysis,” Opt. Express 21, 22194–22205 (2013).
[CrossRef]

K. Bertling, Y. L. Lim, T. Taimre, D. Indjin, P. Dean, R. Weih, S. Höfling, M. Kamp, M. von Edlinger, J. Koeth, and A. D. Rakić, “Demonstration of the self-mixing effect in interband cascade lasers,” Appl. Phys. Lett. 103, 231107 (2013).
[CrossRef]

Y. L. Lim, P. Dean, M. Nikolić, R. Kliese, S. P. Khanna, M. Lachab, A. Valavanis, D. Indjin, Z. Ikonić, P. Harrison, E. H. Linfield, A. G. Davies, S. J. Wilson, and A. D. Rakić, “Demonstration of a self-mixing displacement sensor based on terahertz quantum cascade lasers,” Appl. Phys. Lett. 99, 081108 (2011).
[CrossRef]

P. Dean, Y. L. Lim, A. Valavanis, R. Kliese, M. Nikolić, S. P. Khanna, M. Lachab, D. Indjin, Z. Ikonić, P. Harrison, A. D. Rakić, E. H. Linfield, and A. G. Davies, “Terahertz imaging through self-mixing in a quantum cascade laser,” Opt. Lett. 36, 2587–2589 (2011).
[CrossRef] [PubMed]

Y. L. Lim, K. Bertling, P. Rio, J. Tucker, and A. D. Rakić, “Displacement and distance measurement using the change in junction voltage across a laser diode due to the self-mixing effect,” in Photonics: Design, Technology, and Packaging II, D. Abbott, Y. S. Kivshar, H. H. Rubinsztein-Dunlop, and S. Fan, eds., Proc. SPIE6038, 60381O-1 (2006).

Linfield, E. H.

Lugarà, P. M.

F. P. Mezzapesa, L. L. Columbo, A. Ancona, M. Dabbicco, V. Spagnolo, M. Brambilla, P. M. Lugarà, and G. Scamarcio, “On Line Sensing of Ultrafast Laser Microdrilling Processes by Optical Feedback Interferometry,” Phys. Procedia 41, 670–676 (2013).
[CrossRef]

Mahler, L.

R. P. Green, J.-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Mann, C.

Martos-Levif, D.

D. Giovannacci, D. Martos-Levif, G. C. Walker, M. Menu, and V. Detalle, “Terahertz applications in cultural heritage: case studies,” in Fundamentals of Laser-Assisted Micro- and Nanotechnologies 2013, Proc. SPIE9065, 906510 (2013).
[CrossRef]

May, R. K.

T. Bardon, R. K. May, P. F. Taday, and M. Strlič, “Systematic study of terahertz time-domain spectra of historically informed black inks,” Analyst 138, 4859–4869 (2013).
[CrossRef] [PubMed]

Melinger, J. S.

Menu, M.

Mezzapesa, F. P.

F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, M. S. Vitiello, and G. Scamarcio, “Imaging of free carriers in semiconductors via optical feedback in terahertz quantum cascade lasers,” Appl. Phys. Lett. 104, 041112 (2014).
[CrossRef]

F. P. Mezzapesa, L. L. Columbo, A. Ancona, M. Dabbicco, V. Spagnolo, M. Brambilla, P. M. Lugarà, and G. Scamarcio, “On Line Sensing of Ultrafast Laser Microdrilling Processes by Optical Feedback Interferometry,” Phys. Procedia 41, 670–676 (2013).
[CrossRef]

F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, S. Borri, M. S. Vitiello, H. E. Beere, D. A. Ritchie, and G. Scamarcio, “Intrinsic stability of quantum cascade lasers against optical feedback,” Opt. Express 21, 13748–13757 (2013).
[CrossRef] [PubMed]

Møller, U.

Mourou, G.

Mourou, G. A.

Nikolic, M.

Nishide, K.

S. Shinohara, H. Yoshida, H. Ikeda, K. Nishide, and M. Sumi, “Compact and high-precision range finder with wide dynamic range and its application,” IEEE Trans. Instrum. Meas. 41, 40–44 (1992).
[CrossRef]

Norgia, M.

