## Abstract

We present a novel efficient gradient-based optimization algorithm for the design of diffractive optical elements (DOE’s) for synthetic spectra applications. Two design examples are given. The results demonstrate that the DOE’s obtained by the proposed algorithm can accurately produce the desired intensity spectra at a predetermined diffraction angle.

© 2003 Optical Society of America

## 1. Introduction

Diffractive optical elements (DOE’s) [1] are now becoming increasingly important to a wide range of optical system applications due to their compact configuration, excellent performance, and low cost of production and replication. Such applications include, but are not limited to, microlens array, beam shaping, and spot array generation. Considerable effort has been devoted to the development of numerical procedures for designing DOE’s [2–7]. As a consequence, a number of approaches have been proposed, for example, Gerchberg-Saxton algorithm [2–4], simulated annealing [5] and genetic algorithm [6].

Recently, Sinclair *et al.* reported that DOE’s can also be used to synthesize the infrared spectra of real compounds [8–9]. The DOE’s of this type are considered as the basis for a new class of miniaturized, remote chemical sensing systems based on correlation spectroscopy, which uses these DOE’s rather than reference cells to produce the reference spectra. Thus, many of the difficulties imposed by the use of the real reference materials, for example the material is highly reactive and difficult to handle, can be avoided using synthetic spectra. Furthermore, based on MEMS technology, it is possible to design and fabricate a single programmable DOE, for example electrostatically actuated micromirror array, to create the spectra of a large number of materials. As a result, a correlation spectrometer based on synthetic spectra is extremely compact and can easily be configured to detect a large number of different compounds.

This specific application requires the design of a diffractive element whose diffraction intensity spectrum at a predetermined diffraction angle is identical to the desired target infrared spectrum. Unlike most typical applications of diffractive optics in which monochromatic light is used and spatial distribution of the radiation is of interest, the synthetic spectra application exploits the spectral function of a DOE at a fixed diffraction angle. Sinclair *et al.* also developed a modified Gerchberg-Saxton algorithm for the design of DOE’s that reproduce the infrared spectra of important compounds [9]. Good design results were obtained. However, the phase profile obtained by the modified Gerchberg-Saxton algorithm represents the first approximation of the solution to the design problem. The imposed phase profile by the resulting DOE will exactly match the phase profile determined by Gerchberg-Saxton algorithm only at a specific wavelength within the spectral range of interest (typically chosen to be the wavelength corresponding to the peak of the target spectrum). For other wavelengths, there exist design errors. As indicated in Refs. [8] and [9], in order to obtain an improved result, further optimization with simulated annealing algorithm should be employed. In this paper, an efficient gradient-based optimization algorithm for design of the DOE’s for synthetic spectra application is proposed. Using the proposed algorithm, the above mentioned problem is overcome.

## 2. Design method

Consider a broadband collimated electromagnetic wave strikes a one-dimensional DOE at normal incidence. As shown in Fig. 1, the grating-like DOE consists of *M* micromirror elements each of width Δ, for a total length of *L* (=*M*Δ). The position of each elastically supported micromirror element in the array is controlled precisely by applying a driving voltage between the mirror and the electrode underneath. This programmable, variable-element DOE can be fabricated using microfabrication technology. For simplicity, here we assume that the amplitude of the incident wave does not vary with position or wavelength. The diffracted field just leaving the DOE can be expressed by:

where *A* is the amplitude of the incident wave, λ is the wavelength, and *d*_{m}
is the displacement of the *m*th micromirror element. Working in the Fraunhofer approximation, the diffracted field at a fixed diffraction angle *θ* is described by the Fourier transform [10]:

where C is a constant that does not depend on wavelength λ. Substituting Eq. (1) into Eq. (2), we obtain:

$$\times \sum _{m=1}^{M}\mathrm{exp}\left(-\frac{i4\pi {d}_{m}}{\lambda}\right)\mathrm{exp}\left[-\frac{i2\pi \mathrm{sin}\left(\theta \right)m\Delta}{\lambda}\right].$$

The intensity of the diffracted field for a sampling wave number *u*_{n}
(*u*_{n}
=1/λ
_{n}
, *n*=1, 2, …, *N*) within the spectral range of interest [*u*_{min}
, *u*_{max}
] is given by:

where

and

To evaluate the performance of the DOE for synthetic spectra applications, an error function is introduced to describe the difference between the calculated diffraction spectrum and the desired spectrum:

where ${I}_{n}^{d}$
is desired intensity spectrum, the parameter γ is a scaling factor that scales the diffracted intensity spectrum for a minimum error *E* fit to the target spectrum. It can be derived by setting the partial derivative of *E* with respect to γ equal to 0, resulting in:

After the error function is defined, the design of DOE for synthetic spectra is equivalent to the following optimization problem:

The optimization constraint conditions applied to the displacements of the micromirror elements are:

where *λ*
_{max}=1/*u*
_{min} corresponds to the maximum wavelength of the spectral range of interest.

