Bahareh Haji-saeed,1,* William D. Goodhue,2 Charles L. Woods,1
John Kierstead,3 and Jed Khoury1
1Air Force Research Laboratory, Sensors Directorate, Hanscom Air Force Base, Massachusetts 01731, USA2Photonics Center, Physics Department, University of Massachusetts Lowell, Lowell, Massachusetts 01854, USA
3Solid State Scientific Corporation, Hollis, New Hampshire 03049, USA
The optical limiter is a device whose nonlinear in-put–output transfer function saturates as the inci-dent light increases [1]. Previously, we introduceda new technique, dynamic range compression decon-volution using two-beam coupling [2], and demon-strated that its input–output transfer functionsaturates as a function of light intensity. Two formsof two-beam coupling deconvolution were introduced:spectrally invariant [2] and spectrally variant [3].For spectrally invariant two-beam coupling, it was
necessary to operate the system at very high beamratios in order to perform dynamic range compres-sion deconvolution; working in this regime leads tovery low grating efficiency. To overcome the high
beam ratio problem the spectrally variant two-beamcoupling was introduced. Spectrally variant deconvo-lution performed better image restoration thanspectrally invariant deconvolution; however, thedrawback in spectrally variant deconvolution is thatthe response time at very high frequencies becomesextremely large (slow processing speed), since the re-sponse time of photorefractive crystals is inverselyproportional to the total intensity of light incidenton the crystal [4–6].
Another limitation of both spectrally variant andspectrally invariant two-beam coupling deconvolu-tion is the need for an input spatial light modulator(SLM). This may cause some problems, in particular,for remote sensing applications, in which the light isincoherent. The interface between the SLM and theCCD camera imposes other constraints in systemsfor correcting supersonic turbulence. One way toovercome this limitation is to use an all-optically
addressed SLM in which the pixels are addressedindependently, in parallel. One can also design anall-optically addressed SLM with dynamic rangecompression capability. We have already reportedan all-optically addressed microelectromechanicalsystem (MEMS) device [7,8]. This device exhibitsphase saturation with increasing light intensity. Inthis paper, we introduce a nonlinear optical-limiterMEMS (NOLMEMS) [9] whose input–output trans-fer function saturates with light intensity. The dy-namic range compression saturation nonlinearitywas designed using the membrane saturationdescribed in Refs. [7,8]. In contrast to Horner’s work,in which the deformable MEMS device was used asan SLM within an optical correlator system, and thequadratic phase factor within the MEMS deforma-tion was assumed to be a phase error factor [10],this device takes advantage of the phase error factoras an essential requirement for designing aNOLMEMS.NOLMEMS dynamic range compression has sev-
eral advantages over two-beam coupling deconvolu-tion: (1) The membrane reflectivity is ∼100%; thisovercomes the photorefractive crystal’s low gratingefficiency, which is caused by the high beam intensi-ties necessary to achieve dynamic range compres-sion. (2) The processing time is in microseconds(dictated by the membrane response time), comparedto the millisecond scale associated with two-beamcoupling (dictated by the photorefractive material’sresponse time). (3) There is no response time depen-dency on spatial frequency intensities as in thespectrally variant deconvolution. (4) The saturationlimit is approached faster than in two-beamcoupling, which is more effective in extracting theamplitude information for better image recovery.(5) NOLMEMS works with both coherent and inco-herent light. (6) A NOLMEMS device with opticalamplification capability, integrated with photo-detectors, responds to broadband ultralow lightintensities.This new design (NOLMEMS) enhances both the
high-frequency components of the image, restoringedge features, and the signal-to-noise ratio whenthe noise is stronger than the signal, reducingadditive noise. We have already reported electricalcontrol of these devices [7,8]. A specific combinationof AC and DC electronic bias that stabilizes theMEMS optical limiter controls the NOLMEMS de-vice. The deconvolution orders for this device werealso analyzed using a nonlinear transform method[11]. Computer simulations of image deconvolutionusing the NOLMEMS device input–output transferfunction and the deconvolution orders were alsoperformed.
2. Background
To understand the nonlinear dynamic range com-pression deconvolution approach introduced in thispaper, we present background related to processingphase-only information. Images have spectra, just
like signals. Low-frequency components occur whensignal strength changes slowly, while for images, low-frequency components occur when there is littlechange in gray levels. High-frequency componentsin the signal occur when there are rapid changesin signal strength; for images, high-frequency compo-nents are created by sharp edges. The spectrum of animage has two components, the magnitude spectrumand the phase spectrum. Because the spectrum is ob-tained from a two-dimensional input, the spectrumhas frequency elements in two directions, x (rows)and y (columns). If the combined magnitude andphase spectra are inverse Fourier transformed, theoriginal image appears. If the magnitude spectrumis inverse Fourier transformed, the result showsnothing; however, if the inverse Fourier transformis applied to the phase information with the magni-tude set to unity, an edge-enhanced version of theoriginal image is reconstructed [12]. Therefore, oneconcludes that the phase information of an imageis the more important information in reconstructingthe image. The gray scale of the image can be recov-ered by cascading the phase image with a statisticallowpass filter that has a similar spectral distributionto that of the original image. The quality of thisrecovered image is almost as good as that of the ori-ginal. This method was introduced in 1990 by Hornerand Flavin [13]. Figure 1 illustrates the above discus-sion regarding image recovery using only the phaseinformation of the original input.