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A: Pure Appl. Opt. 4, S283–S294 (2002).
[CrossRef]

Nuss, M.

Osinski, M.

M. Osiński and J. Buus, “Linewidth broadening factor in semiconductor lasers — An overview,” IEEE J. Quantum Electron. 23, 9–29 (1987).
[CrossRef]

Perchoux, J.

Petermann, K.

K. Petermann, Laser Diode Modulation and Noise (Kluwer, Dordrecht, 1991).

Pickwell, E.

E. Pickwell and V. P. Wallace, “Biomedical applications of terahertz technology,” J. Phys. D: Appl. Phys. 39, R301–R310 (2006).
[CrossRef]

Pickwell-MacPherson, E.

S. Y. Huang, Y. X. J. Wang, D. K. W. Yeung, A. T. Ahuja, Y.-T. Zhang, and E. Pickwell-MacPherson, “Tissue characterization using terahertz pulsed imaging in reflection geometry,” Phys. Med. Biol. 54, 149–160 (2009).
[CrossRef]

Picollo, M.

K. Fukunaga and M. Picollo, “Characterisation of Works of Art,” in Terahertz Spectroscopy and Imaging, K.-E. Peiponen, A. Zeitler, and M. Kuwata-Gonokami, (eds.), pp. 521–538, (Springer, Berlin, 2013).

Pierce, I.

P. Spencer, P. Rees, and I. Pierce, “Theoretical analysis,” in Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor LasersD. M. Kane and K. A. Shore, eds. (John Wiley & Sons, 2005), pp. 23–54.
[CrossRef]

Plantier, G.

G. Plantier, C. Bès, and T. M. Bosch, “Behavioral model of a self-mixing laser diode sensor,” IEEE J. Quantum Electron. 41, 1157–1167 (2005).
[CrossRef]

Rakic, A. D.

R. Kliese, T. Taimre, A. A. A. Bakar, Y. L. Lim, K. Bertling, M. Nikolić, J. Perchoux, T. Bosch, and A. D. Rakić, “Solving self-mixing equations for arbitrary feedback levels: a concise algorithm,” Appl. Opt. 53, 3723–3736 (2014).
[CrossRef] [PubMed]

T. Taimre and A. D. Rakić, “On the nature of Acket’s characteristic parameter C in semiconductor lasers,” Appl. Opt. 53, 1001–1006 (2014).
[CrossRef] [PubMed]

A. D. Rakić, T. Taimre, K. Bertling, Y. L. Lim, P. Dean, D. Indjin, Z. Ikonić, P. Harrison, A. Valavanis, S. P. Khanna, M. Lachab, S. J. Wilson, E. H. Linfield, and A. G. Davies, “Swept-frequency feedback interferometry using terahertz frequency QCLs: a method for imaging and materials analysis,” Opt. Express 21, 22194–22205 (2013).
[CrossRef]

K. Bertling, Y. L. Lim, T. Taimre, D. Indjin, P. Dean, R. Weih, S. Höfling, M. Kamp, M. von Edlinger, J. Koeth, and A. D. Rakić, “Demonstration of the self-mixing effect in interband cascade lasers,” Appl. Phys. Lett. 103, 231107 (2013).
[CrossRef]

Y. L. Lim, P. Dean, M. Nikolić, R. Kliese, S. P. Khanna, M. Lachab, A. Valavanis, D. Indjin, Z. Ikonić, P. Harrison, E. H. Linfield, A. G. Davies, S. J. Wilson, and A. D. Rakić, “Demonstration of a self-mixing displacement sensor based on terahertz quantum cascade lasers,” Appl. Phys. Lett. 99, 081108 (2011).
[CrossRef]

P. Dean, Y. L. Lim, A. Valavanis, R. Kliese, M. Nikolić, S. P. Khanna, M. Lachab, D. Indjin, Z. Ikonić, P. Harrison, A. D. Rakić, E. H. Linfield, and A. G. Davies, “Terahertz imaging through self-mixing in a quantum cascade laser,” Opt. Lett. 36, 2587–2589 (2011).
[CrossRef] [PubMed]

A. D. Rakić, “Algorithm for the determination of intrinsic optical constants of metal films: application to aluminum,” Appl. Opt. 34, 4755–4767 (1995).
[CrossRef]

Y. L. Lim, K. Bertling, P. Rio, J. Tucker, and A. D. Rakić, “Displacement and distance measurement using the change in junction voltage across a laser diode due to the self-mixing effect,” in Photonics: Design, Technology, and Packaging II, D. Abbott, Y. S. Kivshar, H. H. Rubinsztein-Dunlop, and S. Fan, eds., Proc. SPIE6038, 60381O-1 (2006).

