## Abstract

In this paper, we present the design of a novel spectral synthesis system that is based on an array of in-plane moving micro gratings, instead of using a conventional out-of-plane micromirror array. Utilizing the unique nondispersive characteristic of the optical phase modulation mechanism based on in-plane movable micro gratings, we demonstrate that the synthetic spectra problem can be greatly simplified and effectively reduced to that of conventional phase retrieval.

©2008 Optical Society of America

## 1. Introduction

Spectral synthesis [1–3] using an array of micromirrors has been previously proposed for remote sensing of chemicals based on correlation spectroscopy. In this spectral synthesis system, the relative height of each micromirror in the array is accurately controlled to generate a diffractive surface relief. When illuminated with a collimated broadband light, the diffractive surface relief produces a desired intensity spectrum at a predetermined diffraction angle. This artificially-generated intensity spectrum - emulating an absorption spectrum of a reference material - is then used in correlation spectroscopy for chemical sensing applications, where it is correlated with the spectral pattern of a light beam transmitted through a sample. Using the microelectromechanical systems (MEMS) technology, it is possible to design and fabricate a single programmable micromirror array to create the spectra of a large number of materials [4–7]. As a result, the correlation spectrometer based on synthetic spectra has the potential to form a new generation of compact multi-component chemical sensing system for field use.

As shown in Fig. 1, considering a broadband collimated electromagnetic wave that strikes a one-dimensional micromirror array at normal incidence, and working in the Fraunhofer approximation, it is easy to show that the diffracted field *U*(*u*) at a fixed diffraction angle *θ* is described by the following equation: [8]

where *u* is the wavenumber of the light, *C*(*u*) is a wavenumber dependent constant, *d _{m}* is the out-of-plane displacement of the

*m*

^{th}micromirror,

*w*is the width of a micromirror element, and

*M*is the total number of elements in the array. The synthetic spectra problem is equivalent to solving the above equation for micromirror displacement

*d*for a given desired intensity spectrum of

_{m}*U*(

*u*). This is a challenging problem and typically, complicated optimization algorithms have to be employed to obtain an approximate solution [2,4,8]. The above-mentioned problem results from the fact that the phase modulation mechanism of a micromirror array is dispersive in nature (the second term of the right-hand side of the equation). The phase profile imposed on the incident light by the micromirror array is wavelength-dependent, thus complicating the problem. In this paper, instead of using a dispersive micromirror array, we report a novel spectral synthesis method using a micro grating array. Using the nondispersive characteristic of the optical phase modulation mechanism based on in-plane movable micro gratings, we demonstrate that the synthetic spectra problem can be greatly simplified and reduced to a conventional phase retrieval problem [9].

## 2. Spectral synthesis using micro gratings

#### 2.1. Operation principle

In our previous work, we have demonstrated that an in-plane translational displacement of a diffraction grating along a direction perpendicular to the grating lines induces a nondispersive phase shift to the diffracted beam [10]. In this paper, the application of this grating based nondispersive optical phase shifting mechanism to spectral synthesis is presented.

Figures 2(a) and (b) show schematically the 3D and top views of the proposed synthetic spectra system using a MEMS driven in-plane movable micro grating array. A compensation grating of the same period is used in conjunction with the micro grating array to cancel out the angular dispersion of an individual grating element. As shown in Fig. 2(b), consider a collimated light beam that illuminates the compensation grating at an incident angle of *θ _{i}*. The grating lines are aligned along the

*z*-axis. The incident light wave is then:

where *U*
_{0}(*u*) is the amplitude of the incident wave and *u* is the wavenumber. At *y*=0, the compensation grating bestows a spatial phase modulation exp[*i*2*πuh*(*x*)] on the incident light wave, which can be expanded in a Fourier series due to its periodicity along the *x*-axis as:

where *h*(*x*) is a function describing the grating surface profile, *k* represents the diffraction order and *p* is the grating period. If we select *k*=-1order beam and direct it to the micro grating array, the light wave diffracted from the compensation grating is then:

where *η*(*u*)=C_{-1}(*u*) is related to the diffraction efficiency and *θ _{d}* is the diffraction angle determined by the grating equation: sin

*θ*=sin

_{d}*θ*- 1/(

_{i}*pu*). At

*y*=l, the light reaches the micro grating array, where it receives a second spatial phase modulation. Here we assume that each grating element has the same grating profile and period as the compensation grating, in addition, the grating lines of different elements are initially aligned collinearly such that the whole array represents a single grating. Consider the spatial phase modulation bestowed by the

