## Abstract

Wavelength modulation imaging (WMI) is capable of determining both spectroscopic and geometrical properties of a target, but the latter is often ignored in spectroscopic studies. This work theoretically and experimentally demonstrates the importance of both in WMI applications. Experiments were performed with an all-digital signal processing approach employing a tunable mid-infrared laser capable of digital wavelength modulation. All three orders of wavelength-derivative images, 0^{th}, 1^{st}, and 2^{nd} are generated simultaneously. Higher order images can reveal or enhance features that are not evident in the 0^{th} order. An example shows a synthetic imaging approach that combines the 2^{nd} order WMI of CO gas with a focal plane array image to allow chemical visualization with minimal background clutter. In another example, fine geometrical features were revealed for a target that has little intrinsic spectroscopic signatures.

© 2004 Optical Society of America

## 1. Introduction

Wavelength modulation imaging (WMI) is a technique to produce wavelength-derivative images *∂I*(*λ*;**r**)/*∂λ*, *∂*
^{2}
*I*(*λ*;**r**)/*∂λ*
^{2}, … *∂*^{n}*I*(*λ*;**r**)/*∂λ*^{n}
that correspond to the intensity image *I*(*λ*;**r**) of a target at wavelength A The target can be a medium, scene, or objects. The purpose is to enhance and detect features of interest for applications such as chemical imaging. The intensity image *I*(*λ*;**r**) can be considered as the zeroth-order image, which can be 2- or 3-dimensional, i.e., **r** = (*x,y*) or (*x,y,z*). A well-known related technique is wavelength modulation spectroscopy (WMS), which aims to enhance spectral feature contrast. Second-order WMS is often applied to gas spectroscopy [1–5]. More generally, WMS of various orders are also applied in condensed matters to sharpen spectroscopic features among a broad background.

It may appear superficially that WMI is generalized WMS with spatial mapping. For example, Kane *et al*. have used WMS absorption to map the spatial distribution of chemical species in a flame [4]. However, WMI is a broader category than WMS on one fundamental aspect: it involves the geometrical features of a target besides spectroscopic features. The term *∂*^{n}*I*(*λ*;**r**)/*∂λ*^{n}
does not always represent spectroscopic signatures. It can also reveal the target geometry via optical effects such as diffraction, interference, reflection, refraction, or generally, scattering, all of which are wavelength-dependent. Furthermore, WMS aims to enhance only the spectral contrast, and is not concerned with spatial background clutter. WMI is concerned with both, and is thus a more general technique.

This paper is a study of the WMI technique, and differs from other spectroscopy works [1–5] with its emphasis on imaging. A key point is that both spectroscopic and geometrical aspects, which offer different information about the target, are essential. The paper describes WMI experiments up to second order, using a laser-based scanning 2-dimensional imaging system with an all-digital signal processing technique that allows simultaneous multi-order imaging. The spectroscopic aspect of WMI was demonstrated with CO gas absorption, showing its relation to WMS.

But as mentioned above, the geometrical aspect is important in WMI. The experiments showed that the geometrical effects can be substantial in the 2^{nd} and likely higher orders. For gas spectroscopy, 2^{nd} order WMS is considered to be of high sensitivity and specificity, as it detects only the curvature of a gas absorption line, and is not susceptible to spurious linear effects that make the 1^{st} order ambiguous. The results here suggest that if WMI is used for detection and imaging of chemicals, consideration of both spectroscopic and geometrical aspects is crucial for correct interpretation. Finally, the paper examines a synthetic multi-spectral imaging approach that combines the WMI technique with a conventional passive imaging technique to enhance the target information. As a specific example, the paper shows the digital fusion of a laser-based mid-IR WMI image of CO gas with a visible passive focal plane array image to create a synthetic image that has a visible CO signature. This approach has a better clutter-rejection capability than that of direct absorption (0^{th} order) imaging [7–9].

The rest of the paper is organized as follow. Section 2 describes a theoretical basis for the interpretation of the spectroscopic and geometrical aspects for the WMI study in this work. Section 3 describes the experimental approach. Section 4 describes the experimental results and discussion; and section 5 provides a summary and conclusion.

## 2. Spectroscopic and geometrical aspects in wavelength modulation imaging study

A fundamental question of an optical measurement of *∂*^{n}*I*(*λ*;**r**)/*∂λ*^{n}
is the origin of this term. The spectroscopic aspect can be readily recognized, but less obvious are the geometrical nature. This section discusses this issue. The approach can be summarized as follow: *i*) WMI is modeled as a scattering problem; *ii*) from this model, the origin of spectroscopic and geometrical effects are described; their meanings are discussed; and *iii*) some calculation are shown to illustrate the discussion; and the detailed background are given in the appendix.

