## Abstract

We study optical spectral filter synthesis with arrays of pistonactuated micro-mirrors. We propose two algorithms for the calculation of the positions of the micro-mirrors, giving us control of both the amplitude and phase of the synthetic filter. Both algorithms for filter synthesis are explored in an analytic version and in numerical searches for the least deviations between the target and the synthesized filter. We measure the quality of the filter both in terms of the deviations and in filter transmissivity, and present results of numerical simulations for a wide selection of target filters. We find that numerical searches can sometimes yield considerable improvement in the filter synthesis compared to the analytic approximation.

©2006 Optical Society of America

## 1. Introduction

The development of micro-electromechanical systems has made possible the fabrication of segmented Diffractive Optical Elements (DOEs), which are essentially holographic diffraction gratings whose shape can be controlled to sub-wavelength accuracy. Moving the segments of a DOE modifies the way light is diffracted by the DOE, and especially the spectral transmission in a given direction when the DOE is illuminated by a broad-band light source. A general analysis of such filters is presented in [1].

Based on this principle, several adjustable filters for visible or near-infrared light have already been fabricated and demonstrated. The polychromator [2] is an array of flat micro-mirrors that makes possible the synthesis of arbitrary filters. The micro-mechanical grating is a similar device, but where several periods of a diffraction grating are patterned on top of the array elements [3]. This makes possible the synthesis of high resolution filters with fewer movable elements, by giving a better control over the center wavelength and spectral width of the filter, and eliminates redundant parts of the filter transmission. The holographic pattern on top of the array elements can also be designed to integrate focusing directly onto the device, simplifying the optical setup [4]. However, control of the amplitude only of the filter transmission was demonstrated with those devices, while the phase was let free to take any value.

Controlling both the amplitude and phase of the pixels composing holograms for the reconstruction of spatial images has long been a subject of interest. Often, only the amplitude or the phase of the hologram transmission can be controlled. To circumvent this problem, several adjacent pixels can be considered as sub-elements of a bigger pixel. There exist several methods to modulate the complex transmission of the bigger pixels by adequately calculating the transmission of the sub-elements, such as [5]. This approach was also used to control the complex transmission of the elements of liquid-crystal spatial light modulators [6] or Deformable Mirror Devices [7]. In this article, we use a similar principle: we divide each element of an array in sub-elements whose phase only can be modulated, thus gaining control over the complex transmission of each element. However, methods for image reconstruction cannot directly be applied to spectral synthesis. In particular, describing polychromatic light requires a formalism that takes into account the wavelength-dependency of the phase modulation of each sub-elements of the modulator. As a consequence, realizing complex-value spectral filter synthesis will also require different devices.

For the synthesis of spectral filters, we will see that leaving spectral phase unconstrained can lead to problems such as excessive distortions in the amplitude of the spectral transmission of the filter, and so controlling spectral phase simultaneously with amplitude is preferred. Control of both the amplitude and phase of the filter transmission was demonstrated in an experiment using Texas Instruments DMD array [8]. However, the principle of operation of the DMD is somewhat different from the devices in [2], [3] and [4], as it functions by turning elements on and off and effectively becomes an amplitude grating instead of a phase grating. As a consequence, filtering with the DMD array suffers from certain restrictions, such as symmetry of the synthetic filter transmission and power losses, while arrays of piston-actuated micro-mirrors such as [2], [3] and [4] cannot fully control spectral phase. A solution for controlling both the phase and amplitude of the reflection on the array elements, is to combine lateral displacement and continuous rotation of elements of a blazed grating array [9]. This makes possible, in theory, full control of the amplitude and phase of the filter transmission.

In this article, we choose to consider piston-actuated arrays of micro-mirrors only. Such devices offer practical advantages, such as a relative simple fabrication process, and the possibility to engineer the filter transmission or to integrate focusing with a holographic pattern on top of the array elements, as already discussed. Micro-mechanical elements of current pistonactuated arrays such as [2], [3] and [4] have a range of motion of at most a few wavelengths. Current methods to find the positions of the array elements use a phase retrieval algorithm [2] or a gradient search algorithm [10], which minimizes the deviation between the amplitude of the synthetic filter and a target filter. They do not allow phase control of the synthetic filter, because the placement of the array elements is subject to two restrictions: their idle positions must be situated at regular intervals in space, and they can only be moved slightly around their idle positions. Although the first restriction would be easy to overcome by changing the geometry of the array, the second restriction would be harder to overcome, as it reflects mechanical limitation in the micro-mirror strokes.