Recur, B.

Rees, P.

P. Spencer, P. Rees, and I. Pierce, “Theoretical analysis,” in Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor LasersD. M. Kane and K. A. Shore, eds. (John Wiley & Sons, 2005), pp. 23–54.
[CrossRef]

Rio, P.

Y. L. Lim, K. Bertling, P. Rio, J. Tucker, and A. D. Rakić, “Displacement and distance measurement using the change in junction voltage across a laser diode due to the self-mixing effect,” in Photonics: Design, Technology, and Packaging II, D. Abbott, Y. S. Kivshar, H. H. Rubinsztein-Dunlop, and S. Fan, eds., Proc. SPIE6038, 60381O-1 (2006).

Ritchie, D. A.

F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, S. Borri, M. S. Vitiello, H. E. Beere, D. A. Ritchie, and G. Scamarcio, “Intrinsic stability of quantum cascade lasers against optical feedback,” Opt. Express 21, 13748–13757 (2013).
[CrossRef] [PubMed]

R. P. Green, J.-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Roberts, J.

Salort, S.

M. Bessou, H. Duday, J.-P. Caumes, S. Salort, B. Chassagne, A. Dautant, A. Ziéglé, and E. Abraham, “Advantage of terahertz radiation versus X-ray to detect hidden organic materials in sealed vessels,” Opt. Commun. 285, 4175–4179 (2012).
[CrossRef]

J.-P. Caumes, A. Younus, S. Salort, B. Chassagne, B. Recur, A. Ziéglé, A. Dautant, and E. Abraham, “Terahertz tomographic imaging of XVIIIth Dynasty Egyptian sealed pottery,” Appl. Opt. 50, 3604–3608 (2011).
[CrossRef] [PubMed]

Scamarcio, G.

F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, M. S. Vitiello, and G. Scamarcio, “Imaging of free carriers in semiconductors via optical feedback in terahertz quantum cascade lasers,” Appl. Phys. Lett. 104, 041112 (2014).
[CrossRef]

F. P. Mezzapesa, L. L. Columbo, A. Ancona, M. Dabbicco, V. Spagnolo, M. Brambilla, P. M. Lugarà, and G. Scamarcio, “On Line Sensing of Ultrafast Laser Microdrilling Processes by Optical Feedback Interferometry,” Phys. Procedia 41, 670–676 (2013).
[CrossRef]

F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, S. Borri, M. S. Vitiello, H. E. Beere, D. A. Ritchie, and G. Scamarcio, “Intrinsic stability of quantum cascade lasers against optical feedback,” Opt. Express 21, 13748–13757 (2013).
[CrossRef] [PubMed]

Servagent, N.

Shinohara, S.

S. Shinohara, H. Yoshida, H. Ikeda, K. Nishide, and M. Sumi, “Compact and high-precision range finder with wide dynamic range and its application,” IEEE Trans. Instrum. Meas. 41, 40–44 (1992).
[CrossRef]

Skryl, A. S.

Spagnolo, V.

F. P. Mezzapesa, L. L. Columbo, A. Ancona, M. Dabbicco, V. Spagnolo, M. Brambilla, P. M. Lugarà, and G. Scamarcio, “On Line Sensing of Ultrafast Laser Microdrilling Processes by Optical Feedback Interferometry,” Phys. Procedia 41, 670–676 (2013).
[CrossRef]

Spencer, P.

P. Spencer, P. Rees, and I. Pierce, “Theoretical analysis,” in Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor LasersD. M. Kane and K. A. Shore, eds. (John Wiley & Sons, 2005), pp. 23–54.
[CrossRef]

Strlic, M.