#### 2.1 Scattering model of WMI

Scattering is a convenient model for discussion. Two common modes of scattering are illustrated in Figs. 1(a) and (b). For both, an incident electromagnetic (EM) field is used to probe a target, and the scattered field is measured. The target is basically a dielectric function *ε*(*λ*;**r**) that causes scattering. Its image is reconstructed from the measured scattered field. This problem, also known as the inverse scattering problem, is most general and applicable to almost any EM scattering problem, including holography and diffraction tomography [10].

The framework for holography and diffraction tomography is the case in Fig. 1(a), which uses an incident plane wave. Here, the scalar diffraction theory as given in Ref. [11] is used, in which, the complex representative of the EM field without time-dependent term *e*^{iωt}
(where *ω* is the frequency, *t* is the time) is described as:

where *U*(*λ*;**r**) is the scalar electric field, which is the sum of the incident plane wave exp(*i*
**k**
_{0} ∙ **r**) and the scattered wave that is represented by the integral, *k*=2*π*/*λ*, and **k**
_{0} is the plane wave k-vector. This mode can be considered as a **k**-space sampling, in which the incident wave is a *δ*-function in **k**-space and infinite in real space. The opposite mode is shown in Fig. 1(b), where the incident wave from a laser beam has a finite spatial extent that is much smaller than the target dimension. Ideally, if the beam at the sampled point is a *δ*-function in **r**-space, the scattered field yields information of *ε*(*λ*;**r**) only at that point. An entire target image can be constructed by sampling all the points; and *∂*^{n}*I*(*λ*;**r**)/*∂λ*^{n}
is strictly a local function, which represents the spectroscopy of *ε*(*λ*;**r**) at **r**.

In practice however, all beams have finite spatial extent within which a target may have substructures, and the geometrical aspect of *ε*(*λ*;**r**) cannot be neglected. This work is based on the scattering mode in Fig. 1(b). A formulation for this mode is to consider the probe beam as a linear combination of plane waves in Eq. (1):

where *A*(**k**
_{0}) represents the beam **k**-space spectrum. c*z*-axis, *A*(**k**
_{0}) =*e*-${k}_{0;\perp}^{2}$
*w*
^{2}/4-*i*
**k**
_{0;⊥}∙**ρ**
_{0}
*δ*(*k*
_{0;z} - √${k}_{0}^{2}$ - ${k}_{0;\perp}^{2}$)∏(${k}_{0}^{2}$ - ${k}_{0;\perp}^{2}$) describes an incident beam with waist *w* at location **ρ**
_{0} =(*x*
_{0},*y*
_{0}). This linear decomposition is the basis for the calculation used later in this section.

#### 2.2 Expression for spectroscopic and geometrical terms in WM

The first-order WMI effect can be obtained by applying the operator *λ∂*/*∂λ* to Eq. (1) and simplifying:

In spite of the complex appearance, the meaning of each term on the right hand side of Eq. (3) can be interpreted. The term in Eq. (3.c) associated with *∂ε*(*λ*;**r**)/*∂λ* is obviously of spectroscopic origin. Switching this term off by setting *∂ε*(*λ*;**r**)/*∂λ* = 0 (i.e., the target has trivial spectroscopic property), leaves the terms in Eqs. (3.a) and (3.b). These two terms are not of equivalent value. The right hand side terms of Eq. 3(a) are directly proportional to the 0^{th} order field and the plane wave, and thus containing no additional information. It simply represents the wavelength scaling behavior in diffraction, which is not interesting for WMI.

The truly meaningful geometrical term is in Eq. (3.b). The integral term involving the product [*ε*(**r**′)-1]**r**-**r**′| can be considered as an approximate measure related to the 1^{st} moment of *ε*(**r**)-1 spatial distribution. In higher order WMI terms, this term becomes
[*ε*(**r**′)-1|**r**-**r**′|^{n} , which is related to the *n*
^{th} moment. They can be interpreted as a measure that provides information with increasingly finer spatial distribution of *ε*(**r**). It is clear that with the geometrical terms, WMI is a broader category than WMS as mentioned in Section 1.

An important issue is the magnitude of these terms. The 1^{st} order wavelength modulating signal amplitude is ∆*U* = *U*[*λ* + ∆*λ*)-*U*(*λ*), which is approximately [∆*λ*/*λ*)(*∆∂U* /*∂λ*), and can be obtained by simply multiplying the factor (∆*λ*/*λ*) to both sides of Eq. (3). For small wavelength modulation ∆*λ*/*λ* ~10^{-5}-10^{-4}, it may appear that these terms may scale as ~(∆*λ*/*λ*)^{n} and thus diminish rapidly vs. order *n*. This scaling rule applies to the wavelength scaling term above, which is the right hand side of Eq. (3.a), and this term is indeed not significant.