After presenting the mathematical background in chapter 2, we show in chapter 3 that full control of spectral amplitude and phase can be achieved with a specially made piston-actuated array where pairs of elements are combined into one element, enabling control of the reflectance amplitude and phase for each element. In other words, the device becomes simultaneously an amplitude and phase grating. As such, it can synthesize both spectral amplitude and phase, and represents a major improvement over current piston-actuated arrays, even though the range of motion can be as small as half a wavelength, just as for [2], [3] and [4]. We present a new algorithm, called array of mirror pairs, to synthesize a filter for such a device. Furthermore, additional benefits can be gained for piston-actuated arrays where the range of motion of the elements is unconstrained, and we present a second algorithm, first introduced in [1] but further optimized in this article, to synthesize a complex-valued filter in that case.

Synthesis of complex-valued filters is crucial for some applications such as pulse-shaping, which is traditionally implemented with liquid-crystal or electro-optic modulators [11]. The use of micro-mirror arrays have several advantages, as they are generally faster, offer better contrast than other light modulators, are less dependent on the wavelength and are well-suited to direct space-to-time pulse shaping [12]. However there are many applications where only the spectral amplitude is important and spectral phase is irrelevant. We will see in the last part of the article that for most filters, the two proposed algorithms yield a lower error between target filter and synthetic filter, compared with placing the array elements using previously demonstrated algorithms such as a phase retrieval algorithm [2] or a gradient search algorithm [10]. Forming quantized surface reliefs with the second algorithm also shows that the power efficiency can be significantly increased. In other words, control of spectral phase of the filter can be very beneficial even when it is not explicitly required for a particular application.

To illustrate the different methods, we have used computer simulations, which were run on a number of different target filters, and we present results in terms of mean and standard deviations obtained by averaging over the different target filters.

## 2. Background

#### 2.1. Implementation of an optical filter with an array of micro-mirrors

We consider the far-field diffraction from a surface relief consisting of an arrangement of micromirrors, as shown in Fig. 1(a). We assume the array of micro-mirrors to be illuminated by a collimated beam of light, propagating parallell with the optical axis and described by a scalar field *U*_{in}
(*v*), which makes the input of the filter. The scalar field *U*_{out}
(*ν*), describing the light that is on the optical axis, but propagating in the opposite direction, makes the output of the filter. Both *U*_{in}
(*v*) and *U*_{out}
(*v*) are complex functions of the wavenumber *v*, defined as the inverse of the wavelength *v*. The *N* micro-mirrors, that we call array elements, are initially assumed to be periodically spaced by a distance

where we have defined the reference wavelength *λ*_{c}
, typically the center wavelength of the filter, and the integer *M*, the diffraction order of the diffractive filter.

Assuming the point of observation of *U*_{out}
to be sufficiently far away, so that the Fraunhofer approximation is valid, we can write

The linear dependence of *U*_{out}
(*v*) on wavenumber reflects the fact that the angular spread of a Fraunhofer diffraction pattern is inversely proportional to the wavenumber. Disregarding this dependence and the scale factor *C*, we can describe the system as a filter of transmission

The quantity *s*_{n}
(*v*) is the complex reflection function of each array element. If all the mirrors are identical and perfectly positioned at perfectly equidistant positions, it is a constant independent of *v* and *n*. The value of the complex reflection of each array element can be modulated by displacing the array elements around their idle positions, which is the object of study of this article. The factor exp(-*j*2*πνnMλ*_{c}
) can be physically understood as an offset in the distance travelled by the light reflected by equidistant array elements. A thorough derivation of the filter transmission for such surface reliefs can be found in [13] and [1].

The array elements shown in Fig. 1(a) are flat mirrors oriented perpendicularly with the optical axis, so that the the exit aperture catches the sum of the specular reflections from each array element. The relative lateral placement of the array elements does not play a major role, as long as the light reflected on the *n*^{th}
element has travelled the distance *L*+*nMλ*_{c}
when reaching the exit aperture, where *L* is a constant. However, the aspect ratio of the array must be small enough so that no shadowing occurs, and the reflectance of each array element seen from the exit aperture must be the same. The shape of the array elements can also be different. For example, if each array element corresponds to several periods of a blazed grating, this would add a filtering effect, whose transmission must be multiplied to *H*(*v*). More details can be found in [3].

In many cases, the amplitude of *s*_{n}
(*v*) is independent of the wavenumber. In this case, we normalize *s*_{n}
(*v*) so that

where *A*_{n}
is the area covered by the *n*^{th}
array element and *A*_{tot}
the total area of the array of micro-mirrors:

With the convention described by (4), we ensure that *H*(*ν*) is a passive filter with a maximum transmission of 1.