T. Bardon, R. K. May, P. F. Taday, and M. Strlič, “Systematic study of terahertz time-domain spectra of historically informed black inks,” Analyst 138, 4859–4869 (2013).
[CrossRef] [PubMed]

Sumi, M.

S. Shinohara, H. Yoshida, H. Ikeda, K. Nishide, and M. Sumi, “Compact and high-precision range finder with wide dynamic range and its application,” IEEE Trans. Instrum. Meas. 41, 40–44 (1992).
[CrossRef]

Taday, P. F.

T. Bardon, R. K. May, P. F. Taday, and M. Strlič, “Systematic study of terahertz time-domain spectra of historically informed black inks,” Analyst 138, 4859–4869 (2013).
[CrossRef] [PubMed]

Taimre, T.

Tredicucci, A.

R. P. Green, J.-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Tucker, J.

Y. L. Lim, K. Bertling, P. Rio, J. Tucker, and A. D. Rakić, “Displacement and distance measurement using the change in junction voltage across a laser diode due to the self-mixing effect,” in Photonics: Design, Technology, and Packaging II, D. Abbott, Y. S. Kivshar, H. H. Rubinsztein-Dunlop, and S. Fan, eds., Proc. SPIE6038, 60381O-1 (2006).

Upadhya, P. C.

Valavanis, A.

Verbeek, B. H.

G. A. Acket, D. Lenstra, A. J. den Boef, and B. H. Verbeek, “The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers,” IEEE J. Quantum Electron. 20, 1163– 1169 (1984).
[CrossRef]

Vitiello, M. S.

F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, M. S. Vitiello, and G. Scamarcio, “Imaging of free carriers in semiconductors via optical feedback in terahertz quantum cascade lasers,” Appl. Phys. Lett. 104, 041112 (2014).
[CrossRef]

F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, S. Borri, M. S. Vitiello, H. E. Beere, D. A. Ritchie, and G. Scamarcio, “Intrinsic stability of quantum cascade lasers against optical feedback,” Opt. Express 21, 13748–13757 (2013).
[CrossRef] [PubMed]

von Edlinger, M.

K. Bertling, Y. L. Lim, T. Taimre, D. Indjin, P. Dean, R. Weih, S. Höfling, M. Kamp, M. von Edlinger, J. Koeth, and A. D. Rakić, “Demonstration of the self-mixing effect in interband cascade lasers,” Appl. Phys. Lett. 103, 231107 (2013).
[CrossRef]

von Staden, J.

Walker, G. C.

G. C. Walker, J. W. Bowen, J. Labaune, J. B. Jackson, S. Hadjiloucas, J. Roberts, G. Mourou, and M. Menu, “Terahertz deconvolution,” Opt. Express 20, 27230 (2012).
[CrossRef] [PubMed]

D. Giovannacci, D. Martos-Levif, G. C. Walker, M. Menu, and V. Detalle, “Terahertz applications in cultural heritage: case studies,” in Fundamentals of Laser-Assisted Micro- and Nanotechnologies 2013, Proc. SPIE9065, 906510 (2013).
[CrossRef]

Wallace, V. P.

E. Pickwell and V. P. Wallace, “Biomedical applications of terahertz technology,” J. Phys. D: Appl. Phys. 39, R301–R310 (2006).
[CrossRef]

Wang, Y. X. J.

S. Y. Huang, Y. X. J. Wang, D. K. W. Yeung, A. T. Ahuja, Y.-T. Zhang, and E. Pickwell-MacPherson, “Tissue characterization using terahertz pulsed imaging in reflection geometry,” Phys. Med. Biol. 54, 149–160 (2009).
[CrossRef]

Weih, R.

K. Bertling, Y. L. Lim, T. Taimre, D. Indjin, P. Dean, R. Weih, S. Höfling, M. Kamp, M. von Edlinger, J. Koeth, and A. D. Rakić, “Demonstration of the self-mixing effect in interband cascade lasers,” Appl. Phys. Lett. 103, 231107 (2013).
[CrossRef]

Wilson, S. J.