Figure 1A is the original image, Fig. 1B is the ori-ginal image with no phase information, and Fig. 1C isthe original image with no amplitude information.Figure 1B shows that if the phase information is ex-tracted from the original, the recovered image showsno information from the original, while Fig. 1C de-monstrates the recovered image from the phase-onlyinformation, which is the edge-enhanced version ofthe original input. Applying a generic form of low-pass filter to the image in Fig. 1C recovers a gray-level image with a quality comparable to the original.The filtered image is shown in Fig. 1D. The genericlowpass filter used here was of the following form:
where u and v are the frequency coordinates, M ×Npixels is the array size, andD0 is the cutoff frequency.This filter will be used for gray-level image recoveryin this paper.
3. MEMS Optical Limiter
Figure 2 shows the proposed architecture for a singlepixel of a MEMS optical limiter. In the proposedNOLMEMS, a plane wave is incident on the mem-brane mirror of a back-illuminated device and, asthe membrane deflects, its parabolic shape focusesthe light to its diffraction limit. The electronic biasacross the device can be set such that the membrane
mirror, when it is fully deflected, can focus the beamon the pinhole. Appropriate pinhole and microlensarrays will collimate each of the saturated MEMSoutput beams, as shown in Fig. 2. If the device is ac-curately fabricated, uniform illumination on thephotoconductive layer will produce a plane-wave out-put accompanied by weak off-axis higher-order dif-fraction beams. For an individual MEMS elementwith very little light incident on the photoconductivelayer, the reflected beam from the MEMS mirrorwill be very small, while very strong illuminationof the photoconductive layer produces an intensereflected beam.
4. Background Theory
The input–output nonlinear transfer function of theNOLMEMS device, which shows the relationship be-tween the light transmitted through the pinhole tothe light intensity incident on the back side of the
device, was derived by Haji-saeed [14]. It was foundthat the device has a saturation limit given by
GðyÞ ∝�
AyAyþ B
�2; ð2Þ
where the parameters are defined as
A ¼ 8λTs2; ð2aÞ
B ¼ r61V2L2ω2π3ε30; ð2bÞ
y ¼ ðaþ bðΔnþ nÞ2Þ; ð2cÞ
a ¼ L2ω2π2r41ε20; ð2dÞ
Fig. 1. A, original image; B, image recovery from the amplitude-only information; C, image recovery from the phase-only information;D, gray-level recovery via application of a lowpass filter to C.
In the above equations, λ is the back-illuminationwavelength, ε0 is the permittivity of free space, 2r1is the diameter of a pixel, s is the depth of the well,T is the membrane tension, V is the total voltageapplied across the MEMS, q is the electronic charge,μn is the carrier mobility, L is substrate thickness, Adis the area of each pixel, and ω is the angularfrequency. n andΔn are the substrate carrier concen-tration and the photogenerated carrier concentra-tion, respectively.Δn can be described as a function of light intensity,
I0, incident on the back of the device:
Δn ¼ ατnI0hνAd
; ð3Þ
where α is the absorption coefficient, τn is the carrierlifetime, h is Planck’s constant, v is the lightfrequency, and Ad is the illuminated area.The optical-limiter nonlinear transfer function,
GðyÞ, as a function of back-illumination intensity,I0, is shown in Fig. 3. The plot shows the device sa-turation (GaAs substrate) as a function of light
intensity. Note that, as the beam intensity increases,the total reflected light intensity saturates to unity.
In this plot the parameters are as follows:
a. For calculating the Ad parameter, λ ¼ 980nm,s ¼ 5 μm, and T was calculated using
hd ¼ ε0r21V2
32Ts2:
All the parameters for this equation are known fromthe experimental results. The maximum deflectionhd for this particular device was almost 0.4 for50V applied bias with the pixel radius r1 ¼ 1mm,and ε0 is 8:85 × 10−12 F=m.
b. For calculating the B parameter, L ¼ 250 μmand ω ¼ 10kHz.
c. For calculating y, the a and b first need to befound. All parameters for a are known and for b,q ¼ 1:6 × 10−19 C, μn ¼ 8500 cm2=ðV-sÞ, and Ad ¼πr12.