However, this scaling rule does not apply to the spectroscopic and geometrical terms, and it is precisely the reason that they are relevant in WMI. The spectroscopic term has been treated extensively in literature [1–5] and will not be considered here. It is relatively straight forward. In the case of absorption, let the absorption change be ∆*I* for wavelength modulation of ∆*λ*, then the magnitude of the 1^{st} order image relative to the 0^{th} order is simply ∆*I*/*I*. For the 2^{nd} order, assuming that the absorption line is symmetric vs. wavelength, then the signal change is also ∆*I* for both ±∆*λ*, and its magnitude relative to the 0^{th} order is 2∆*I*/*I* at the peak. Thus, the 2^{nd} order signal is comparable to the 1^{st}, and not scaled by ∆*λ*/*λ*. The sharp absorption peak curvature gives WMS its ability to reject linear background for sensitive detection. A value of ∆*I*/*I* of 10^{-5}-10^{-8} or even smaller is detectable in low-noise system.

For the geometrical terms, their magnitudes depend on the optical effects. In the following, three examples are considered: the diffraction of a 1-dimensional Gaussian beam obscured by a perfectly conducting infinite half-plane, the etalon transmission effect, and Fraunhofer diffraction of an annulus. The first two have direct bearing on the experiments. The following describes only the calculation results, the details are given in the Appendix.

#### 2.3 Calculation examples

The expression in Eqs. (3) is for the 1^{st} order *λ*-derivative of the field amplitude, which is
relevant in experiments involving both phase and amplitude. Here, only intensity |*U*|^{2} ≡ *I* was measured, and the corresponding 1^{st} and 2^{nd} order terms *∂I*(*λ*;**r**)/*∂λ* and *∂*
^{2}
*I*[*λ*;**r**)/*∂λ*
^{2} are calculated in the following examples. The calculation uses coarse step ∆*λ*/*λ*~10^{-4} to cover a wide wavelength range without excessive computation time. The result can be proportionally scaled for experimental results with ∆*λ*/*λ* ~ 1-2×10^{-5}.

### 2.3.1 Gaussian beam diffraction by an edge

An important effect with direct bearing on the experiments is the diffraction of a finite beam focused on an edge or a boundary, where the wavelength-modulated signal can be substantial. The experiments here correspond to the scattering mode in Fig. 1(b). An idealized model is used for computation, involving one-dimensional Gaussian beam obscured by a perfectly conducting half plane. The treatment in the Appendix uses the approach in Eq. (2) with Sommerfeld’s exact or accurate solutions [12]. The main result is that the diffracted power
can have substantial wavelength dependence. Figure 2 shows the 0^{th} order diffraction pattern relative to the peak transmitted intensity, which consists of a main lobe and a number of side diffraction fringes (spatial intensity modulation) on the unblocked half (x<0). The diffraction pattern varies vs. wavelength, resulting in WM signals shown in Fig. 3. The figure shows that the main lobe and the diffraction fringes have quite different behaviors as discussed in Eqs. (3) above. The diffraction fringes exhibit “wavelength scaling” behavior as in Eq. (3.a), which is ~∆*λ*/*λ* for the 1^{st} and become negligible in the 2^{nd} order pattern. The main lobe however, which corresponds to the geometrical term in Eq. (3.b), exhibits oscillatory wavelength-dependent behavior on a fine scale. Their magnitudes are ~10^{-3} for ∆*λ*=2.5x10^{-4}, which would be substantial in a system designed to detect, say 10^{-5} - 10^{-6} absorption.

The total diffracted power exhibits wavelength dependence behavior as shown in Fig. 3 (right). The reason that the ~(∆*λ*/*λ*)
^{n}
scaling does not apply here is clear: fine oscillatory behavior vs. wavelength. The obscuration optical cross section of a beam of finite size is analogous to the scattering optical cross section of a finite object. For objects with geometrical dimensions that are not excessively larger than the wavelength, the exact Mie’s scattering solution shows that the scattering cross section has pronounced oscillatory λ-dependence [13] similarly to the results here, and deviates significantly from the geometrical cross section.

### 2.3.2 Etalon transmission effect

This case appears to be nearly trivial but is a highly relevant example of optical interference that can be quite common at anywhere in a target that involves two parallel optical surfaces, such as windows or thin films. It can be represented as a case of multiple scattering with longitudinal geometrical effects. Even for low-index material such as CaF_{2} used in this work, the 0^{th} order effect itself has a mild λ-dependent modulation, but the associated WMI signals can have substantial wavelength-dependence modulation. A calculation result with parameters corresponding to the experimental conditions is shown in Figs. 4. Figure 4(a) shows the magnitude of the 0^{th}, 1^{st}, and 2^{nd} order terms as a function of wavelength, using the expression given in the Appendix. As shown, both the 1^{st} and 2^{nd} terms have significant amplitude with ∆*λ* = 10^{-4}. It will be seen in the experimental results that the variation of the 2^{nd} order WM signal vs. window thickness *L* in Fig. 4(b) here corresponds to the effect of the window being a wedge, giving rise to fringes that were not obvious in the 0^{th} order but observable in the 2^{nd} order images as the laser was scanned across the window.