The partial derivative of *E* with respect to *d*_{m}
is given by:

Since:

with:

where *δ*_{nk}
is the discrete Kronecker delta function, defined as:

Substituting Eqs. (13) and (14) into Eq. (12) results in:

Using Eq. (4) we obtain:

where

and the asterisk denotes the complex conjugate. Substituting Eq. (18) into Eq. (17), and then substituting Eqs. (17) and (16) into Eq. (11) results in:

With Eq. (19), we can compute the gradient directions of the objective function. Therefore, gradient-based optimization techniques can be applied to this DOE design problem.

The optimization technique we used here is the so-called Davidon-Fletcher-Powell (DFP) method [11], which is well known for its good convergence property. A brief description of the method is given below. Consider the optimization problem: minimize *F*(**Φ**) with respect to **Φ**, where **Φ** is the vector of design variables and *F*(**Φ**) is the objective function, the basic recursion formula for the DFP method is:

where the step length *s*_{k}
is determined by a line search in the direction **d**
_{k}
:

The search direction **d**
_{k}
is determined iteratively by following formulas:

and

where ∇*F*(**Φ**
_{k}
) is gradient vector of the objective function, and **I** is the identity matrix. The correction matrix **A**
_{k}
is derived from information collected during the last iteration, i.e., from the change in the variable vector:

and the change in the gradient vector:

it is given by:

More details of the DFP method can be found in Ref. [11].

## 3. Design examples and results

Figure 2 shows the desired intensity spectrum of the first design example. The programmable grating-like DOE consists of 2048 micromirror elements each of width 4.5 µm for a total length of 9.2 mm. The design goal is to determine the displacements of each micromirror so that the diffraction intensity spectrum at a diffraction angle of 15° is identical to the intensity spectrum shown in Fig. 2. In our design, the number of the sampling wavelengths within the spectral range of interest, from 3600 cm^{-1} to 4200 cm^{-1}, is 256 chosen at equal intervals. The coefficients *G*_{nm}
in Eq. (5) do not change from iteration to iteration. Therefore, to minimize the processing time, they are generated and stored at the beginning of the run. In DFP search procedures, if a new parameter generated is outside the specified boundary, which is defined by the Inequality (10), it will be set to the parameter close to the nearest boundary. The calculations were performed on a 2GHz Pentium computer with 256MB memory. The design started from a randomly chosen initial point and terminated when the error defined by Eq. (7) converged or a predetermined number of iterations (300 iterations was chosen here) was reached. Under these conditions, the design required approximately 15 minutes. Figure 3 shows a part of the surface profile of the resulting DOE with the displacement of each micromirror set to the designed value. Only the first 64 of the 2048 micromirror elements are shown. Figure 4 shows the diffracted intensity spectrum generated by the programmable grating-like DOE shown in Fig. 3. In order to investigate the efficiency of the DOE, the following parameter *η* is defined:

where *I*(*u*_{p}
) is the diffraction intensity produced by the DOE for a specific wave number *u*_{p}
(typically the wave number corresponding to the peak of the target spectrum) at the given diffraction angle, and *I*
_{0}(*u*_{p}
) is the diffraction intensity for the same wave number *u*_{p}
observed at 0° diffraction angle, when the displacements of the micromirror elements are all set to 0 (a flat surface profile, for this setting, the DOE is actually a reflective mirror). The value of *η* in this case is 0.7%.

The target diffraction intensity spectrum of the second design example is shown in Fig. 5. It consists of three extremely sharp lines located at 3800, 3900, and 4000 cm^{-1}. We also use the same programmable DOE to generate the desired intensity spectrum shown in Fig. 5 at a diffraction angle of 15°. After the optimization, the displacements of the micromirrors in the DOE are set to the resulting design values, and the diffraction intensity spectrum at the specific diffraction angle is calculated. A part of the surface profile of the resulting DOE (first 64 of the 2048 micromirror elements) is shown in Fig. 6, and the calculated spectrum is shown in Fig. 7. The value of *η* obtained in this case is 13%.

It is observed that the efficiency of the DOE’s for synthesis spectra applications is relatively low as compared with the DOE’s for single wavelength applications such as beam shaping. Fortunately, for applications such as correlation spectroscopy, the accuracy of the synthetic spectra is more important for sensitive and selective sensing and analysis of chemicals.

## 4. Conclusions

In this paper, a novel efficient algorithm for the optimization design of DOE’s for synthetic spectra applications is proposed. The design examples demonstrate that the DOE’s obtained by the proposed algorithm can accurately produce the desired intensity spectra at a predetermined diffraction angle.

## References and links

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