TheΔn can be calculated using Eq. (3), where α forGaAs substrate is 2 cm−1, τn ¼ 10−8 s, the Planck’sconstant h is 6:63 × 10−34 J, v ¼ c=λ, wherec ¼ 3 × 108 m=s, and I0 is from 0 to 10mW. The nparameter was chosen to be an intrinsic carrier con-centration for GaAs by a value of 2 × 106 cm−3.
5. NOLMEMS Joint Fourier Processor
Figure 4 shows a schematic diagram of a NOLMEMSjoint Fourier processor. Reference (R) and signal (S)information are joint Fourier transformed by a lens,L, into an all-optically addressed MEMS deformablemirror located at one facet of a cubic beam splitter.Reference and signal correspond to a very remote ob-ject (a point source) and the target of interest passingthrough the atmosphere, respectively. Therefore, thejoint input image consists of the impulse response ofthe atmosphere (the blur) and the blurred target in-put image. The joint information before addressingthe device is high-pass filtered for emphasizing
higher frequencies and avoiding very high beam in-tensity requirements to achieve dynamic range com-pression. Otherwise, the high spatial frequencyintensities will be degraded due to the power eightdependency within the saturation function inEq. (2). This technique makes the joint spectra illu-mination more uniform. The high-pass filter infor-mation is fed to the SLM located right before theNOLMEMS device. The deformable mirror functionsas an array of parabolic mirrors. The focal point ofeach of these parabolic mirrors is dependent onthe joint spectra back-illumination intensity. The sa-turation focal plane is located within distance f s fromthe deformable mirror.A plane wave propagating counter to the direction
of the joint Fourier spectrum of the input images, isincident to the front side of the deformable mirror.The plane wave is reflected out of the parabolic mi-cromirror to a focused array of wavelets. Dependingon the joint spectra back-illumination intensity, eachof the wavelets is focused on a different focal plane.The minimum focal length of each of the microde-formable mirrors is always greater than or equalto f s.A planoconcave lens is located at the other facet of
the beam splitter, imaging the saturation focal planeinto an array of pinholes. The pinhole array isaligned to the center of the converging wavelets.Only a portion of the converging wavelets is trans-
mitted through the pinhole array. An array of wave-let sources is generated after transmission withinthe pinhole array. The array of point sources fromthe pinhole array is collimated to a plane wave usinga lenslet array. The superposed wavefront of plane-wave arrays from the micromirror represents a dy-namic range compressed version of the joint spectraof the reference and the signal information. Thesuperposed plane wave consists of a collimated wave-let array, which is spatially low-pass filtered by usingan SLM. After spatial filtering, the output is Fouriertransformed by a lens, L, into a CCD camera, therebyproducing the required joint Fourier transformsignal processing functionality.
6. Device Theory
In the joint Fourier processor, Δn corresponds to thenumber of carriers generated by the light intensitythat has the joint spectra information and is equal to
Δnðνx; νyÞ ¼ατnI0jRþ Sj2hνADðλf Þ2
; ð4Þ
where I0 is the total beam intensity transmitted withthe reference and the signal information, λ is thewavelength, and f is the lens focal length. Incorpor-ating the joint spectra into Eq. (2) and expanding thenonlinear transfer function to several orders gives us
Fig. 4. (Color online) Schematic diagram of a NOLMEMS joint Fourier processor.
where y0 is the separation factor between the refer-ence and signal information, ΦR and ΦS are thereference and signal phases, respectively, andHkðvx; vyÞ is the deconvolution order given by
Hkðvx; vyÞ ¼ 1 − εkikþ1
��2mð0ÞJA×
1 − f 2 þffiffiffiffiffif 1
p�
k�Bm2ð0Þ
Ab
�1ffiffiffiffiffif 1
p�−B2m4ð0Þ4ðAbÞ2
�f 2 þ f 3f 1
ffiffiffiffiffif 1
p�
−kB2m4ð0Þ4ðAbÞ2
�2 − f 2
ð1 − f 2Þf 1 þ f 1ffiffiffiffiffif 1
p��
−
�2mð0Þy
−1 − f 2 þ ffiffiffiffiffie1
p�
k�Bm2ð0Þ
Ab
�1ffiffiffiffiffie1
p�−B2m4ð0Þ4ðAbÞ2
�f 3 − f 2e1
ffiffiffiffiffie1
p�
−kB2m4ð0Þ4ðAbÞ2
�f 2
ð−1 − f 2Þe1 þ e1ffiffiffiffiffie1
p���
;
ð6Þ
where εk is the Neumann factor (εk ¼ 1 for k ¼0 andεk ¼ 2 for k > 0), and
JA−¼ ðR − SÞ2; ð6aÞ
JAþ ¼ ðRþ SÞ2; ð6bÞ
JA×¼ RS; ð6cÞ
JI¼ ¼ R2 þ S2; ð6dÞ
f 2 ¼ imð0ÞðJIþ þ nÞ; ð6eÞ
f 3 ¼ m2ð0ÞðJAþ þ nÞðJA− þ nÞ; ð6f Þ
f 1 ¼ 1 − 2f 2 − f 3; ð6gÞ
e1 ¼ 1þ 2f 2 − f 3; ð6hÞ
m2ð0Þ ¼ AbaAþ B
: ð6iÞ
The joint spectra coefficients in Eq. (6) are complexnumbers and complex coefficients introduce phase
distortion to the output signal. In order to evaluatethe effect of the severity of this phase distortion, theabsolute value, real, and imaginary part of the com-plex numbers and the argument of the joint spectracoefficient [Eq. (6)] are plotted and are shown inFigs. 5–7. In these plots R and S are assumed tobe equal to unity.
