## Abstract

We present a simplified all-reflective Fourier transform spectrometer with a split-mirror configuration for use over a broad spectral range with spatially coherent sources. The device is particularly well suited for measurement of broadband laser-like light, with resolution limited by beam size and collimation. Spectra are taken in the near-UV and the mid-IR, a total span of 4.6 octaves, including an octave spanning spectrum. Potential sources of error are investigated both theoretically and experimentally.

©2007 Optical Society of America

## 1. Introduction

Fourier transform spectrometers (FTSs) have been used for some time to determine the spectrum of an electric field based on coherence measurements [1]. Whereas a traditional FTS sweeps the delay of one arm in a Michelson interferometer and records an interference pattern over time [2], stationary FTSs [3] form the interferogram in the spatial domain by introducing a spatial deviation between two overlapping beams. Published stationary FTS designs have been based on common-path, misaligned Michelson [4, 5] and Sagnac interferometers, Wollaston prisms [6, 7], Savart plates [8], and liquid crystals [9]. Common to each of these is the inclusion of a beam splitter to separate an input beam into two, which restricts the wavelength range over which the instrument may be used. All-reflective designs eliminate the highly chromatic components, non-stationary implementations of which have been demonstrated using catseye reflectors [10, 11].

In a recent development, a pinhole pair spectrometer (PPS) was developed that demonstrated spectral measurements of a coherent beam of broad bandwidth radiation [12]. The Young’s interferogram from the pinhole pair is Fourier transformed to yield the spectrum with simultaneous measurement of the spatial coherence [13]. The PPS is highly achromatic with the spectral response limited solely by the detector. Increased resolving power in the PPS is limited because smaller and more separated pinholes are required, which severely limits the light collection for the spectral measurement.

In this article, we present a simplified double-mirror stationary FTS [14] that uses an all-reflective geometry, for use with spatially coherent fields such as those emitted by lasers and laser-like sources, including nonlinear optical conversion. The elimination of all spectrally-restrictive transmissive optics allows the spectrometer to be used over an extremely broad wavelength range, and we present results from the mid-IR (10*μ*m) to the near-UV (400 nm), a total span of some 4.6 octaves. In order to maintain achromaticity over the broadest spectral range, a scanning slit is employed to sample the interferogram. Moreover, a slit of order of *λ* may be used, which will be substantially smaller than the pitch of an array detector. An off-axis curved mirror is used to collect the highly astigmatic beam after the slit and focus it onto the detector. Finally, the most common sources of measurement error are discussed.

## 2. Theory

The spectrometer is based on the formation of a spatial interferogram from which the input beam spectrum may be extracted. The interferogram is formed by two closely set mirrors that split a collimated, spatially coherent input beam of diameter 2*w*
_{0} into two halves, and overlaps them with angle 2*α* in the crossing plane. The crossing plane lies a distance *z _{c}*=

*a*/2sin

*α*from the split mirror, where

*a*is the distance between the beam centers (equal to

*w*

_{0}if the mirror separation is zero).

In the space–frequency domain, the fields of the two halves can be written as

where *A*
_{1,2} is a constant such that the peak of *A*(*t*) is normalized to unity, *ω*
_{0} is the central angular frequency of the input beam, and *U*
_{1,2}(*x*) are the (real) spatial profiles of the beam halves in the crossing plane (which have incurred diffraction from the mirror edges). The spatial frequency, *f _{x}*=sin

*α*/

*λ*is due to the crossing angle of the beams that leads to the spatial interference pattern. We assume no spatio-temporal distortions are present, which allows us to separate the spatial envelope

*U*

_{1,2}(

*x*) from the spectral

*Ã*(

*ω*) envelope.

We can rewrite the phase of Eq. (1) as 2*πf _{x}x*=

*ωτ*, with the position-dependent relative delay

*τ*(

*x*)=

*x*sin

*α*/

*c*, as can be seen by a simple geometric construction. Fourier transforming yields the space–time field,

The time-averaged intensity interferogram of this field as observed by a square-law detector in the crossing plane is now given as usual by *I*(*x*)=〈|*E*
_{1}(*x*,*t*)+*E*
_{2}(*x*,*t*)|^{2}〉, or

The terms *I _{j}*(

*x*)=

*I*

_{0j}.|

*U*(

_{j}*x*)|

^{2}are the intensities due to the individual beam halves (

*j*∈{1,2}), with the peak intensity of the beam halves given by ${I}_{{0}_{j}}=\frac{1}{2}{\epsilon}_{0}\mathrm{nc}{\mid {A}_{j}\mid}^{2}$ . Γ(2

*τ*)=〈

*A*(

*t*)

*A*

^{*}(

*t*-2

*τ*)〉≡

*τ*̄

_{p}

*γ*(2

*τ*) is the field autocorrelation, assuming that the field

*A*(

*t*) is time-stationary. We define

*τ*̄

*=∫*

_{p}^{∞}

_{-∞}|

*A*(

*t*)|

^{2}d

*t*such that γ(2

*τ*) is the normalized field autocorrelation.