*m*

^{th}grating element in the array. As shown in Figs. 2(a) and (b), a local coordinate system is defined by the

*x’y’z’*axes with its origin located at the center of the array. It is noted that the micro gratings are reversed relative to the compensation grating. Hence, the local axes

*x’*and

*y’*are designated to point to the -

*x*and -y directions, respectively. In the local coordinate system

*x’y’z’*, the light field just leaving the grating element can be expressed as:

where *d _{m}* is the in-plane displacement of the grating element driven by its MEMS actuator for phase modulation. Using the Fourier expansion:

and again taking *k*=-1, we get:

The above equation implies that parallel beams of different wavelengths incident on the first grating are still parallel when they leave the second grating. Additionally, the phase shift introduced to the output light beam by the in-plane movement of the grating element along a direction perpendicular to the grating lines is dependent only on the ratio of the in-plane displacement to the grating pitch and is independent of wavelength. The total diffracted field that just leaves the micro grating array can then be expressed by:

where *Δ* is the width of the element along the *z’* axis, and 2*M*+1 is the total number of grating elements. Again, working in the Fraunhofer approximation, the light wave diffracted to a predetermined direction specified by the direction cosines (-sin*θ _{i}*, cos

*θ*, sin

_{i}*α*) in the

*x’y’z’*coordinate system as shown in Fig. 2(a) is given by [11]:

where *A* is a constant that does not depend on wavenumber *u*. Substituting Eq. (8) into Eq. (9), we obtain:

$$\phantom{\rule{.8em}{0ex}}\phantom{\rule{.8em}{0ex}}\phantom{\rule{.8em}{0ex}}\phantom{\rule{.6em}{0ex}}\phantom{\rule{.6em}{0ex}}\times \sum _{m=-M}^{M}\mathrm{exp}\left(\frac{i2\pi {d}_{m}}{p}\right)\mathrm{exp}\left[-i2\pi mu\Delta \phantom{\rule{.2em}{0ex}}\mathrm{sin}\alpha \right],$$

where *B* is a wavenumber-independent constant related to the width of the light beam along the *x’* axis (Here we assumed that the micro grating elements are long enough to cover the incident beam width along the *x’* direction.). It should be noted that in order for the proposed spectral synthesis method to work properly the angle *α* must be non-zero. Otherwise the term exp[-*i*2*πmu*Δsin*α*] in Eq. (10) will be equal to one, and then the spectrum of the outgoing light field *U*
_{3}(*u*) will be independent of the phase shifts imparted by the in-plane grating displacements *d _{m}*.

We define a set of discrete sampling wavenumbers *u _{n}* within the spectral range of interest [

*u*,

_{min}*u*]:

_{max}where *n*=0,±1, …, ±*M*, and *l* is an integer. We further define:

Taking the modulus and then squaring both sides of Eq. (10) and utilizing Eqs. (11), (12), and (13), we obtain:

where *DFT* stands for the operation of discrete Fourier transform. The synthetic spectra problem is now equivalent to solving the above equation for phase modulation *ϕ _{m}* for a given desired intensity spectrum

*I*or |

_{n}*U*(

*u*)|

_{n}^{2}. The synthetic spectra problem using an array of nondispersive phase modulators has therefore now been reduced to a conventional phase retrieval problem [12–13] - more specifically, recovering phases from two known intensity constraints, i.e. |

*U*(

*u*)|

_{n}^{2}and 1. A number of approaches have been proposed for this latter problem, including the Gerchberg-Saxton algorithm [14–15], simulated annealing [16] and genetic algorithm [17].

#### 2.2. Spectral resolution and bandwidth

According to Eq. (11), the spectral resolution of the proposed spectral synthesis system is inversely proportional to (2*M*+1)Δsin*α*, which is actually the total length of the grating array along the *z’* axis projected onto the propagation direction of the outgoing light of interest. In addition, from Eqs. (10), (11) and (13), it is clear that the amplitude spectrum |*U*(*u*)| is periodic with a period of 1/(Δsin*α*). Therefore, for practical design considerations of the proposed system, one can simply determine the width of the grating element in the array based on the reciprocal of the spectral bandwidth of interest, i.e.,

and determine the total length of the array and consequently, the total number of elements, from the reciprocal of the required spectral resolution *δu*,

The relationships are displayed in Fig. 3.