Figures 5A, 5B, 5C, and 5D show plots of the abso-lute, the real, the imaginary, and the argumentvalues of H1ðvx; vyÞ, respectively. The same para-meters that have been used for Fig. 3 plot are usedfor these plots. The only unknown parameters arethe joint spectra parameters. Since R ¼ S ¼ 1, theseparameters are found to be JA− ¼ 0, JAþ ¼ 4,JIþ ¼ 2, and JA× ¼ 1. As is evident from Fig. 5A,the H1ðvx; vyÞ value has been increased by simplyincreasing the intensity I0 from 3mW to 9mW.Comparing almost identical absolute and real plotsof the H1ðvx; vyÞ in Fig. 5 shows that the main contri-bution of the H1ðvx; vyÞ magnitude is from its realvalue component. This also can be verified from al-most identical imaginary and phase plots. Eventhough the phase has been increased by 2 ordersof magnitude (from 10−11 to 10−9) by an increase from1mW light intensity to 10mW, yet the imaginarycomponent has an insignificant contribution to theH1ðvx; vyÞ magnitude. The phase and the imaginaryfactors can be identical if and only if the imaginarycomponent of a complex number is extremely small.Since the phase of the first-order joint spectra coeffi-cient (H1ðvx; vyÞ) is insignificant, it is possible toignore the phase distortion and utilize this coefficientfor both correlation and convolution.
Figure 6 shows the same sequence of plots as Fig. 5except for theH2ðvx; vyÞ coefficient. It is evident fromthese plots that the magnitude of the imaginary com-ponent of this coefficient is much larger than its realcomponent. The real values have a small declinefrom 1 to 0.99985 as the intensity increases from1mW to 8mW, and the imaginary values increaseslightly from −85 to −83 as the intensity increasesfrom 2mW to 10mW; hence, the imaginary compo-nent has the most contribution of the H2ðvx; vyÞ
absolute value rather than the real component. Thisis evident by comparing Figs. 6A and 6C.The insignificant changes in theH2ðvx; vyÞ real and
imaginary component plots lead to nearly a constantphase. According to Fig. 6D, the H2ðvx; vyÞ phase de-creases slightly from 1.5591 to 1.5587 as the inten-sity increases from 1mW to 10mW. Since thephase is almost a constant, the phase distortioncan be negligible for utilizing this coefficient in cer-tain signal processing criteria. In contrast to thefirst-order coefficient H1ðvx; vyÞ, where its absolutevalue was dominated from its real component, herethe absolute value of the second-order H2ðvx; vyÞ isdominated by its imaginary component.Figure 7 shows the same sequence of plots as
Figs. 5 and 6 for the third-order coefficientH3ðvx; vyÞ. Similar to the first-order coefficient theabsolute value of the H3ðvx; vyÞ is mainly dominatedby its real component. This is evident when compar-ing plots of H3ðvx; vyÞ absolute values (Fig. 7A) andthe real values (Fig. 7B). Both curves almost areidentical and drop from 1:44 × 107 to 1:38 × 107 asthe intensity increases from 1mW to 10mW. Theimaginary values drop from −1 to −25 as the inten-
sity increases from 1mW to 10mW, which are insig-nificant compared to the real values. Although theH3ðvx; vyÞ phase factor increases from 0 to1:8 × 10−6, it is insignificant enough to cause anyphase distortion.
7. Computer Simulation
The effectiveness of the proposed image recovery wastested with computer simulation. A 128 × 128 pixelLena face located in the middle of a 512 × 512 nullarray was used for the input image. The blurred im-age was generated by convolving the original inputwith common blur functions: motion, atmosphericturbulence, and misfocusing. Random noise was alsoadded to the blurred image. The noise was created bya random-number generator that produced whiteGaussian noise with zero mean and a variance of1, and was added to the blurred joint input.