### 2.3.3 Fraunhofer diffraction through an annulus

Although this case is not specifically related to an experimental result, it is useful to illustrate the high-spatial-frequency aspect associated with high-order WMI images discussed earlier. Fraunhofer diffraction theory is not accurate since it uses the aperture geometrical cross section rather than the exact optical cross section, and thus grossly underestimates the geometrical effects. But it is analytically well-known and simple for this discussion. Here, we consider the image as the diffracted light intensity distribution. Expressions of the 0^{th} through the 2^{nd} order are given in the appendix, and they are plotted in Fig. 5 as a function of tan θ_{y}, tan θ_{y} of polar angles θ_{x}, θ_{y}. Higher order images also differ from the 0^{th} order in the regard that they can have negative sign. Here, their absolute values are plotted. Compared with the 0^{th} order, higher order images in Fig. 5 clearly have increasingly higher spatial frequency components. The extrema of higher order images can be very sharp functions of position, allowing accurate determination of their zeros. In this sense, higher order WMI can help determine the dimension of the diffractive structure more accurately than the 0^{th} order alone.

The above examples help the discussion of the experimental results in section 4, and illustrate a key point of this paper, which is that WMI is a general technique not necessarily restricted to spectroscopy, and that even when spectroscopy is the main interest, the geometrical effects should be taken into consideration.

## 3. Experimental approaches

#### 3.1 Scanning imaging system and data acquisition

The experimental setup is diagrammed in Fig. 6. A laser beam from a broadly tunable, mid-infrared quantum cascade (QC) semiconductor laser [6], (which will be described in section 3.3 below) was coupled into an X-Y galvanometer optical scanner (Cambridge Tech. Inc. M6880), which is capable of high-speed and accurate raster scanning over a target. The beam can be set collimated or gently focused (low numerical aperture) on the target. Prior to the scanner, a 60-40 beam splitter was used to split off the beam for monitoring. The detector could be set-up in the backward or forward scattering mode, although all targets reported in this work were studied in the forward mode. The scattered beam was collected by a 2.5-inch *f*/1 Si lens, attenuated by 20 dB to simulate a low return signal, and focused onto an InSb detector (Judson Tech. J10D).

The 2-D images were acquired by correlating the beam scanned position {*i*,*j*} with detected power. The scanning field-of-view was ~ 0.14x0.14 rad. In general, different WMI images can be obtained by placing the receiver at different vantage points. For strongly scattering targets, similar to diffraction tomography, these can provide essential information to reconstruct the target. The two targets reported in this work do not scatter strongly, and only the on-axis beam was captured. The first target is a strong spectroscopic target with high specificity, which consists of CO gas mixed with the atmosphere confined in a 15-cm long, 2.5-cm diameter cell with CaF_{2} windows orthogonal to its axis. As mentioned earlier, the gas itself is not the subject of interest, but only serves as a template material for the study of the WMI technique. The second target serves as a low-spectroscopic-signature object, consisting of a plastic sheet (transparency printed medium) printed with different inks and dyes.

Both signals from the power monitoring detector and the receiver detector were fed into a multi-channel analog-to-digital converter (ADC) capable of 40-MHz sampling rate (Analog Device AD9042). The digitized signals were subsequently buffered into a First-In-First-Out (FIFO) board and then processed with a digital signal processor (DSP) (Motorola DSP56L307), which can perform digital filtering, and adaptively process the data with its maximum 160-MHz main frequency. The multi-channel ADC, FIFO, and DSP broads were home-built and allow flexibility in control and programming.

#### 3.2 Signal processing, system noise, and system detectivity

As the laser was operated in pulsed mode, the basic unit of measurement is the received pulse energy, obtained by digital integration. Since the laser was set up on a separated optical table that was not mechanically coupled with the scanner, which resulted in opto-mechanical noise, the received pulse energy was also normalized to the monitoring signal.