A. D. Rakić, T. Taimre, K. Bertling, Y. L. Lim, P. Dean, D. Indjin, Z. Ikonić, P. Harrison, A. Valavanis, S. P. Khanna, M. Lachab, S. J. Wilson, E. H. Linfield, and A. G. Davies, “Swept-frequency feedback interferometry using terahertz frequency QCLs: a method for imaging and materials analysis,” Opt. Express 21, 22194–22205 (2013).
[CrossRef]

Y. L. Lim, P. Dean, M. Nikolić, R. Kliese, S. P. Khanna, M. Lachab, A. Valavanis, D. Indjin, Z. Ikonić, P. Harrison, E. H. Linfield, A. G. Davies, S. J. Wilson, and A. D. Rakić, “Demonstration of a self-mixing displacement sensor based on terahertz quantum cascade lasers,” Appl. Phys. Lett. 99, 081108 (2011).
[CrossRef]

Xu, J.-H.

R. P. Green, J.-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Yeung, D. K. W.

S. Y. Huang, Y. X. J. Wang, D. K. W. Yeung, A. T. Ahuja, Y.-T. Zhang, and E. Pickwell-MacPherson, “Tissue characterization using terahertz pulsed imaging in reflection geometry,” Phys. Med. Biol. 54, 149–160 (2009).
[CrossRef]

Yoshida, H.

S. Shinohara, H. Yoshida, H. Ikeda, K. Nishide, and M. Sumi, “Compact and high-precision range finder with wide dynamic range and its application,” IEEE Trans. Instrum. Meas. 41, 40–44 (1992).
[CrossRef]

Younus, A.

Zhang, Y.-T.

S. Y. Huang, Y. X. J. Wang, D. K. W. Yeung, A. T. Ahuja, Y.-T. Zhang, and E. Pickwell-MacPherson, “Tissue characterization using terahertz pulsed imaging in reflection geometry,” Phys. Med. Biol. 54, 149–160 (2009).
[CrossRef]

Ziéglé, A.

M. Bessou, H. Duday, J.-P. Caumes, S. Salort, B. Chassagne, A. Dautant, A. Ziéglé, and E. Abraham, “Advantage of terahertz radiation versus X-ray to detect hidden organic materials in sealed vessels,” Opt. Commun. 285, 4175–4179 (2012).
[CrossRef]

J.-P. Caumes, A. Younus, S. Salort, B. Chassagne, B. Recur, A. Ziéglé, A. Dautant, and E. Abraham, “Terahertz tomographic imaging of XVIIIth Dynasty Egyptian sealed pottery,” Appl. Opt. 50, 3604–3608 (2011).
[CrossRef] [PubMed]

Analyst (1)

T. Bardon, R. K. May, P. F. Taday, and M. Strlič, “Systematic study of terahertz time-domain spectra of historically informed black inks,” Analyst 138, 4859–4869 (2013).
[CrossRef] [PubMed]

Appl. Opt. (7)

Appl. Phys. Lett. (4)

K. Bertling, Y. L. Lim, T. Taimre, D. Indjin, P. Dean, R. Weih, S. Höfling, M. Kamp, M. von Edlinger, J. Koeth, and A. D. Rakić, “Demonstration of the self-mixing effect in interband cascade lasers,” Appl. Phys. Lett. 103, 231107 (2013).
[CrossRef]

R. P. Green, J.-H. Xu, L. Mahler, A. Tredicucci, F. Beltram, G. Giuliani, H. E. Beere, and D. A. Ritchie, “Linewidth enhancement factor of terahertz quantum cascade lasers,” Appl. Phys. Lett. 92, 071106 (2008).
[CrossRef]

Y. L. Lim, P. Dean, M. Nikolić, R. Kliese, S. P. Khanna, M. Lachab, A. Valavanis, D. Indjin, Z. Ikonić, P. Harrison, E. H. Linfield, A. G. Davies, S. J. Wilson, and A. D. Rakić, “Demonstration of a self-mixing displacement sensor based on terahertz quantum cascade lasers,” Appl. Phys. Lett. 99, 081108 (2011).
[CrossRef]

F. P. Mezzapesa, L. L. Columbo, M. Brambilla, M. Dabbicco, M. S. Vitiello, and G. Scamarcio, “Imaging of free carriers in semiconductors via optical feedback in terahertz quantum cascade lasers,” Appl. Phys. Lett. 104, 041112 (2014).
[CrossRef]