The motion impulse response was a 2 × 50 pixelrectangle. The atmospheric turbulence was [15]
Hðνx; νyÞ ¼ e−βðν2xþν2yÞ5=6 ð7Þ
Fig. 5. (Color online) H1ðvx; vyÞ plots: A, absolute value; B, real value; C, imaginary value; and D, phase.
for β ¼ 0:0025. The misfocusing impulse responsewas a circle of ones with a diameter of 20 pixels.The nearly hard-clipping nature of the NOLMEMS
device saturation nonlinearity fully clips the jointspectra high frequencies, even for low lightintensities.Figures 8–15 show the simulation results based on
the first-order nonlinear transform method expan-sion presented in Eq. (6). Figures 8, 10, 12, and 14represent the results of deconvolution for the follow-ing blur functions respectively: motion, atmospheric,misfocusing, and phase, while A is the noisy blurredinput with SNR ¼ 5; A′ is the corresponding recov-ered image. The B and C rows are sequentially thesame as the A row for SNR ¼ 1 and SNR ¼ 0:1, re-spectively. Figures 9, 11, 13, and 15 represent a com-parative study of the deconvolution via a NOLMEMSdevice with both of theWiener and the inverse filters.We used the following equation for the Wiener filter:
where Hðu; vÞ is the blur function, H�ðu; vÞ is thecomplex conjugate of Hðu; vÞ, jHðu; vÞj2, jFðu; vÞj2,
and jNðu; vÞj2 are the power spectrum of the distor-tion function, the undegraded image, and the noise,respectively. In this approach, the goal is to find anestimated recovered image of the uncorrupted imagein such a way that the mean square error betweenthem is minimized.
In all of these figures, A represents the recoveredimage with spectrally variable joint information forSNR ¼ 1, B is the recovered image using the Wienerfilter, with the expected value of F (an estimated re-covered image), C is the recovered image using in-verse filtering, and D is the recovered image usingthe Wiener filter, with the exact value of F (theuncorrupted image).
As it is evident in the C figures, the inverse filterwas incapable of either recovering or even detectingthe image. In these results noise is covering thewhole array. By comparing the As with the Ds, itis clear that, for all the blur functions tested here,the image recovery of a noisy blurred signal withSNR ¼ 1 via the spectrally variant NOLMEMS de-convolution was always better (with a generic shapeof the lowpass filter or the expected value of F) thanthe Wiener filter with the exact value of F. When theexact value of F was replaced by a generic shape of
Fig. 6. (Color online) H2ðvx; vyÞ plots: A, absolute value; B, real value; C, imaginary value; D, phase.
the lowpass filter or the expected value of F, theWiener filter failed, as is evident in B for Figs. 9,11, 13, and 15.As it was discussed previously, this large difference
in the power requirement is attributed to the powereight dependency of the input–output nonlineartransfer function of this device on the incident opticalintensity. For blurred signals, from the signal proces-sing point view, power eight dependency leads forinfinitesimally small high frequency intensitieswithin the input–output nonlinear transfer function[Eq. (2)]. This requires a high power to obtain outputsaturation. Therefore, the use of a highpass filter forpreprocessing is essential to release the constraintsfor unrealistically high power beam intensities.The dynamic range compression deconvolution
technique is associated with noise conversion fromlow frequencies to high frequencies, which is the ran-dom distribution of delta functions (salt and peppernoise) [16–19]. This frequency conversion leads tofurther enhancement of the signal-to-noise ratiothrough spreading the noise over the entire array,while in image recovery via the Wiener filter, thenoise remains nearly concentrated in the originalnoisy image area.
It is clear from the simulation results that dynamicrange compression deconvolution via NOLMEMS iscapable of both noise reduction and image restora-tion embedded in a severe noise environment fromaberrations such as motion, atmospheric, misfocus-ing, and phase. A signal-to-noise ratio of 1 meansthat the probability of signal detection is ∼50% witha 0.1 average false alarm rate (FAR).
8. Conclusion
This paper proposes dynamic range compressiondeconvolution using a newly designed NOLMEMSdevice, and shows that it has several advantages overtwo-beam coupling deconvolution. This device candefeat the low grating efficiency problem in two-beam coupling deconvolution. The processing timefor the NOLMEMS device is in the microsecondrange. Unlike the two-beam coupling deconvolutionthat processes only coherent light, this device canalso work with incoherent light. It is possible to de-sign a MEMS optical limiter that can detect and pro-cess broadband ultralow light intensities. Computersimulation results showed excellent image recoveryfrom three different blur functions in a high noiseenvironment.
Fig. 7. (Color online) H3ðvx; vyÞ plots: A, absolute value; B, real value; C, imaginary value; and D, phase.
Fig. 8. Motion aberration indirect simulation results: A, noisyblurred input with SNR ¼ 5; A′, corresponding recovered image.The B and C rows are sequentially the same as the A row forSNR ¼ 1 and SNR ¼ 0:1, respectively.
Fig. 9. Deconvolutionresult for themotionaberrationcomparedtoWiener and inverse filters result: A, recovered image for SNR ¼ 1;B,recoveredimageusingtheWienerfilter,withtheexpectedvalueofF;C,recoveredimageusinginversefiltering,andD, recoveredimageusing Wiener filter, with the exact value of F.