The wavelength modulation was applied to each laser pulse. It was not continuous, but digitized. Each sequence is a set of 4 pulses of different wavelengths: *λ*-Δ*λ*, *λ*, *λ*+Δ*λ*, and *λ*, not necessarily in that order. Let *S*_{ij}
denote the normalized measured pulse energy for pixel {*i,j*}, the 0^{th}, 1^{st}, and 2^{nd} order images were obtained as:

where *S*_{ij}
(λ;1) and *S*_{ij}
(λ,2) denote the two different measurements for the same wavelength at two different times (one can be optionally dropped for faster measurements). In DSP programming, all three orders of images can be simultaneously generated from the raw data set {*S*_{ij}
(*λ* - Δ*λ*),*S*_{ij}
(*λ*;1),*S*_{ij}
(*λ* + Δ*λ*),*S*_{ij}
(*λ*;2)}via dot-products with the three digital filters 1/4{1,1,1,1},1/2{-1,1,1,-1}, and {1,-1,1,-1}. The DSP board allowed much faster computation than the computer. Since the laser wavelength can be randomly modulated, any sequence can be used such as pseudo-noise codes to avoid interference.

The advantage of this scheme is that it uses the minimum number of sampling points needed to obtain the highest orders of image of interest, which is 2^{nd} in this case. It is also more efficient than the analog approach of harmonic continuous wavelength modulation. The latter scheme, in a discrete sampling representation with *N* points {*S*_{ij}
(*λ*_{m}
)${\}}_{m=0}^{N-1}$, would also require a dot product with 1/√*N*{*e*
^{i2πmp/N}${\}}_{m=0}^{N-1}$ for signal of order *p*, which is the discrete Fourier transform. Analog lock-in amplifier would lose information of other harmonics. In terms of signal-to-noise ratio (SNR), there is no advantage for finer wavelength sampling than needed. In an additive white Gaussian noise system, the noise of an order is simply:

if all *S*’s have the same noise. This shows that choice of the number of sampling points *N* is not crucial with respect to the SNR, provided that the total energy of the set {*S*_{ij}
(*λ*_{m}
)${\}}_{m=0}^{N-1}$ is the same. However, it is clear that with regard to DSP (data buffering and computation) it is far more efficient to use the minimum number of points needed.

The system noise level determines the detection sensitivity. The current system is limited by the laser relative intensity noise (RIN). The normalized laser pulse energy has a Gaussian distribution with a RIN power spectral density (PSD) of -100 dB/Hz, which is flat (white noise) from 10 Hz to 5 kHz. Since the system is all-digital, the noises of all orders as given in Eqs. (4) obey the statistics represented in Eq. (5) without any analog processing noises:

and have the same PSD shape. The magnitude of the 1^{st} and 2^{nd} order WMI signals can be expressed relative to the 0^{th} order as *I*
^{(1 or 2)}/*I̅*
^{(0)} provided the average *I̅*
^{(0)} ≠ 0, ≫ *I*
^{(1,2)}, then their noise power are simply proportional to *I*
^{(0)} RIN, which are: *N*_{S-RIN}
and 4 *N*_{S-RIN}
, respectively, where *N*_{S-RIN}
is the normalized laser pulse RIN PSD. The system sensitivity is
then 10^{-5}/√*Hz* and 2 × 10^{-5}/√*Hz* for the 1^{st} and 2^{nd} WMI signal. This figure, however does not truly reflect the system, which is limited by the wavelength modulation rate. A more indicative measure is with regard to total received energy: 1.4 and 2.8×10^{-6}√μJ.

The image acquisition rate was limited by the 2.5-kHz wavelength modulation rate, as each pixel requires 4 pulses of different wavelengths. Thus, the laser was limited to 10 kHz even though it was capable of multi-MHz rate. The wavelength modulation rate is discussed further in the next section. The system is capable of 256×256-pixel resolution, which was limited by the DSP on-board memory, but 100×100 pixels was sufficient here. The minimum time for a 100×100 pixel-image is about 4 sec, and can be up to 40-80 sec for better SNR. A video-rate (30 Hz) acquisition will require a wavelength modulation rate of >300 kHz.

#### 3.3 Laser transmitter

A key component in this work is a broadly tunable, wavelength-modulatable mid-infrared quantum cascade (QC) semiconductor laser. There have been interests in mid-infrared lasers in recent years, partially for spectroscopic sensing [14–21]. Basically, the laser is an external cavity laser ~4.85-4.9 μm; the details have been published elsewhere [6, 21–23]. The feature essential to this work is that the laser employs different mechanisms for broad wavelength tuning and for fast wavelength modulation, which is also isolated from the power control. The wavelength modulation was done with a thermo-optic phase segment, which allows a modulation rate up to ~30 kHz and amplitude up to ~2 GHz (~0.16 nm). However, there is a trade-off; large Δ*λ* requires longer cooling time, reducing usable modulation rate to 2.5 kHz in this work. An electro-optic phase segment in the future can allow multi-MHz modulation. Most experiments were performed with Δ*λ*~0.09 nm (Δ*λ*/*λ*~1.8×10^{-5}). Since the phase segment is isolated from the gain segment, its *λ*-modulation does not affect the laser power. This avoids the problem of simultaneous power/wavelength modulation in many diode lasers, which confounds the use of the first order WMS because of a large background. Furthermore, in many cases of direct modulation, the relation of laser power vs. current is not perfectly linear, resulting in a background for even the 2^{nd} order. For this laser, as shown in a previous work [6], the 2^{nd} order background is negligible, estimated ~< 10^{-5}. These properties were crucial for the experiments in this work. The laser yielded a peak power up to 4 mW, and was operated with 100-ns pulse length. This power level was actually more than needed, since the work was interested in evaluating the technique under high return loss condition, and the receiver was actually attenuated by 20 dB.