#### 2.2. Properties of the filter transmission

### 2.2.1. Phased array

We assume that each array element can be described by a complex reflection function *s *_{n}
that does not depend on the wavenumber *v*, so that

where *r*_{n}
is the amplitude of *s*_{n}
and *φ*_{n}
the phase of the reflection. The filter transmission becomes

The filter transmission is periodic with a period 1/(*Mλ*_{c}
). Therefore, we define the signal window or spectral range of the filter, as the period of the filter transmission centered on *ν*_{c}
:

Having array elements consisting of several periods of a blazed grating would be equivalent to multiplying the filter transmission by the filtering function due to each array element. In practice, this principle can be used to attenuate the filter transmission that is repeated outside the signal window, as it was done in [3].

We define the relative bandwidth of the filter

where *v*_{c}
=1/*λ*_{c}
is the center wavenumber of the filter and Δ*ν* the bandwidth of the filter -i.e. the spectral width of the signal window, as shown in Fig. 1(b).

A discrete filter transmission, represented by a vector **H** of dimension *N*, can be calculated by computing the discrete Fourier transform of *s*_{n}
at discrete wavenumbers *v*_{m}
:

### 2.2.2. Time-delay array

There is to our knowledge no simple way of controlling the phase of light reflected on micromirrors independently of its wavenumber with MEMS-technology. But it is possible to displace the array elements around their idle positions, so that time delays are added to the reflected light, as shown in Fig. 2(a). If the array elements are displaced parallell to the optical axis, a reflection phase shift of 2*π* requires a displacement of half a wavelength (note that the phase shift is wavelength-dependent!). Such a displacement is easily achieved by means of electrostatic actuation. Under this condition, the complex reflection functions of the array elements become

where *d*_{n}
is the optical path modulation due to the displacement of the *n*^{th}
array element. The corresponding time delay is *t*_{n}
=*d*_{n}
/*c*, where c is the velocity of light. The complex reflection function of the array elements *s̃*_{n}
(*v*) now depends on the wavenumber *v*. We consider the discrete filter transmission

at the wavenumbers *v*_{m}
defined by (10), and the corresponding continuous filter transmission

defined on the interval given by (8). An example is shown in Fig. 2(b).

#### 2.3. Synthesis of optical filters

### 2.3.1. Inverse problem

We have described the spectral filtering resulting from the diffraction by an array of micromirrors. The inverse problem is to find the positions of the array elements, so that the associated filter transmission *H*(*v*) is as close as possible to a scaled version of a target *T*(*v*).

### 2.3.2. Error

An array of *N* elements can only synthesize filter transmissions sampled at *N* or fewer regularly spaced wavenumbers defined by (10) in the signal window defined by (8). Therefore, we assume the target filter to be defined as *N* samples *T*_{m}
. The relation between the sampled synthetic and target filter transmissions, here written as vectors of dimension *N*, is then

where *γ* is a scaling factor and **e** an error vector that should be minimized. We define the normalized mean square (NMS) error *ξ* as the normalized variance of the error vector **e**, for the value of *γ* that minimizes this variance. This gives

which is obtained for *γ*=<**H**, **T**>/<**H**, **H**>, and where <, > denotes the inner product defined in appendix .1.

The NMS error is null for *γ*
**H**=**T** and has a maximum value of 1 if **H** and **T** are orthogonal. The NMS error as defined by (16) is the inverse of the signal-to-noise ratio (SNR), where <**T**, **T**> is the desired signal intensity and <**e**, **e**> the noise intensity.

### 2.3.3. Power efficiency

Another important characteristic of the filter transmission is the power efficiency defined by

With this definition, the maximal value of the power efficiency is *N*, which corresponds to a filter transmission of 1 at every sampling point.

For phased arrays, the filter transmission **H** is equal to the discrete Fourier transform of the complex reflection function s, so that it is possible to relate the power efficiency to the modulus of s, using Parseval's theorem. This gives

In our analysis, we have implicitly assumed that the entrance aperture illuminates the micromirror array with a constant intensity and that no light falls outside the array. In practice, this would be very difficult to achieve. An easy implementation is to use a coherent aperture such as a single-mode fiber, with a very small size. This would ensure a constant illumination over the micro-mirror array, but most of light would fall outside the array, resulting in a much lower power efficiency than given by (18). The same considerations can be made for the exit aperture. Other alternatives than narrow coherent apertures are possible, but the effect of the finite size of the apertures must then be taken into account in the analysis, as the filter transmission might be modified [14].