Fig. 10. Atmospheric turbulence indirect simulation results:A, noisy blurred input with SNR ¼ 5; A′, corresponding recoveredimage. The B and C rows are sequentially the same as the A rowfor SNR ¼ 1 and SNR ¼ 0:1, respectively.
Fig. 11. Deconvolution result for the atmospheric turbulencecompared to theWiener and inverse filters result: A, recovered im-age for SNR ¼ 1; B, recovered image using the Wiener filter, withthe expected value of F; C, recovered image using inverse filtering;D, recovered image using the Wiener filter, with the exact valueof F.
Fig. 12. Misfocusing aberration indirect simulation results:A, noisy blurred input with SNR ¼ 5; A′ the corresponding recov-ered image. The B and C rows are sequentially the same as the Arow for SNR ¼ 1 and SNR ¼ 0:1, respectively.
Fig. 13. Deconvolution result for the misfocusing aberration com-pared to the Wiener and inverse filters result: A, recovered imagefor SNR ¼ 1; B, recovered image using the Wiener filter, with theexpected value of F; C, recovered image using inverse filtering; andD, recovered image using Wiener filter, with the exact value of F.
Fig. 14. Phase-only aberration indirect simulation results:A, noisy blurred input with SNR ¼ 5; A′, the corresponding recov-ered image. The B and C rows are sequentially the same as the Arow for SNR ¼ 1 and SNR ¼ 0:1, respectively.
Fig. 15. Deconvolution result for the phase-only aberration com-pared to the Wiener and inverse filters result: A, recovered imagefor SNR ¼ 1; B, recovered image using the Wiener filter, with theexpected value of F; C, recovered image using inverse filtering; andD, recovered image using Wiener filter, with the exact value of F.
The nonlinear transformmethod was used to analyzethe nonlinear transfer function in Eq. (2). For thisanalysis, the nonlinear transfer function was ex-panded to a superposition of four infinitesimally dif-ferent complex negative saturation nonlineartransfer functions as follows:
f ðEÞ ¼ 11� iGpðEþ E0Þ
; ðA1Þ
whereGp is a generic parameter, E is the energy, andE0 is a constant energy parameter.In the analysis of any nonlinear system, generally
any generic function in the form of f ðEÞ is expandedin terms of its frequency spectrum FðωÞ, where ω isits complement frequency variable.The relationship between the input nonlinear
transfer function and its frequency spectrum canbe expressed as
FðωÞ ¼Z
∞
−∞
f ðEÞ expð−iωEÞdE; ðA2Þ
f ðEÞ ¼ 12π
Z∞
−∞
FðωÞ expðiωEÞdω: ðA3Þ
In a joint Fourier processor, let us assume that thescene transmittance function is sðx; yþ y0Þ and thereference transmittance function is rðx; y − y0Þ. yo isthe separation between the reference and the signalinformation. In the Fourier plane, the joint intensityspectrum is
where R and S are the Fourier transforms of r and s,respectively. ΦR and ΦS are the reference and thesignal phases and vx and vy are the spatial frequencycoordinates.Substituting Eq. (A3) into Eq. (A4) yields
f ðEÞ ¼ 12π
Z∞
−∞
FðωÞ exp½iωðR2
þ S2Þ� exp½i2ωRS cosð2y0νy þ ϕR − ϕSÞ�dω:ðA5Þ
The exponential factor containing the cosine termis expanded using the Jacobi–Anger formula:
f ðEÞ ¼ 12π
Z∞
−∞
FðωÞ exp½iωðR2
þ S2Þ��X∞
k¼−∞
ikJkð2ωRSÞ cos½2ky0νy
þ kðϕR − ϕSÞ��dω; ðA6Þ
where Jk is the Bessel function of the first kind,order k.
The integral form of f ðEÞ in Eq. (A3) is thenconverted into a summation series:
f ðEÞ ¼X∞k¼0
Hkðνx; νyÞ cos½2ky0νy þ kðϕR − ϕSÞ�; ðA7Þ
where the weighting factors Hk are given by
Hkðνx; νyÞ ¼εk2π i
k
Z∞
−∞
FðωÞ exp½iωðR2
þ S2Þ�Jkð2ωRSÞdω; ðA8Þ
and εk is the Neumann factor (εk ¼ 1 for k ¼ 0 andεk ¼ 2 for K > 0).