## 4. Experimental results and discussion

#### 4.1 Spectroscopic target with narrow spectral line: CO gas

### 4.1.1 Spectroscopic signature

Any gas with a narrow spectral line can offer a strong spectroscopic target to test the system. Commercially purchased CO gas was mixed with the atmosphere in the cell as described above. The 0^{th} and 2^{nd} order images for different CO concentration were shown in Fig. 7. The center wavelength was at the peak of the absorption line at 4.88693 μm. Each image consists of 100x100 pixels. The total received energy for each pixel is 160 pJ, and for the entire image, 1.6 μJ. From left to right, the absorptance varies from 0.57 to 0 as indicated, measured by taking the ratio of the transmitted power to the background without gas. The 2^{nd} order WMI images on the top row have a reverse brightness relationship with the 0th order images on the bottom row as expected. Some details of WMS measurement of this line was reported in a previous work [6]. The signal-to-noise ratio for the 2.8×10^{-3}-absorptance image is actually much better than its darkness may indicate. The reason it was dark is simply because it was compared with the much stronger signals on the same brightness scale. The actual 2^{nd} -order signal was not even a full 2.8×10^{-3}-absorptance, but only ~6×10^{-4} because the wavelength modulation amplitude was smaller than the absorption linewidth (it is still ~3 times above the relative noise level 2×10^{-4}). Compared with the corresponding 0^{th} order images, which is quite ambiguous to determine the presence of the gas because of clutter, the 2^{nd} order images are clearly more sensitive, which is well-known in WMS studies. Particularly relevant to imaging, it is also more specific in terms of discrimination of background clutter.

However, on an image with a narrower contrast scale, even the 2^{nd} order WMI in Fig. 8(a) itself has ambiguity. The grayish region marked in the dashed green outline has no gas, but its signal clearly does not vanish. The signal is not purely spectroscopic. Many geometrical features are obvious. This geometrical issue is a key interest in this work.

### 4.1.2 The essence of imaging

Without geometrical effects, the system is sensitive enough to detect absorptance even lower than 2.8×10^{-3}. By subtracting the 0-absorptance image Fig. 8(b) from the 2.8×10^{-3}-absorptance image Fig. 8(a) to remove some of the geometrical features, the resulting image Fig. 8(c) and its 3D plot Fig. 8(d) show that there is significant spectroscopic signal in the gas cell. But the residual geometrical features in Fig. 8(c) are also meaningful. An issue is how some features, such as the clamp holding the gas cell [cf. Fig. 10] was so distinctive even as its signal average is 0, which is the same as its surrounding. The reason is that the two regions have different noise amplitudes: one region has larger 0^{th} order signal (higher laser signal) than the other (no laser signal); (It should be noted that the 2^{nd} order noise is proportional to the 0^{th} order signal magnitude). A single point measurement in each region would not provide any meaningful information. But noise coupled with certain spatial pattern in an image actually contains meaningful information as in Figs. 8(c) and (d). This is a fundamental essence of imaging that distinguishes WMI from point-sampling WMS.

### 4.1.3 The geometrical aspect

The geometrical effects are not significant for strong absorptance in Fig. 7. But an interest in WMI is for high-sensitivity detection. For 160-pJ received power per pixel in this case, the expected noise equivalent of 2^{nd} order signal is 2×10^{-4} relative to the maximum 0^{th} order signal. In a high power and fast modulation system, e.g., 1 μJ/pixel, the expected noise equivalent signal would be 2.8×10^{-6}. But it is clear that the geometrical effects will be an issue long before such a sensitivity level can be reached. The optical geometrical effects discussed in Section 2 can account for some of the geometrical features, of which two are considered here: the etalon fringe effect and the edge effect.