#### 3. Synthesis of arbitrary filters with adjustable arrays of micro-mirrors

### 3.1. Complex filter synthesis with an “array of mirror pairs”

### 3.1.1. Principle

The Grating Light Valve is an array of mirror elements where the amplitude of the reflection on each element can be controlled by adjusting the relative positions of several reflecting ribbons of which each element consist [15]. We propose to exploit this principle of amplitude control in a device that we call an array of mirror pairs, shown in Fig. 3. It is somewhat like a Grating Light Valve where each reflecting array element consists of only two ribbons or micro-mirrors. We call the micro-mirrors sub-elements. Phase control can be achieved bymoving each element as a whole, although the capability was not implemented in [15]. It is important to notice that for the device to work, the sub-elements must be placed such that light reflected on the subelements belonging to the same element has travelled the same distance when reaching the exit aperture (this is when the sub-elements are at their idle positions). As a consequence, devices such as [3], [7] and [15] can not be used without modifications. To implement an array of mirror pairs, it might be easier to place the sub-elements side by side, thus forming a two-dimensional array. Such an implementation is described in [14].

Assuming, in a first approximation, that the device can be described by a phased array, the complex reflection function ${s}_{n}^{\left(\mathit{\text{AMP}}\right)}$ of each array element can be written as the sum of the contributions of the two sub-elements, each associated with a phase shift Φ and Ψ. This gives

with

The factor 1/*N* in the expression of **r**
^{(AMP)} is to satisfy (4) when all the mirror pairs are in phase (when Φ_{n}=Ψ_{n} for every *n*).

In order to be able to modulate both the amplitude and phase of an array element, the micromirrors must have stroke, when moved parallel with the optical axis, of

which is typical of current MEMS devices, for wavelengths in the visible or near-infrared region.

The resulting device is then effectively an amplitude and phase grating. Emulating the phase of the array elements can be done using an algorithm that we call phase emulation, first used with the polychromator [2], and further analyzed in [1]. In the following we present a new algorithm called amplitude and phase emulation, which computes the positions of the elements of an array of mirror pairs, to emulate not just the phase of each element, but also the amplitude.

### 3.1.2. Calculation of the positions of the micro-mirrors with an analytical method

**Amplitude and phase emulation** As we have seen, moving the sub-elements in space adds time-delays to the reflected light. For a time-delay array, the complex reflection function of each array element is given by

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.4em}{0ex}}=\frac{1}{N}\mathrm{cos}\left(2\pi v\frac{{d}_{n}^{\left(1\right)}-{d}_{n}^{\left(2\right)}}{2}\right)\mathrm{exp}\left(-j2\pi v\frac{{d}_{n}^{\left(1\right)}+{d}_{n}^{\left(2\right)}}{2}\right).$$

To find the positions **d**
^{(1)} and **d**
^{(2)} of the sub-elements, so that a given filter is synthesized, we first calculate a reference complex reflection functions **s**
^{(ref)}=**r**
^{(ref)} exp (*jφ*
^{(ref)}) using (6) and (11). Then, we retrieve the relevant phase shifts induced by the two sub-elements Φ and Ψ by setting **r**
^{(AMP)}=**r**
^{(ref)} and *φ*
^{(AMP)}=*φ*
^{(ref)} in (20). Finally, the optical path modulations obtained by displacing the two sub-elements **d**
^{(1)} and **d**
^{(2)} are obtained with

where *v*_{c}
is the center wavenumber of the filter. We call this analytical method amplitude and phase emulation, as both the amplitude and phase of the complex reflection function are emulated by moving the two sub-elements.

**Analytical expression of the NMS error** Taking a discrete Fourier transform of **s**
^{(ref)} gives exactly the target filter transmission, within a scaling constant. But this is not the case of **s̃**
^{(AMP)}. Indeed, at wavenumbers other than *v*_{c}
, **r̃**
^{(AMP)}(*v*) is not equal to **r**
^{(ref)}, and we call the resulting NMS error in the synthetic filter transmission the amplitude emulation error, which we denote *W*^{r}
. Similar considerations can be made about the phase shift *φ˜*
^{(AMP)}(*ν*), which is only equal to *φ*
^{(ref)} at the wavenumber *v*_{c}
. We call the corresponding NMS error the phase emulation error, which we denote *W*^{φ}
. The total NMS error in the synthetic filter, as defined by (16), becomes

which relates to the relative bandwidth of the filter Δ*ν*/*ν*_{c}
and two characteristics of the shape of the target filter, *T**
_{φ}
and *T**
_{r}
. This relation is derived in appendix .2.