The relationship between *τ* and *x* allows us to transform this intensity pattern to the frequency domain. Introducing symbols ℱ and ⊗ to represent Fourier transform and convolution, we obtain finally

where *D*(*ν*)=ℱ{(*I*
_{1}(*τc*/sin*α*)+*I*
_{2}(*τc*/sin*α*))} represents the Fourier transform of the slow-varying interferogram background due to the beam half distributions, and the spectrometer “response” function
$R\left(v\right)=\mathcal{F}\left\{{2\overline{\tau}}_{p}\sqrt{{I}_{{0}_{1}}{I}_{{0}_{2}}}{U}_{1}\left(\frac{\mathrm{\tau c}}{\mathrm{sin}\alpha}\right){U}_{2}\left(\frac{\mathrm{\tau c}}{\mathrm{sin}\alpha}\right)\right\}$
. The normalized power spectrum *S*(*ν*) was obtained using the Wiener–Khintchine theorem [15], *S*(*ν*)=∫^{∞}
_{-∞}
*γ*(*τ*)exp[*i*2*πvτ*]d*τ*. Examination of Eq. (4) reveals that the Fourier transform of the interferogram consists of a peak centered around zero frequency with a width given by *D*(*ν*), and on either side, at ±*ν*
_{0}=±*ω*
_{0}/2*π*, the normalized power spectrum convolved with the response function *R*(*ν*).

To obtain an accurate spectrum, we must ensure the fringes in the interferogram of Eq. (3) are properly resolved and that the entire autocorrelation is captured. These conditions can be expressed in terms of the limiting sampling step *δx*, which may be interpreted as the width of a scanning slit or pixel size of an array detector, and Δ*X*, the scan or array length. The Nyquist–Shannon theorem [16] holds that each fringe must be sampled at least twice per period to prevent aliasing. The sample size must thus satisfy *δx*≤(1/2)(*λ*
_{min}/2sin*α*), where the fringe periodicity for the shortest wavelength *λ*
_{min} to be captured is evident from Eq. (3). It is also reasonable to limit the sample size to half the maximum wavelength, *λ*
_{max}/2≤*δx*, as transmission decreases rapidly for wavelengths above this limit [17]. The observable octave rage with this slit device is given by log_{2}[*λ _{max}*/

*λ*]. If both conditions are met in the limiting case, the octave span is given by log

_{min}_{2}[1/(2sin

*α*)]. While the slit width sets the boundary wavelengths, the octave span depends only on the split mirror angle. For an angle of 2.5°, the octave range is ~ 3.5.

The spectral resolution of the spectrometer is determined by both the spatial extent of the beam halves projected into the scanning slit plane, *w*
_{0}/cos*α*, and the periodicity of the interferogram fringes, set by the crossing angle, *x*
_{0}=*λ*
_{0}/2sin*α*. The narrowest spectral feature that can be resolved is given by Δ*ν*=*c*/(*w*
_{0}tan*α*) and implies that the resolving power of the instrument is given by Δ*λ*/*λ*
_{0}=*λ*
_{0}/(*w*
_{0}tan*α*). The finite temporal coherence, *τ _{c}*, of a broad bandwidth spectrum will restrict the autocorrelation to a region of Δ

*X*≈

*c*/Δ

*ν*sin

*α*. Provided that the projected beam radius exceeds this width, the entire autocorrelation trace will be captured so that the spectrum is fully resolved.

## 3. Experiment

#### 3.1. Design

With the above criteria, we designed a spectrometer suitable for measuring wavelengths between the near-UV and the mid-IR. To cover this broad range, we used different scanning slits (National Aperture, Inc.), from 5-*μ*m in the visible wavelengths to 25-*μ*m at the long wavelength end. To capture the highly astigmatic beam transmitted by the apertures, which typically had a long dimension of ~ 4 mm, a short focal length spherical mirror at a large incidence angle of 46° was used to create a near-symmetric focus at the detector. For detection in the visible and near-IR, we used a fast Si photodiode (Thorlabs), while in the mid-IR, a single-element, liquid-nitrogen cooled HgCdTe detector (IR Associates) was used. The remainder of the setup, including the split angle and the focusing optic, remains unchanged. The choice of scanning slit and interchangeable detectors maintains functionality over the widest spectral range, allowing the spectrometer to be used in spectral regions where other spectrometers are unavailable or costly. If interchanging slits and detectors is undesirable, it is possible to capture the same range demonstrated below by using a 5*μ*m slit with a pyroelectric detector to measure the UV and MIR simultaneously.