The Fourier transform of the complex negativesaturation nonlinear transfer functions describedin Eq. (A1) is
FðωÞ ¼ −i2π�Gp
e−ω=�GpeiωE0 : ðA9Þ
Substituting Eq. (A9) into Eq. (A8) yields theweighting factors Hkðvx; vyÞ as
Hkðvx;vyÞ¼εk2πi
k
Z∞
0
−i2π�Gp
e−ω=�GpeiωðJIþÞeiωE0Jkð2ωJA×Þdω
¼−εkikþ1
�Z∞
0e−rcosðrGpðJIþ
þE0ÞÞJkð2rGpJA×Þdr
þiZ
∞
0e−rsinðrGpðJ2
Iþ
þE0ÞÞJkð2rGpJA×Þdr�;
ðA10Þ
where
� r ¼ ω�Gp
; ðA10:1Þ
and JA−, JAþ, JA×, and JIþ are the joint spectra mo-mentums defined as
In above equations, ε0 is the permittivity of freespace, 2r1 is the size of one pixel pattern, s is thedepth of the well, T is the membrane tension, V isthe total voltage applied across the MEMS, q isthe electronic charge, μn is the carrier mobility, nis the carrier concentration under dark illuminationconditions, L is the substrate thickness, A is the areaof each pixel, Δn is the increase in carrier concentra-tion due to light illumination, and ω is the frequency.For subsequent presentation of the joint spectra
coefficient in a general standard presentation, it isrelevant to write Eq. (A18) in different format withnew parameters. Equation (A18) can be rearrangedin the following format:
haA
½AaþB� þ bA½AaþB� ðxþ x0Þ2
i2
½Aaþ B�2h1þ bA
AaþB ðxþ x0Þ2i2
¼h
α1βð0Þ þ γ2ð0Þðxþ x0Þ2
i2
β2ð0Þ½1þ γ2ð0Þðxþ x0Þ2�2
¼ ½γ1ð0Þ þm2ð0Þðxþ x0Þ2�2β2ð0Þ½1þm2ð0Þðxþ x0Þ2�2
; ðA19Þ
α1 ¼ aA; ðA19:1Þ
α2 ¼ Ab; ðA19:2Þ
βð0Þ ¼ aAþ B; ðA19:3Þ
γ1ð0Þ ¼α1βð0Þ ¼
AaaAþ B
; ðA19:4Þ
γ2ð0Þ ¼α2βð0Þ ¼
AbaAþ B
¼ m2ð0Þ: ðA19:5Þ
In a joint Fourier processor, x corresponds to thenumber of carriers that is generated by the light in-tensity that has the joint spectra information and isequivalent to
xðνx; νyÞ ¼ατnI0jRþ Sj2hνADðλf Þ2
; ðA20Þ
where α is the absorption coefficient, τn is the carrierlifetime, I0 is the total beam intensity transmittedwith the reference and the signal information, h isPlanck’s constant, v is the light frequency, AD isthe detector area, λ is the wavelength, and f is thelens focal length.Incorporating the joint spectra into Eq. (A19)
: ðA21cÞFor simplicity of the analysis at this stage, theanalysis using Eq. (A18) is considered. Throughthe analysis, the parameters were defined inEqs. (A19)–(A21) will be replaced.
The input–output nonlinear transfer function inEq. (A1) is a positive saturation function. For the pur-pose of the analysis, it was found from prior workthat, in general, it is simpler to analyze a negativesaturation function. Therefore, this function can bewritten as
GðyÞ ¼ A2y2
ðAyþ BÞ2 ¼ 1 −2AByþ B2
ðAyþ BÞ2 : ðA22Þ
The second term in the right-hand side of Eq. (A22) isa negative saturation function. This function is stillnot easy to analyze using the nonlinear transformmethods. The fractional expansion is done in twostages. The first stage is for the second-order expan-sions (i.e., y expansion) and the second stage is for thefirst-order fraction expansion (i.e., x expansion).After expansion to the first order,. the nonlineartransform method is used to derive the contributionof each of the expansion to the correlation and theconvolution factors. Then the contribution from allthe fractional orders was superimposed.