**Estimation of W^{φ}
and W^{r}
from simulation results** We have simulated filter synthesis with an array of mirror pairs, by computing the value of the filter transmission when the complex reflection of the array elements is given by (22), for a set of test filters that is shown in Fig. 4. The NMS error

*ξ˜*

^{(AMP)}obtained for the set of test filters is shown in Fig. 5(a). The validity of (24) was verified by comparing the predicted value of the NMS error, given by (24), with the simulated errors. The result is shown in Fig. 4(b). Generally, the NMS error is very high on large spectral ranges, where the analytical expression given by (24) is not valid. We also noticed that, for the test filters we used,

*T**

_{r}and

*T**

_{φ}were independent and that the NMS error resulting from emulating the amplitude of s was much higher than the phase emulation error, so that in practice we had

### 3.1.3. Improvement of the error with a numerical search

We have run a numerical search algorithm that, starting form the values of **d**
^{(1)} and **d**
^{(2)} found with (23), minimizes the NMS error *ξ˜*
^{(AMP)} by modifying **d**
^{(1)} and **d**
^{(2)}. We have used a multidimensional unconstrained nonlinear minimization of the error (Nelder-Mead), which is the standard global optimization method of Matlab. This numerical search improves substantially the amplitude and phase emulation error, but requires much longer computing time. The error vector **c**
^{(AMP)} resulting from the convergence of the numerical search is given by

The value of the improvement *ξ˜*
^{(AMP,NS)}/ξ˜^{(AMP)}, for the test filters, is plotted in Fig. 6. On average, the NMS error is divided by a factor 5 to 10 and the improvement is maximal for medium bandwidths. A reason might be that the numerical search converges in some local minima for filters on large bandwidths, whose errors are initially very high.

We have fitted the convergence NMS error of the numerical search *C*
^{(AMP)} with a function 𝒜^{(AMP)}+𝓑^{(AMP)} (Δ*ν*/*ν*_{c}
)^{2}, for each test filter. The quality of the fit is shown in Fig. 7(a).

The values of 𝒜^{(AMP)} and 𝓑^{(AMP)} for each filter are shown in Fig. 7(b). We see that 𝒜^{(AMP)} is almost null and that 𝓑^{(AMP)} tends to increase with the filter characteristic *T**
_{φ}
+*T**
_{r}
, even though the correlation is poor. This gives a total NMS error

### 3.1.4. Power efficiency

Each array element of the array of mirror pairs can attenuate the light and the maximum value of *r*_{n}
is 1/*N*. This gives, using Parseval's theorem

Simulation results run on the set of test filters for time-delay arrays are shown in Fig. 8. They show that (28) is also valid for time-delay arrays, and that there is virtually no difference in power efficiency between the analytical algorithm and the numerical search.

There is an intrinsic limitation in the power efficiency of a filter implemented with an array of mirror pairs: the power efficiency is 1 in the best case, while we have seen that the theoretical maximal value of the power efficiency is *N*. For many filters, the power efficiency might be much lower than 1. This can be improved by engineering the shape of the spectral filter, as this is done for holograms in the space domain [16]. However this is achieved at the price of the addition of an error in the shape of the synthetic filter. We have not implemented such a solution.

### 3.2. Complex filter synthesis with a “quantized surface relief”

### 3.2.1. Principle

In this section, we consider a device similar to an array of mirror pairs, with two micro-mirrors per array element, but let the micro-mirrors be moved from an array element to the other (see Fig. 9). The amplitudes of each array element, described by reflection coefficients **r**
^{(QSR)}, are then the sum of the number of mirrors belonging to each array element. The algorithm for the calculation of the positions of the micro-mirrors presented in this part were first developed in [1], but here we have further optimized the positions of the micro-mirrors using optimization methods.

The number of micro-mirrors *p*_{n}
than must be moved to the the *n*^{th}
array element is given by

where **r**
^{(ref)} is calculated with (6) and (11). There are *P* micro-mirrors to be distributed between the *N* array elements (in the simulation we used *P*=2*N*). The function round gives the closest integer of its argument and *ρ* is found so that

As the micro-mirrors can now be moved all the way from the first to the last array element, they must have a stroke

However, in practice, such a stroke will be required only for an extremely small fraction of possible filters, and the typical stroke of the mirrors will be

Because of the possibility to obtain surface reliefs with high aspect ratio, there is with this method a risk that some micro-mirrors might shadow others. Reducing the aspect ratio with wider micro-mirrors (and thus a larger device) or inserting gaps in between the micro-mirrors would reduce the effect of shadowing, but the last solution would result in an additional power loss.