In a practical spectral synthesis system shown in Fig. 2, a lens together with a pinhole is employed to select the light beam diffracted from the array to the desired direction. It is noted that the use of a lens and pinhole will affect the spectral resolution of the synthesized spectrum. The effect of the lens and pinhole however may be minimized to a negligible level when a long-focal-length lens coupled with a small pinhole (for example smaller than the diffraction-limited spot size) is used.

#### 2.3. Efficiency considerations

Using Parseval’s theorem on Eq. (14), we obtain:

This equation implies that the loss of the system is proportional to the complexity of the desired spectrum. For example, if the desired spectrum contains only a single non-zero value at a certain wavenumber *u*
_{1}, then the efficiency of the system can be potentially very high. This is true because we separate the input beam into multiple identical channels and recombine them to form a single output. If we configure the system such that the subdivided light beams emerging from all channels are in phase at wavenumber *u*
_{1}, then the light of this particular wavenumber can be completely transmitted through the system, thereby achieving high efficiency. However, if the desired spectrum contains a set of non-zero values at different wavenumbers, then the in-phase condition has to be compromised to achieve the desired intensity spectrum. According to Eq. (17), the values of |*U*(*u _{n}*)|

^{2}at these targeted wavenumbers will generally reduce since the total amount is fixed. As a result, at each wavenumber, the light is attenuated by the system.

In addition, by combining Eq. (17) and Eq. (13), it is clear that the values of *η*(*u*) and sinc(*u*Δsin*α*) must be maximized in order to minimize the loss and achieve maximum efficiency, because it is the outgoing light wave *U*
_{3}(*u*) we are interested in. For *η*(*u*), the solution is obvious, that is to utilize gratings blazed at a certain wavenumber within the spectral range of interest [*u _{min}*,

*u*]. For a MEMS-based programmable spectral synthesis system capable of producing a large number of different spectral patterns, it is convenient to set the grating blazing wavenumber at the center of the spectral band of interest. For sinc(

_{max}*u*Δsin

*α*), the solution is not so straightforward. In fact, this term is closely related to the Fraunhofer diffraction pattern of a slit aperture with a width of Δ at the diffraction angle

*α*. As the diffraction angle α increases, the value of sinc(

*u*Δsin

*α*) decreases and hence the loss of the system increases. To overcome this problem, an additional diffractive surface profile has to be utilized. As shown in Fig. 4(a), consider the spatial phase modulation bestowed by the

*m*

^{th}grating element in the array’s local

*x’y’z’*coordinate system. In addition to the phase modulation along the

*x’*axis, we assume that the grating element also induces a periodic spatial phase modulation along the

*z’*axis with a period of

*g*. The element is now actually a two-dimensional (2D) crossed-grating with a primary grating profile whose periodicity runs along the

*x’*axis for phase-shifting and an auxiliary grating profile whose periodicity runs along the

*z’*axis for improving the efficiency. It is noted that continuous surface profiles (for example blazed gratings) can be implemented using gray-scale mask technology [18] or approximately using binary optics technology [19]. For binary optics, the fabrication process consists of a set of discrete steps, each of which comprises of aligned photolithography and etching processes, to form a staircase-like approximation to the continuous surface profile. Considering the secondary grating profile, the overall phase modulation bestowed by the

*m*

^{th}grating element is given by:

$$\phantom{\rule{10.em}{0ex}}\phantom{\rule{8.em}{0ex}}\times \{\sum _{k}{D}_{k}\left(u\right)\mathrm{exp}\left[\frac{i2\pi k\left(z\prime -m\Delta \right)}{g}\right]\},$$

where *h*(*x*’) and *f*(*z*’) are the functions describing the grating surface profile. Here, we assume that the surface profiles of the grating elements are identical and the micro gratings are confined to move only along the *x*’ axis to introduce phase shifts. Selecting the negative and positive first-order terms from the first and second Fourier expansions respectively on the right-hand side of Eq. (18) and assuming Δ/*g* is an integer (each element has exact multiple full periods of the auxiliary grating profile), we find that instead of Eq. (10), the outgoing light beam from the grating array to the predetermined direction is now given by:

$$\phantom{\rule{.8em}{0ex}}\phantom{\rule{.8em}{0ex}}\phantom{\rule{.8em}{0ex}}\phantom{\rule{.8em}{0ex}}\phantom{\rule{.8em}{0ex}}\phantom{\rule{.8em}{0ex}}\phantom{\rule{.8em}{0ex}}\phantom{\rule{.8em}{0ex}}\phantom{\rule{.6em}{0ex}}\times \sum _{m=-M}^{M}\mathrm{exp}\left(\frac{i2\pi {d}_{m}}{p}\right)\mathrm{exp}\left[-i2\pi mu\Delta \phantom{\rule{.2em}{0ex}}\mathrm{sin}\phantom{\rule{.2em}{0ex}}\alpha \right],$$

where *ξ*(*u*)=*D*
_{1}(*u*). Following the same procedure described in section 2.1, we define:

We find that the main equation for spectral synthesis, ie. Eq. (14) remains unchanged. Comparing Eq. (20) with Eq. (13), it is noted that in addition to a new term *ξ*(*u*), the maximum of the sinc function in Eq. (20) is shifted. We can utilize this to minimize the loss of the system, for example, by setting the period *g* of the auxiliary grating to fulfill the following equation:

where *u*
_{0} is a chosen wavenumber, typically the center of the spectral band *u*
_{0}=(*u _{max}*+

*u*)/2, while at the same time, choosing a blazed profile at wavenumber

_{min}*u*

_{0}for the auxiliary grating so as to maximize the value of

*ξ*(

*u*). From Eq. (21), we see that the function of the auxiliary grating profile is to diffract the light of wavenumber

*u*

_{0}with high efficiency from normal incidence to the predetermined angle α. The increase in the sinc function values using an auxiliary grating profile is illustrated in Fig. 4(b).

In conclusion, in order to achieve high efficiency, the first compensation grating must be blazed at a certain wavenumber within the spectral range of interest, usually at the center of the band. In addition, each grating element in the second phase-shifting array must be a 2D crossed-grating that can be decomposed into two orthogonal grating profiles with one identical to that of the compensation grating and the other blazed at a selected wavenumber (for example the center of the spectral band) to the predetermined direction.

#### 2.4. Design method

We have shown in the above sections that the synthetic spectra problem using an array of MEMS-driven in-plane-moving gratings is reduced to a conventional phase retrieval problem. In this section, we will develop a gradient research method to optimize the design of the proposed spectral synthesis system.

Based on the system requirements including the spectral range of interest, spectral resolution, and diffraction angle *α*, we can determine the width of each grating element as well as the total number of required elements in the array using Eqs. (15) and (16). The period of the primary grating for phase shifting can be determined by considering the maximum stroke of the driving MEMS microactuator. The period of the auxiliary grating is determined by the required diffraction angle *α* and a chosen wavenumber. Next, the detailed grating profiles and consequently, the efficiency terms *η*(*u*) and *ξ*(*u*), can be obtained by properly selecting blazing wavenumbers within the spectral range of interest. Based on the desired intensity spectrum of the outgoing light wave |*U*
_{3}
^{d}(*u*)|^{2}, and using Eq. (20), we can then determine the targeted intensity constrain *I ^{d}_{n}*=|

*U*(

^{d}*u*)|

_{n}^{2}for the Fourier transform Eq. (14).

To evaluate the system performance, we define an error function to describe the difference between the calculated intensity spectrum *I _{n}* (from the phase shifts

*ϕ*using

_{m}*DFT*) and the desired intensity

*I*:

^{d}_{n}where the parameter γ is a scaling factor that scales *I _{n}* for a minimum

*E*fit to

*I*. It can be derived by setting the partial derivative of

^{d}_{n}*E*with respect to γ equal to 0, resulting in:

After the error function is defined, the phase retrieval problem is equivalent to the following optimization problem:

Using the method reported in Ref. [8], we find:

where the function Im[.] returns the imaginary part of its argument. It is noted that the summation term in Eq. (25) is actually an inverse discrete Fourier transform operation, i.e. *IDFT*{2*γ*(*γI _{k}*-

*I*)

^{d}_{k}*U*(

*u*)}. With Eq. (25), we can compute the gradient direction of the objective function, allowing gradient-based search techniques including the Davidon-Fletcher-Powell (DFP) method [20] to be applied to solve this optimization problem.

_{k}A block diagram showing the proposed gradient search method for spectral synthesis using an array of non-dispersive MEMS grating phase shifters is given in Fig. 5. Compared with the method we developed earlier for spectral synthesis based on micromirrors [8], the advantages of the current approach are obvious. It can easily handle a larger number of sampling points with less memory requirements yet has a faster computational speed, since the gradient of the error function is now obtained by a fast Fourier transform (FFT) and an inverse FFT instead of matrix operations. Using more samples generally results in enhanced spectral resolution and increased resolving power in the spectral synthesis system.