Appl. Spectrosc. (1)

IEEE J. Quantum Electron. (5)

G. A. Acket, D. Lenstra, A. J. den Boef, and B. H. Verbeek, “The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers,” IEEE J. Quantum Electron. 20, 1163– 1169 (1984).
[CrossRef]

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982).
[CrossRef]

M. Osiński and J. Buus, “Linewidth broadening factor in semiconductor lasers — An overview,” IEEE J. Quantum Electron. 23, 9–29 (1987).
[CrossRef]

G. Plantier, C. Bès, and T. M. Bosch, “Behavioral model of a self-mixing laser diode sensor,” IEEE J. Quantum Electron. 41, 1157–1167 (2005).
[CrossRef]

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
[CrossRef]

IEEE Trans. Instrum. Meas. (1)

S. Shinohara, H. Yoshida, H. Ikeda, K. Nishide, and M. Sumi, “Compact and high-precision range finder with wide dynamic range and its application,” IEEE Trans. Instrum. Meas. 41, 40–44 (1992).
[CrossRef]

J. Appl. Phys. (1)

S. Donati, “Laser interferometry by induced modulation of cavity field,” J. Appl. Phys. 49, 495–497 (1978).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

G. Giuliani, M. Norgia, S. Donati, and T. Bosch, “Laser diode self-mixing technique for sensing applications,” J. Opt. A: Pure Appl. Opt. 4, S283–S294 (2002).
[CrossRef]

J. Phys. D: Appl. Phys. (1)

E. Pickwell and V. P. Wallace, “Biomedical applications of terahertz technology,” J. Phys. D: Appl. Phys. 39, R301–R310 (2006).
[CrossRef]

Laser Photon.Rev. (1)

P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging — Modern techniques and applications,” Laser Photon.Rev. 5, 124–166 (2011).
[CrossRef]

Mater. Today (1)

A. G. Davies, A. D. Burnett, W. Fan, E. H. Linfield, and J. E. Cunningham, “Terahertz spectroscopy of explosives and drugs,” Mater. Today 11, 18–26 (2008).
[CrossRef]

Opt. Commun. (1)

M. Bessou, H. Duday, J.-P. Caumes, S. Salort, B. Chassagne, A. Dautant, A. Ziéglé, and E. Abraham, “Advantage of terahertz radiation versus X-ray to detect hidden organic materials in sealed vessels,” Opt. Commun. 285, 4175–4179 (2012).
[CrossRef]

Opt. Express (5)

Opt. Lett. (3)

Phys. Med. Biol. (1)

S. Y. Huang, Y. X. J. Wang, D. K. W. Yeung, A. T. Ahuja, Y.-T. Zhang, and E. Pickwell-MacPherson, “Tissue characterization using terahertz pulsed imaging in reflection geometry,” Phys. Med. Biol. 54, 149–160 (2009).
[CrossRef]

Phys. Procedia (1)

F. P. Mezzapesa, L. L. Columbo, A. Ancona, M. Dabbicco, V. Spagnolo, M. Brambilla, P. M. Lugarà, and G. Scamarcio, “On Line Sensing of Ultrafast Laser Microdrilling Processes by Optical Feedback Interferometry,” Phys. Procedia 41, 670–676 (2013).
[CrossRef]

Other (7)

K. Petermann, Laser Diode Modulation and Noise (Kluwer, Dordrecht, 1991).

P. Spencer, P. Rees, and I. Pierce, “Theoretical analysis,” in Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor LasersD. M. Kane and K. A. Shore, eds. (John Wiley & Sons, 2005), pp. 23–54.
[CrossRef]

K. Fukunaga and M. Picollo, “Characterisation of Works of Art,” in Terahertz Spectroscopy and Imaging, K.-E. Peiponen, A. Zeitler, and M. Kuwata-Gonokami, (eds.), pp. 521–538, (Springer, Berlin, 2013).