### 3.2.2. Power efficiency

If we have enough micro-mirrors so that we can approximate **r**
^{(QSR)}≈**r**
^{(ref)}, the power efficiency of the device is given by

where *T**
_{N}
is the so-called filter complexity characteristic introduced in [1], given by

Simulations results plotted in Fig. 10(a) shows that the quantized surface relief yields a much better power efficiency than the array of mirror pairs, as the simulated mean power efficiency was close to 4. The maximal achievable power efficiency with a quantized surface relief is *N*, which is obtained in the case of an array where only a single array element as a non-null value (in this case, its complex reflection has an amplitude of 1). This trivial case corresponds to looking at the specular reflection from a mirror. Figure 10(b) shows that (33) is a good approximation, even for time-delay arrays.

### 3.2.3. Quantization error

We consider a phased array where the amplitude of the complex reflection function is given by (29) and the phase shift *φ*
^{(ref)} induced by each array element is calculated with (6) and (11). The complex reflection of the array element is then given by

The deviation between **r**
^{(QSR)} and **r**
^{(ref)} results in a quantization error in the amplitude of the complex reflection function of the array elements. This is described by the error vector **q**, which relates to the synthetic and target filter transmission with

It is possible to relate the quantization NMS error to the filter characteristic *T**
_{N}
and the total number of mirrors *P* with

The derivation of this relation is similar to appendix .2. The simulated results obtained for the test filters are shown in Fig. 11(a) and 11(b).

### 3.2.4. Time-delay array with phase emulation

As for the array of mirror pairs, reflection phase shifts can be emulated by moving the array elements around their idle positions. The displacements of the array elements are calculated so that they result in optical path modulations

The complex reflections of the time-delay array elements become

The deviation between -2*πν*
**d**
^{(QSR)} and ${\phi}_{n}^{\left(\mathit{\text{ref}}\right)}$, when *ν*≠*ν*_{c}
, generates a phase emulation error, described by the vector **w**
^{φ}
. This error vector adds to the quantization error vector, so that the relationship between synthetic and target filters becomes

We assumed that the mean values of the error vectors are negligible and that they are uncorrelated. Furthermore we assume that **q** is small enough so that we can approximate **r**
^{(QSR)}≈**r**
^{(ref)} for the calculation of *W*^{φ}
. Under these conditions the total NMS error becomes

The total NMS error for the quantized surface relief, computed with simulations of the test filters, is plotted in Fig. 12(a), and a comparison of (41) with simulated values is shown in Fig. 12(b).

### 3.2.5. Improvement of the error with a numerical search

There are two strategies for reducing the error with numerical search algorithms. If we move all the micro-mirrors belonging to an array element together, we ensure that the power efficiency remains unchanged. It is then described by (33). The NMS error becomes a convergence error of the numerical search *C*
^{(QSR,NS)}. We can further improve the error to *C*
^{(QSR,NS2)} by letting the numerical search adjusting the positions of the *P* mirrors independently. However, this will degrade the power efficiency as the light coming from each array element will not interfere reconstructively. The value of the improvement associated with the numerical searches are shown in Fig. 13(a). For the first method, we see that on average the NMS error is devided by a factor 1.6 at Δ*ν*/*ν*_{c}
=1 and there is virtually no improvement on narrow bandwidths. In figure 13(b), we see that there is almost no decrease in power efficiency. With the second method the NMS error is on average devided by a factor 3.7 at Δ*ν*/*ν*_{c}
=1 and is decreasing to a factor 1.8 at Δ*ν*/*ν*_{c}
=1/15. The power efficiency is divided by a factor of nearly 2 on large bandwidths.

We have fitted a function 𝒜^{(QSR,NS)}+𝓑^{(QSR,NS)} (Δ*ν*/*ν*_{c}
)^{2} to the error *ξ˜*
^{(QSR),NS} and *ξ˜*
^{(QSR),NS2} for each test filter (see Fig. 14). If the mirrors of each array element are moved together, we see in Fig. 15(a), that 𝒜^{(QSR,NS)}≈*Q* and that only the phase emulation error is reduced, as shown in Fig. 15(b). Letting the numerical search move the *P* mirrors independently results in a reduction of both the quantization error and the phase emulation error.

#### 4. Advantage for “amplitude” filter synthesis

### 4.1. Synthesis of amplitude filters

For some applications, such as spectroscopy, only the amplitude of the synthetic filter is relevant, not its phase. The error of interest is then calculated between the amplitudes of the synthetic and target filter transmissions, and we defined the amplitude NMS error as

with

Any algorithm capable of synthesizing complex filters is also capable of synthesizing amplitude filters. In this case the target filter **T**
^{|} is real and the synthetic filter transmission **H** would also be real if there were no error. We observed that, when synthesizing a complex-value filter with zero phase, the error computed on the filter amplitude was virtually similar to the error computed with the complex value of the filters. In other words, *ξ*≈*ζ*.