The accurate determination of the crossing angle *α* is difficult in practice. Our spectrometer was calibrated using a narrow-bandwidth 773.4-nm source from a cw Ti:sapphire oscillator, shown in Fig. 2(a). With *z _{c}* set at 10 cm, this source gave a fringe spacing of 8.7

*μ*m, from which we derive

*α*≈2.5°. The source was then mode-locked and half of the split mirror translated in the

*z*direction to place time-zero at the center of the beam halves. The calibration was verified using 632-nm HeNe and 532-nm frequency-doubled Nd:YAG laser sources, with the errors in calibration < 0.5%. Keeping a constant

*α*allows the spectrometer to be calibrated, subject to the errors described in section 4, with a well-characterized narrowband source, with the calibration unchanged when the slit and detector are exchanged for a different wavelength range.

#### 3.2. Results

Figure 2 shows the measured spectra for a number of sources. For the measurements in the visible and near-UV, the FTS measurements are compared to spectra measured with conventional grating-based spectrometers (OceanOptics). Spectra from a cw and mode-locked Ti:sapphire oscillator (KM Labs) are shown in Figs. 2(a) and 2(b), respectively. When mode-locked, the oscillator produces ~ 20-fs pulses, with a corresponding FWHM spectral bandwidth of 47 nm. The excellent agreement between FTS and reference verifies that the spectral envelope is correctly retrieved.

One common limitation of grating-based spectrometers is multiple diffraction orders, which prevent the measurement of octave-spanning spectra unless bandpass filters are installed in the spectrometer. The same limitation does not apply to the FTS. We generated an octave-spanning spectrum by frequency-doubling part of the Ti:sapphire radiation in a 4-mm thick, type-I KDP crystal placed near the focus of the Keplarian beam expansion telescope used to enlarge the oscillator beam. The resulting octave-spanning spectrum with components both at the fundamental 800 nm and frequency-doubled 400nm is shown in Fig. 2(c). From the acceptance bandwidth of KDP [18], we expect a spectral bandwidth in the near-UV of ~ 3 nm. In this case, the reference spectra were recorded using two separate grating-based spectrometers.

To test the device operation in a different spectral region, we exchanged slit and detector as described above and measured the spectrum from a CO_{2} laser (Synrad). The laser produced narrow-band pulses centered at 10.8 *μ*m. The spatial period of the fringes in the interferogram was 120 *μ*m, and transform yielded a linewidth of 0.16 *μ*m. The spectrum shown in Fig. 2(d) is taken after the interferogram has been processed to remove spatial envelope variations due to the beam profile, as described in the next section.

## 4. Errors

#### 4.1. Edge diffraction

The constraint on Δ*X* in the previous section does not take into account the diffraction caused by the edges of the split mirror. As can be seen from Eq. (3), the envelope of the oscillatory structure of the interferogram is given by the product of the spatial beam profile and the autocorrelation. The transform of this product becomes a convolution of the power spectrum and the response function *R*(*ν*) of the device. For large-diameter beams with a large bandwidth (small *τc*), an appropriate choice of relative *z* positions of the split mirror places the interference fringes in the center of the pattern, minimizing the edge diffraction contribution. For cases in which the interference pattern is broad or the spectrum is narrow, we can eliminate some structure from the response function by dividing out the term *U*
_{1}(*x*)*U*
_{2}(*x*) from the interferogram. This removal can also eliminate the effect of a poorly centered input beam, which would could cause asymmetry in the envelope. If a measurement of the beam halves is unavailable, the expected form could be fit to the interferogram to facilitate removal, as described by Harimoto et al [19].

Figure 3 shows experimental data for the response correction. The raw interferogram of the narrow-band CO_{2} laser with a central wavelength of *λ*
_{0}=10.8 *μ*m is shown in Fig. 3(a). The beam width is clearly too narrow to capture the entire autocorrelation, and the superposition of the autocorrelation fringes and the edge diffraction is clearly visible. In order to eliminate this edge diffraction for the autocorrelation trace, we record the intensity distributions of each of the two beam halves in the same apparatus (Fig. 3(b)), blocking the other half in turn. We first use a high-pass filter to remove the dc offset, and the slow diffraction oscillations. The interferogram after this removal is shown in Fig. 3(c). The spectrum produced by transform of the interferogram both before and after envelope removal is shown in Fig. 2(d). An analogous technique can be applied in cases where the spatial distribution of the beam to be measured has structure more rapid than the total width of the autocorrelation.

#### 4.2. Beam divergence

In the derivations of section 2, we have assumed that the phase difference between the two beams halves is due solely to the path length difference introduced by the relative tilt of the split mirror. Practical implementations of this spectrometer may be subject to errors in collimation and beam pointing.