As discussed in Section 2, the etalon effect is everywhere in this system, from the coating on lenses, optics, to the detector window. The net result is a product of all the etalon effects along the beam path. With a likely random alignment of the phases of the transmission curves, many of these fringes can cancel each other out. However, some fringes on the gas cell are visible in all the 2^{nd} order WMI images in Figs. 7 and 8, and more faintly on the 0^{th} order images. These fringes as likely associated with the cell CaF_{2} windows. The intensity along the red lines of the 2^{nd} and 0^{th} order images in Fig. 7 are plotted in Fig. 9(a), showing the fringes across the 2.5-cm diameter window. The 0^{th} order fringes have opposite phase to those of the 2^{nd} order, as indicated by the arrows, similarly to Fig. 4. The spacing of the fringes is consistent with the window thickness and wedge angle ~0.2-0.4 mrad. The fringe depth is ~1×10^{-4}, which is ~10 times smaller than the perfect fringes calculated in Fig. 4, with Δ*λ*/*λ* adjustment included.

Also obvious are the 2^{nd} order signals at edges and boundaries. It is clear that different regions in the image are distinguished not only by their amplitudes, but also by the signal at the boundaries. Figure 9(b) shows spikes along the red lines in Fig. 8(b), marking the boundary. This edge spikes occur with both polarity, with magnitude ranging from ~10^{-5} to 10^{-4}, which are consistent with the edge diffraction discussion in Section 2.3.1 (Fig. 3).

#### 4.2 Multi-modal imaging: digital fusion of passive and active images

Since the 2^{nd} order images have better sensitivity and clutter-rejection capability than the 0^{th} order image, one approach for gas imaging is to use a synthetic imaging technique that combines passive images acquired with focal plane array with laser-based 2^{nd} order WMI images. This is different from other non-wavelength-modulation gas sensing imagers that rely only on the 0^{th} order absorption image [7–9]. The concept is illustrated in Fig. 10(a). A synthetic image obtained by digital fusion of a CCD camera image of the gas cell with the 5%-absorptance 2^{nd} -order WMI images in Fig. 7 is shown in Fig. 10(b). The new image *I*_{S;ij}
is simply a function *f*|${I}_{\mathit{\text{ij}}}^{\left(2\right)}$,*J*_{ij}
|, which can be selected with various imaging processing algorithms and techniques (details are outside the scope of this paper). Here, the intensity image *I*
^{(2)} was processed with a simple threshold-alarm function and displayed with false-color attribute. More generally, a system with many wavelengths can simultaneously detect different spectroscopic signatures and use color-coding to show different species.

A crucial task is to handle the geometrical features, which involves image congruence alignment on one hand, and subsequent removal the geometrical effects from the final spectroscopic image on the other hand. The laser can be tuned off line or another laser can be used at random but near the spectroscopic line wavelength to generate geometrical features for comparison and processing.

#### 4.3 Target with low spectroscopic signature

WMI is a general technique not just for gas spectroscopy. Many common materials do not have strong spectroscopic signatures like light-weight molecule gases. For them, the geometrical and spectroscopic effects can be comparable. The target in Fig. 11(a) serves as an example of this condition. The three orders of WMI images are shown in Fig. 11(b). At the measured wavelength of 4.887 μm, FTIR spectra of the material and various features do not show any sharp curvature for small ∆*λ*/*λ*~1.8×10^{-5}. The existence of a strong 2^{nd} order image that has comparable magnitude with the 1^{st} order underscores the need to consider WMI in a broader context than spectroscopy. Both geometrical effects discussed above can account for many of the features observed. The plastic film itself is a lossy etalon. Edges and borders are quite sharp, in fact the higher order images can be used as edge detectors. As discussed above, each order can offer unique information of the target. It is beyond the scope of this paper to perform information-content analysis and apply algorithms to extract maximum information from these images. Nevertheless, an example is useful to demonstrate its potential. New images obtained by Gram-Schmidt ortho-normalization are shown in Fig. 12(a). The fringes in the 2^{nd} -order processed image may indicate film thickness variation, for example. The printed lines in higher-order images have much sharper contrast than those in the 0^{th} order. Also, two dye spots indicated with the circles are more visible in higher order images than in the 0^{th} order. Figure 12(b) is a false-color combination of the three images in Fig. 12(a). This example qualitatively illustrates the capability of WMI to provide different information of a target with different orders, and the information is not necessarily purely spectroscopic.

## 5. Summary and conclusion

Wavelength modulation imaging is a general technique that can be applied to study the spectroscopic and geometrical features of a target. This work examines both aspects theoretically and experimentally. The theoretical basis was discussed in the context of scattering, showing that diffraction and interference have strong *λ*-dependence. Experiments were demonstrated using an all-digital approach, employing a tunable mid-infrared laser that is capable of wavelength modulation independently of power, allowing negligible background in wavelength modulation signals. An X-Y raster scanning system was used to scan targets and generate images. With the DSP approach, all three orders of image 0^{th}, 1^{st}, and 2^{nd} can be simultaneously generated. Both strong and weak spectroscopic targets were studied.