#### 2.5. Alternative setup

Alternatively, the synthetic spectra using micro gratings can also be configured as shown in Fig. 6. Two lenses in a 4-f configuration are inserted between the parallely aligned compensation grating and micro grating array. The 4-f setup reverses the sign of the angles so that the beams will hit the second grating at the same angle as they leave the first one. This can also effectively cancel out the angular dispersion of the MEMS grating elements.

## 3. Design examples and results

Here, we will investigate designs reported in Ref. [8]. Our spectral range of interest is from 3600 cm^{-1} to 4200 cm^{-1}. The spectrum of the incident light beam is assumed to be uniform over the spectral range of interest. First, we set our spectral resolution to be around 2.3 cm^{-1}, which corresponds to about 256 points in the spectral range of interest. In this work, instead of using 2048 micromirrors, we use only 256 MEMS-driven in-plane moving micro-grating elements. The design goal is to determine the phase shift or equivalently, the in-plane displacement of each grating element in the array so that the diffraction intensity spectrum of the light that passes through the spectral synthesis system shown in Fig. 2 and diffracts to a predetermined direction (*α*=15°) is identical to a desired intensity spectrum. Based on the spectral range of interest and diffraction angle *α*, we determine the width of each grating element to be 60 µm. The total length of the array is hence around 15.4 mm. The periods of the primary and auxiliary gratings are both set at 10 µm. All gratings are blazed at the center wavenumber *u*
_{0} of the spectral band of interest, i.e. around 3922 cm^{-1}. Under thin grating assumption, the efficiency terms in Eq. (20) are approximately given by: *η*
^{2}(*u*)=*ξ*
^{2}(*u*)=sinc^{2}(1-*u*/*u*
_{0}) [21].

In the first design example, the targeted intensity spectrum is shown in Fig. 7(a). The design started from a randomly-chosen initial point and terminated when the error defined by Eq. (22) converged or a predetermined number of iterations is reached. The design took only a few seconds on an ordinary PC, and the results are shown in Fig. 7(b). To investigate the efficiency of the spectral synthesis system, the following parameter *ζ* is defined:

where *I*(*u _{p}*) is the intensity of the light for a specific wave number

*u*(typically the wave number corresponding to the peak of the target spectrum) passes through the grating-array-based spectral synthesis system and diffracts to the predetermined direction, and

_{p}*I*

_{0}(

*u*) is the intensity of the same incident light that passes through a double-mirror system and propagates to a direction with 0

_{p}^{0}diffraction angle.

*I*

_{0}(

*u*) can be estimated using Eq. (10) by setting

_{p}*η*(

*u*) to 1, diffraction angle

*α*to 0, and at the same time all displacements

*d*to 0. The value of

_{m}*ζ*in this case is about 1.1%.

The target intensity spectrum of the second design example is shown in Fig. 8(a). It consisted of three extremely sharp lines located at 3800, 3900, and 4000 cm^{-1}. We used the same spectral synthesis system to produce this spectrum and again, the design again took only a few seconds to be completed. The results shown in Fig. 8(b) give a *ζ* value of 33.5%. From these two design examples, we noted that, as compared with the conventional micromirror-based spectral synthesis systems, the proposed system using an array of in-plane moving gratings leads to improved results with fewer elements needed in the array. In addition, with reduced memory requirements and shorter computational time, the optimization algorithm can easily handle a large number of sampling points within the spectral range of interest, hence making high-resolution spectral synthesis possible.

To demonstrate this, we re-investigated the second design. We now set the total number of sampling points in the spectral range of interest to be 1024 instead of 256. Since the spectral range of interest and diffraction angle *α* are kept the same, the system parameters including the width of grating element, grating periods, and blazing wavelength remain unchanged. We simply increased the number of grating elements to 1024, resulting in a spectral resolution of about 0.6 cm^{-1}. The optimization process took less than a minute and results are shown in Fig. 9. The value of *ζ* obtained in this case is around 40%.

## 4. Conclusion

In this paper, we have presented a novel design for a spectral synthesis system using an array of in-plane moving micro gratings wherein the phase modulation mechanism is nondispersive. Hence, it greatly facilitated the solution of the synthetic spectra problem. The proposed system has been shown theoretically to perform well and has high potential of being used in many applications including correlation spectroscopy and optical wavelength filtering.

## Acknowledgments

Financial support by the Ministry of Education’s AcRF Tier 1 funding under Grant No. R-265-000-235-112 and R-265-000-211-112/133 is gratefully acknowledged.

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