Letting the phase of a spectral filter free to take any values introduces distortions in the filter amplitude in between the sampling points, as described in [3]. Setting the phase of the filter to zero at the sampling points eliminates this problem. In this article, to take into account this phenomenon, we have computed the value of the NMS error at 10 times the number of sampling points *N*, and interpolated the target filters with a low-pass filter. In practice, using over-sampled versions of the target and synthetic filters does not change the value of the error and power-efficiency, if the complex-value of the filter is controlled. If the phase of the filter is let free to take any value, the new over-sampled error will take into account the oscillations occurring in between the sampling points.

### 4.2. An algortihm for amplitude filter synthesis: the Gradient Search Algorithm

It is possible to synthesize the amplitude of arbitrary filters with an array of micro-mirrors, where each array element consist of a single micro-mirror that can be displaced slightly around its idle position, as it was shown in Fig. 2(a). Zhou et al. have proposed the use of a Gradient Search Algorithm (GSA) that calculates the positions of the array elements, minimizing the NMS error between the intensities of a target and synthetic filter [10]. They have calculated an analytical expression of the partial derivatives of the NMS error, which we have modified in order to minimize the NMS error between the amplitudes of the filter transmissions. Knowing the partial derivatives makes the algorithm converge must faster than the numerical search we have used for complex filter synthesis.

In this case, each array element consists of a single micro-mirror and the maximal stroke needed in order to operate the device, when the mirrors are moved parallel to the optical axis, is

The GSA converges towards a solution to the time-delay array problem, with a convergence error defined by

For each test filter, we have found fit coefficients 𝒜^{(GSA)} and 𝓑^{(GSA)} so that

The NMS error obtained with the GSA is plotted in Fig. 16(a) and the result of the fit can be seen in Fig. 16(b).

The device described in this section consists of array elements where the amplitude of the complex reflection function is the same for every array element. We normalize to ${r}_{n}^{\left(\mathit{\text{GSA}}\right)}$=1/*N* so that (4) is valid. Parseval Theorem then gives

Even though the approximation can become rough at small orders, simulations on the test filters show that (47) remains a good approximation for time-delay arrays, as shown in Fig. 17.

### 4.3. Comparison of the algorithms

In Figures 18(a) and 18(b) are plotted the mean amplitude NMS error obtained on the set of test filters with different methods. We see that the mean NMS error obtained using the GSA is substantially higher than the NMS error obtained with algorithms synthesizing complex-value filters (to make the comparison, the phase of the target test filters was set to zero). The reason is that algorithms for complex filter synthesis enable full control of the complex reflection function of each array element. With the GSA, only the phase of the complex reflection function can be modulated, so that the algorithm converges towards a solution that is close, but cannot be an exact replica of the target filter.

On the other hand, the GSA can be used to calculate the positions of the elements of a device consisting of *N* independently movable elements, instead of 2*N* for the array of mirror pairs and *P* for the quantized surface relief for the same filter resolution. Intuitively, we would expect that increasing the number of array elements in the array would in general make it possible to reduce the filter synthesis NMS error. But increasing the number of array elements for the GSA would not improve substantially the error, as its main effect would be to over-sample the filter. A better strategy is to increase the number of sub-elements at each of the original equidistant positions, as we have done with the two algorithms for complex filter synthesis, and which results in a significant improvement of the error.

Analytical expressions of the complex NMS error and power efficiency obtained using analytical methods for the calculation of the positions of the micro-mirrors are shown in table 1. Figures 19(a) and 19(b) show plots of the mean complex NMS error, while Fig. 20(a) and 20(b) show the mean power efficiency over the test filters, for both analytical and numerical methods.

#### 5. Conclusion

To summarize, we have presented two algorithms that make possible the synthesis of complexvalue filters with arrays of piston-actuated micro-mirrors. To allow phase control, we have done several things. First, we have doubled the number of independently movable micro-mirrors to get the adequate number of degrees of freedom. In addition, with the first algorithm, we place the mirrors by pairs, instead of periodically in space. With the second algorithm, we allow the micro-mirrors to be displaced without restriction in the mirror stroke.

The first method for complex filter synthesis, that we called “array of mirror pairs”, also gives an advantage in terms of error for the synthesis of filters, whose amplitude only is relevant, compared with existing piston-actuated micro-mirror arrays. We believe the method to be a simple and low-cost solution for applications which require amplitude and phase control of the filter, or a low error in the synthetic filter transmission, but when at the same time a relatively low through-put of the filter can be tolerated. The second method, called “quantized surface relief”, yields even better error and shows theoretically that the power efficiency of optical filters implemented with arrays of micro-mirrors can be significantly improved.