D. Giovannacci, D. Martos-Levif, G. C. Walker, M. Menu, and V. Detalle, “Terahertz applications in cultural heritage: case studies,” in Fundamentals of Laser-Assisted Micro- and Nanotechnologies 2013, Proc. SPIE9065, 906510 (2013).
[CrossRef]

Y. L. Lim, K. Bertling, P. Rio, J. Tucker, and A. D. Rakić, “Displacement and distance measurement using the change in junction voltage across a laser diode due to the self-mixing effect,” in Photonics: Design, Technology, and Packaging II, D. Abbott, Y. S. Kivshar, H. H. Rubinsztein-Dunlop, and S. Fan, eds., Proc. SPIE6038, 60381O-1 (2006).

S. Donati, Electro-Optical Instrumentation: Sensing and Measuring with Lasers (Prentice Hall, Upper Saddle River, 2004).

D. P. Kroese, T. Taimre, and Z. I. Botev, Handbook of Monte Carlo Methods (Wiley, New York, 2011).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Three mirror model for a laser under feedback. Light recirculates in the laser cavity between mirrors M0 and Ms — solid arrows in laser cavity of length L. Light exits through Ms, is reflected from the external target Mext, and is reinjected into the laser cavity — solid arrows in external cavity of length Lext.

Fig. 2
Fig. 2

Systematic sensitivity of the system when η ≈ 0 and ηΓ 0, in terms of equivalent change in amplitude reflection coefficient Δ R A and phase-shift on reflection Δ θ R A, for a range of laser wavelengths (frequencies). Blue contours are lines of constant λ0γ. Red contours are lines of constant γ.

Fig. 3
Fig. 3

Sensitivity of the system to random variations in ( R A, θ R A) for a range of plastics. 95% confidence intervals are drawn for each plastic, with Var ( R A ) 0.00316 (Var( R A) = 10−5) and Var ( θ R A ) 0.00316 rad (Var( θ R A) = 10−5 rad2), at ν = 2.59 THz with Var ( ν ) = 10 kHz (Var(ν) = 108 Hz2), and an external cavity of Lext = 500 mm. Black contours are lines of constant n. Red contours are lines of constant k. Plastics at 2.59 THz are POM (dot), PA6 (circle), PVC (×), HDPE (cross), PTFE (star), PMMA (square), HDPE Black (diamond), and PC (triangle).

Fig. 4
Fig. 4

Sensitivity of swept-frequency self-mixing signals to systematic and random changes. (a) Systematic change in external cavity length (ΔL) of 0, 250, and 500 wavelengths on Lext = 500 mm, for fixed RA = 0.01 and θ R A= 0.05 rad. (b) Systematic change in RA from 0.001, 0.01, and 0.1, for fixed Lext = 500 mm and θ R A= 0.05 rad. (c) Systematic change in θ R Afrom 0.01 rad, 0.05 rad, and 0.1 rad, for fixed Lext = 500 mm and RA = 0.01. (d)–(f) As (a)–(c), with additive zero-mean Gaussian variation in Lext with standard deviation 10−4 λ 0 m; R Awith standard deviation 10−2; and θ R Awith standard deviation 10−2 rad.

Equations (39)

Equations on this page are rendered with MathJax. Learn more.