Poor collimation of the beam incident on the split mirror will introduce an extra phase term due to the curvature of the phase front to the field in the crossing plane. In a Gaussian beam, this extra phase term can be written as
${\varphi}_{\mathrm{div}}=\frac{\frac{1}{2}{k\left(x\pm {x}_{0}\right)}^{2}}{R\left(z\right)}$
, where *k*=2*π*/*λ* is the wavenumber, *R*(*z*) is the phase front radius of curvature at the crossing plane, and *x*
_{0} is the phase front center in the crossing plane. This extra phase causes the oscillatory component of Eq. (3) to be cos{2*ω*
_{0}
*τ*(*x*)[1+*w*
_{0}/2*R*(*z*)sin*α*]}, where *x*
_{0} has been set at *w*
_{0}/2. The uncollimated beam causes a fractional shift of Δ*ν*
_{0}/*ν*
_{0}=*w*
_{0}/2*R*(*z*)sin*α* in the deduced central frequency of the incident field spectrum. In the event that the beam is not centered on the mirror, the phase term proportional to *x*
^{2}
_{0} will no longer cancel exactly, and a slight asymmetry will be introduced into the interferogram. In either case, this extra phase will show up as a shift in the central frequency in the spectral domain.

To test this experimentally, we intentionally introduced divergence errors to a 532-nm, frequency doubled Nd:YAG which was measured simultaneously with a well-collimated 632-nm reference HeNe laser. We introduced this divergence by adjusting the telescope spacing by Δ*z* from the ideal separation *z _{f}*, indicated in Fig. 1, for an initially collimated, narrow-band beam at

*λ*

_{0}=532 nm. The effect of the telescope misalignment on the beam at the crossing plane was modeled using the

*ABCD*matrix formalism [20], using the measured beam radius,

*w*

_{0}=5 mm, at the entrance of the telescope for the initial complex beam parameter

*q*

_{0}. Figure 4(a) shows a comparison between the shifts in central wavelength as predicted by this model and measured values as a function of phase front curvature, taken over a range of offsets Δ

*z*. In practice, the divergence from phase front curvature in excess of |1/

*R*(

*z*)|>0.1 m

^{-1}is visible by eye, reducing the error to ≈ 1% for a visually collimated beam.

#### 4.3. Pointing error

A similar error can arise if the split beam halves travel different distances to the crossing plane. In the case that the incident beam is at some angle Θ relative to the normal of the crossing plane, the spatial frequencies of the beams halves are now *f*
_{x1}=sin*α*-Θ and *f*
_{x1}=-sin*α*+Θ, giving a fringe spacing of *x*=*λ*/2sin*α*cosΘ. The functional form taken by the fringe spacing is now that of *x*
_{0} projected onto a scanning plane rotated by Θ, a situation analogous to the pointing error described above. This change in the fringe spacing shows up as a shift in the central frequency in the frequency domain. The fractional shift in this central wavelength is given by secΘ-1. Fig. 4(b) shows a comparison between this expected functional form and a simulation.

As both the above errors result in a shift of the central frequency, calibration can compensate for both in some cases. When calibrating *in situ*, any error in pointing or divergence would become part of the apparent crossing angle, and thus not affect spectral measurements. However, when spectra are to be measured on a beam other than that use to calibrate, care must be taken to ensure accurate collimation and alignment.

## 5. Conclusions

A simple stationary FTS, employing a double-mirror to split the beam, a slit to sample the interferogram, and an off-axis curved optic to maximize the sensitivity, is used to measure spectra across 4.6 octaves, including a demonstration of octave-spanning spectra. Potential measurement error sources are identified, and their effect on a spectral measurement is calculated. Due to its simple construction and wide applicability to measure spatially-coherent beams, we believe this spectrometer should be useful in spectral regions where contemporary spectrometers are not available. Of particular interest are measurement of extreme ultraviolet (EUV) pulses created using High-Harmonic Generation (HHG) [21, 22], where optics and gratings operate over very narrow spectral ranges and suffer from poor performance. In the EUV, a grazing-incidence split mirror spectrometer could be implemented using an MCP or a CCD. The device is also well suited for measurement of other sources of broadband pulses, such as ultrafast MIR sources [23], supercontinuum generation with photonic crystal fibers [24, 25] and pulses generated with noncollinear optical parametric amplifiers [26, 27].

## Acknowledgments

The authors gratefully acknowledge support from the NSF CAREER Award ECS-0348068, ONR Young Investigator Award, the Beckman Young Investigator Award, and Sloan Research Fellowship support for R.A.B. We acknowledge the loan of the Synrad CO_{2} laser from Margaret Murnane and Henry Kapteyn.

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