The algorithms that calculate the positions of the micro-mirrors make the approximation that the shifts in the phase of the light reflected on each micro-mirror are not wavelength-dependent. These algorithms converge towards synthetic filters which differ from the corresponding target filters by an error, which can be substantially improved by running numerical search algorithms that further optimize the positions of the micro-mirrors. We also show that it is possible to predict the error and power efficiency of the synthetic filters from characteristics of the target filters, at the exception of the improvements resulting from the numerical searches, which were rather inconsistent.

Although existing devices, such as the micromechanical grating, cannot be used for complex filter synthesis in their current forms, only a slight increase in the mechanical complexity would be required to enable practical operation as an array of mirror pairs. However, implementing a quantized surface relief would need some mirror strokes that are out of reach at the current state of micro-technologies.

## Appendix

#### .1. Notations

We define the inner product of two complex-value vectors **a** and **b** as the real number

where Re and Im are the real and imaginary part of a complex-value variable.

#### .2. Phase and amplitude emulation errors W^{φ} and W^{r} for the array of mirror pairs

We consider the error in the synthetic filter transmission of an array of mirror pairs, given by the error vector

Defining the filter transmission **H**
^{(ref)} associated with the phased array of complex reflection function **s**
^{(ref)}, we have

In order to write (50), we assumed the mean of the error vector to be negligible, so that $\gamma =\frac{\overline{\mathbf{T}}}{\overline{{\stackrel{~}{\mathbf{H}}}^{\left(\mathit{AMP}\right)}}}=\frac{\overline{\mathbf{T}}}{\overline{{\mathbf{H}}^{\left(\mathit{ref}\right)}}}$ , where (̅) denotes the mean value of a vector.

After some calculations, we obtain the expression of the error vector

$$\phantom{\rule{.9em}{0ex}}\frac{{w}_{m}^{\phi}}{\gamma}\{\begin{array}{c}-j\frac{{v}_{m}-{v}_{c}}{{v}_{c}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\\ \phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\sum _{n=0}^{N-1}\left[{r}_{n}^{\left(\mathit{ref}\right)}\mathrm{exp}\left(j{\phi}_{n}^{\left(\mathit{ref}\right)}\right){\phi}_{n}^{\left(\mathit{ref}\right)}\mathrm{exp}\left(-j2\pi n\frac{m-\frac{N}{2}}{N}\right)\right]\end{array}$$

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.4em}{0ex}}\frac{{w}_{m}^{r}}{\gamma}\{\begin{array}{c}\phantom{\rule{.9em}{0ex}}-\frac{{v}_{m}-{v}_{c}}{N{v}_{c}}\sum _{n=0}^{N-1}[2\pi {v}_{c}\frac{{d}_{n}^{\left(1\right)}-{d}_{n}^{\left(2\right)}}{2}\mathrm{sin}\left(2\pi {v}_{c}\frac{{d}_{n}^{\left(1\right)}-{d}_{n}^{\left(2\right)}}{2}\right)\\ \mathrm{exp}\left(j{\phi}_{n}^{\left(\mathit{ref}\right)}\right)\mathrm{exp}\left(-j2\pi n\frac{m-\frac{N}{2}}{N}\right)].\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\end{array}$$

The first sum is the phase emulation error vector **w**^{φ}
. We call the second sum amplitude emulation error vector **w**^{r}
. The total error vector becomes

In addition, we assume <**T**, **T**>≈*γ*
^{2} <**H**, **H**>, which is true for ${w}_{m}^{\phi}$
+${w}_{m}^{r}$
≪*T*_{m}
, and <**w**^{φ}
+**w**^{r}
, **w**^{φ}
+**w**^{r}
>=<**w**^{φ}
, **w**^{φ}
>+<**w**^{r}
, **w**^{r}
>, which is true if **w**^{φ}
and **w**^{r}
are uncorrelated. With those assumptions, the NMS error becomes

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\approx {W}^{\phi}+{W}^{r}\approx \left({T}_{\phi}^{\u066d}+{T}_{r}^{\u066d}\right){\left(\frac{\Delta v}{{v}_{c}}\right)}^{2},\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}$$

which depends on the the filter characteristics

and

where 𝒟𝓕𝒯 represents the discrete Fourier transform.

## Acknowledgments

This work was partially funded by the Norwegian Research Council under the project 133675/I30 “Micro-opto-electromechanical and Micro-electromechanical Systems”. The first author would like to thank Professor Solgaard for welcoming him in his group at Stanford University as a visiting scientist.

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