R = ( n 0 n ) 2 + k 2 ( n 0 + n ) 2 + k 2 , θ R = arctan ( 2 n 0 k n 0 2 n 2 k 2 ) ,
n = 1 R 1 + R 2 R cos ( θ R ) , k = 1 R sin ( θ R ) 1 + R 2 R cos ( θ R ) .
φ ( t ) = φ 0 + Φ Δ T t θ R ,
φ S φ FB = C sin ( φ FB + arctan α ) ,
V = V 0 + β cos ( φ FB ) ,
C C ( α ) = τ ext τ L κ ext 1 + α 2 ,
κ ext = ε R R s ( 1 R s ) .
R M = C 1 + α 2 = ε ( 1 R s ) n in L in R s L ext R A ,
θ R M = θ R A 4 π c ν 0 L ext ,
R M = a R + b R R A
θ R M = a θ + b θ θ R A ,
R 1 A = a R + b R R 1 M , R 2 A = a R + b R R 2 M , θ R , 1 A = a θ + b θ θ R , 1 M , θ R , 2 A = a θ + b θ θ R , 2 M .
a R = R 1 A R 2 M R 1 M R 2 A R 1 M R 2 M , b R = R 1 A R 2 M R 1 M R 2 M ,
a θ = θ R , 1 A θ R , 2 M θ R , 1 M θ R , 2 A θ R , 1 M θ R , 2 M , b θ = θ R , 1 A θ R , 2 A θ R , 1 M θ R , 2 M .
φ + Δ φ = 4 π c ( ν 0 + Δ ν ) ( L ext + Δ L ) ( θ R A + θ R A ) ,
C + Δ C = ε 1 + α 2 ( 1 R s ) n in L in R s ( L ext + Δ L ) ( R A + Δ R A ) .
θ R M = θ R A + Δ θ R A 4 π c ( ν 0 L ext + ν 0 Δ L + Δ ν L ext + Δ ν Δ L ) .
R M = ε ( 1 R s ) n in L in R s ( L ext + Δ L ) ( R A + Δ R A ) .
θ R M = θ R A + Δ θ R A 4 π { Γ + γ + η ( Γ + γ ) } .
Δ θ R A 4 π = 1 4 γ + η ( Γ + γ ) Δ θ R A 4 π + 1 4 .
R M = ε ( 1 R s ) n in L in R s ( Γ + γ ) λ 0 ( R A + Δ R A ) .
( R M θ R M ) ~ N ( μ , Σ ) ,
μ = ( 𝔼 R M 𝔼 θ R M ) , and Σ = ( Var ( R M ) Cov ( R M , θ R M ) Cov ( R M , θ R M ) Var ( θ R M ) ) .
𝔼 R M = ε ( 1 R s ) n in L in R s 𝔼 R A 𝔼 L ext , 𝔼 θ R M = 𝔼 θ R A 4 π c 𝔼 ν 𝔼 L ext .
Var ( R M ) = ( ε ( 1 R s ) n in L in R s ) 2 ( Var( L ext ) Var ( R A ) + ( 𝔼 L ext ) 2 Var ( R A ) + ( 𝔼 R A ) 2 Var ( L ext ) ) ,
Var ( θ R M ) = Var ( θ M A ) + ( 4 π c ) 2 ( Var ( L ext ) Var ( ν ) + ( 𝔼 L ext ) 2 Var ( ν ) + ( 𝔼 ν ) 2 Var ( L ext ) ) ,
Cov ( R M , θ R M ) = ε ( 1 R s ) n i n L i n R s 4 π c 𝔼 R A 𝔼 ν Var ( L e x t ) .
( R A θ R A ) = A ( R M θ R M ) + b ~ N ( A μ + b , A Σ A T )
( n k ) = g ( R M , θ R M ) ~ approx N ( g ( A μ + b ) , J g A Σ A T J g T ) ,
( n ^ k ^ ) = g ( R M ¯ , θ R M ¯ ) , R M ¯ = 1 N i = 1 N R M i , θ R M ¯ = 1 N i = 1 N θ R , i M ,
( R M ¯ θ R M ¯ ) ~ N ( μ , Σ / N ) .
R M i = β 0 + β x ( x i x 0 ) + β y ( y i y 0 ) ,
A = ( 1 x 1 x 0 y 1 y 0 1 x 2 x 0 y 2 y 0 1 x N x 0 y N y 0 ) , β = ( β 0 β x β y ) ,
( A T A ) β ^ = A T v .
v i ˜ = v i β x ^ ( x i x 0 ) β y ^ ( y i y 0 ) , i = 1 , , N .
J g = ( n R n θ R k R k θ R , )
n R = 2 ( 2 R + ( 1 + R ) cos ( θ R ) ) ( 1 + R 2 R cos ( θ R ) ) 2 , n θ R = 2 R ( 1 + R ) sin ( θ R ) ( 1 + R 2 R cos ( θ R ) ) 2 ,
k R = 2 ( 1 + R ) sin ( θ R ) ( 1 + R 2 R cos ( θ R ) ) 2 , k θ R = 2 R ( 2 R + ( 1 + R ) cos ( θ R ) ) ( 1 + R 2 R cos ( θ R ) ) 2 .
θ * argmin θ Θ S ( θ ) , with S ( θ ) = k = 1 n w k ( V k V ( t k ; θ ) ) 2 ,

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