The results show the equal importance of both spectroscopic and geometrical aspects in the WMI technique. The geometrical aspect, which can have their own applications such as microscopy and diffraction tomography, was neglected in earlier WMS works, but is shown here to be crucial for correct interpretation and application of the WMI technique regardless of the primary application interest.

This work also explores the potential of synthetic imaging involving WMI. An example of fusing the 2^{nd} order WMI image of CO gas with conventional passive focal plane image shows the multimodal feasibility to detect and visualize spectroscopic target. This approach is more sensitive and more capable of clutter-rejection than direct 0^{th}-order imaging. The technique is not restricted to strong spectroscopic targets, but was shown to be generally applicable to even targets with weak spectroscopic features.

From the practical application viewpoint, a key feature in this work is the use of advanced semiconductor lasers and the DSP approach. Semiconductor lasers have well-known advantages of power efficiency, compactness, and potential cost-effectiveness. Recent high-power performance [22] suggests their capability for short-range stand-off sensing. The DSP approach will be more practical than analog for system with advanced algorithms.

## Appendix

## A.1 Obscuration of a beam

The problem is idealized as a one-dimensional Gaussian beam blocked by a perfectly conducting half plane. This requires Sommerfeld’s exact solution for plane wave [12]:

for the forward diffracted transverse electric field (z<0), where${k}_{x}^{2}$ + ${k}_{z}^{2}$ =*k*
^{2} , and the integration path for *μ* encircles the singularity at -1 in the upper half, and at -*k*_{x}
/*k* in the lower half of the complex plane. The solution for an incident 1-D Gaussian beam can be obtained by integrating Eq. (A.1) with a Gaussian k-space spectrum as explained in Eq. (2):

This solution can be numerically computed, but requires excessive time. An alternate approach is to use Sommerfeld’s approximated plane wave solution [11] in lieu of Eq. (A.1):

where (*r,θ*) is the polar coordinate in lieu of (*x,z*), and *α* is the incident angle off normal axis. This solution is useful for large *kr* and allows faster computation. The issue is whether it is sufficiently accurate for the calculation of small geometrical effects. The approach is to perform computation for both solutions at only one wavelength over a range of few *kr* for comparison. The result was that the integrated power of the approximate solution deviates from the exact solution by <10^{-5} for *kr* >100, which is smaller than the geometrical effects, typically 10^{-4}-10^{-3}. Numerical Romberg’s integration was used, with residual relative errors <10^{-2}. Results for a beam waist of 200 μm, exactly half-blocked, are shown in Figs. 2 and 3.

## A.2 Etalon effects

The transmission of a plane wave through a window of thickness *L* and index *n* and with parallel surfaces is given by [25]:

and *α* is the absorption coefficient. For convenience, the weak spectroscopic wavelength-dependences of *n* and *α* (dispersion) are neglected. The 1^{st} and 2^{nd} order terms are:

There are no wavelength scaling terms since there is no lateral diffraction. These are used to plot Figs. 4 with *n*=1.4 (CaF_{2} at 5 μm), *α*=0, and *L*=3 mm [for Fig. 4(a)].

## A3 Fraunhoffer diffraction of an annulus

Here, if we consider the image as the diffracted light intensity distribution, the 0^{th} order is (normalized to the center peak intensity):

*ξ* = *ak*sin*θ*, *η* = (*a* + *b*)*k*sin*θ*, *u* = *a*
^{2} /⌊(*a* + *b*)^{2} -*a*
^{2}⌋, *v* = (*a* + *b*)^{2} /⌊(*a* + *b*)^{2} - *a*
^{2}⌋, *a* is the inner radius, and *b* is the width of the annulus. The first-order image is:

The first term, -2(∆*λ*/*λ*)*I*
^{(0)} on the right side Eq. (A.8) is the wavelength scaling term in Eq.(3.a). It contains no useful additional information. The second term -8(∆*λ*/*λ*)*F*
_{0}
*F*
_{1} is the geometrical term in Eq. (3.b) containing higher spatial frequency information through Bessel function *J*
_{2} in *F*
_{1}. Likewise, the 2^{nd} order pattern is:

which also has a wavelength scaling term and a geometrical high spatial frequency term. Figure 5 were plotted for *a*=25 *λ* and *b*= *λ*.

## Acknowledgments

We wish to acknowledge the support of the US DARPA/ARO under contract #DAAD-190010361, the US DARPA/Air Force Research Laboratory under contract #F33615-02-C-1139, and the State of Texas THECB under an ATP program, and the Texas Center for Superconductors and Advanced Materials at University of Houston for this work.

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