At first, the second term in the right-hand side ofEq. (A22) is fractionally expanded as follows:
2AByþ B2
ðAyþ BÞ2 ¼ limε→0
2AByþ B2
ðAyþ Bþ εÞðAyþ B − εÞ : ðA23Þ
The right-hand side of Eq. (A23) can be fractionallyexpanded to
Equation (A26) consists of the summation ofnonlinear transfer functions described in Eq. (A1).Therefore, combining the contribution from bothmð�εÞ and −mð�εÞ and through replacing x to its cor-respondence equation [Eq. (A21)] gives us
The actual value of HCkðvx; vyÞ can be found fromtaking the limit when ε approaches zero:
HCkðvx; vyÞ ¼ limε→0
½HCkðvx; vy;mðεÞÞ
þHCkðvx; vy;mð−εÞÞ�: ðA29Þ
A Taylor expansion equation for the secondorder yields that
Hkðvx; vyÞ ¼ 1 − εkikþ1
��2mð0ÞJA×
1 − f 2 þffiffiffiffiffif 1
p�
k
×�Bm2ð0Þ
Ab
�1ffiffiffiffiffif 1
p�−B2m4ð0Þ4ðAbÞ2
�f 2 þ f 3f 1
ffiffiffiffiffif 1
p�
−kB2m4ð0Þ4ðAbÞ2
�2 − f 2
ð1 − f 2Þf 1 þ f 1ffiffiffiffiffif 1
p��
−
�2mð0Þy
−1 − f 2 þ ffiffiffiffiffie1
p�
k�Bm2ð0Þ
Ab
�1ffiffiffiffiffie1
p�
−B2m4ð0Þ4ðAbÞ2
�f 3 − f 2e1
ffiffiffiffiffie1
p�
−kB2m4ð0Þ4ðAbÞ2
�f 2
ð−1 − f 2Þe1 þ e1ffiffiffiffiffie1
p���
; ðA30Þ
where
f 2 ¼ imð0ÞðJIþ þ x0Þ; ðA30:1Þ
f 3 ¼ m2ð0ÞðJAþ þ x0ÞðJA− þ x0Þ; ðA30:2Þ
f 1 ¼ 1 − 2f 2 − f 3; ðA30:3Þ
e1 ¼ 1þ 2f 2 − f 3: ðA30:4Þ
Substituting parameters from Eqs. (A19)–(A21)into Eq. (A30), the weighting factors equation yieldsto
Hkðvx; vyÞ ¼ 1 − εkikþ1
��2CmJA×I0
Bf ð1 − f 2 þffiffiffiffiffif 1
p Þ
�k
�BC2
m
B2f Ab
�1ffiffiffiffiffif 1
p�−
B2C4m
4B4f ðAbÞ2
�f 2 þ f 3f 1
ffiffiffiffiffif 1
p�
−kB2C4
m
4B4f ðAbÞ2
�2 − f 2
ð1 − f 2Þf 1 þ f 1ffiffiffiffiffif 1
p��
−
�2CmJA×I0
Bf ð−1 − f 2 þ ffiffiffiffiffie1
p Þ�
k�BC2
m
B2f Ab
�1ffiffiffiffiffie1
p�
−B2C4
m
4B4f ðAbÞ2
�f 3 − f 2e1
ffiffiffiffiffie1
p�
−kB2C4
m
4B4f ðAbÞ2
�f 2
ð−1 − f 2Þe1 þ e1ffiffiffiffiffie1
p���
; ðA31Þ
where
f 2 ¼ iCm
BfðJIþ þ x0Þ; ðA31:1Þ
f 3 ¼�Cm
Bf
�2ðJAþ þ x0ÞðJA− þ x0Þ; ðA31:2Þ
f 1 ¼ 1 − 2f 2 − f 3; ðA31:3Þ
e1 ¼ 1þ 2f 2 − f 3; ðA31:4Þ
Cm ¼ mð0ÞBf : ðA31:5Þ
References1. Y. Huang, G. Siganakis, M. G. Moharam, and S-T. Wu, “Broad-
band optical limiter based on nonlinear photoinduced aniso-tropy in bacteriorhodopsin film,” Appl. Phys. Lett. 85,5445 (2004).
2. B. Haji-saeed, S. K. Sengupta, W. Goodhue, J. Khoury, C. L.Woods, and J. Kierstead, “Nonlinear dynamic range compres-sion deconvolution,” Opt. Lett. 31, 1969–1971 (2006).
3. B. Haji-saeed, S. K. Sengupta, W. Goodhue, J. Khoury, C. L.Woods, and J. Kierstead, “Spectrally variable two-beamcoupling nonlinear deconvolution,” Appl. Opt. 46, 8244–8249 (2007).
4. H. J. Eichler, P. Gunter, and D. W. Pohl, Laser InducedDynamic Gratings (Springer-Verlag, 1986).
5. T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photo-refractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
6. P. Gunter and J.-P. Huignard, Photorefractive Materials andTheir Applications, Vols. I and II (Springer-Verlag, 1988).
7. B. Haji-saeed, R. Kolluru, D. Pyburn, R. Leon, S. K. Sengupta,M. Testorf, W. Goodhue, J. Khoury, A. Drehman, C. L. Woods,and J. Kierstead, “Photoconductive optically driven deform-able membrane under high frequency bias: fabrication, char-acterization and modeling,” Appl. Opt. 45, 3226–3236 (2006).
8. J. Khoury, C. L. Woods, B. Haji-saeed, S. K. Sengupta, W.Goodhue, and J. Kierstead, “Optically driven microelectrome-chanical-system deformable mirror under high-frequency ACbias,” Opt. Lett. 31, 808–810 (2006).
9. J. Khoury, B. Haji-saeed, W. D. Goodhue, C. L. Woods, and J.Kierstead, “MEMS-based optical limiter,” Appl. Opt. 47,5468–5472 (2008).
10. J. L. Horner and P. D. Gianino, “Effects of quadratic phase dis-tortion on correlator performance,” Appl. Opt. 31, 3876–3878 (1992).
11. W. B. Davenport andW. L. Root, An Introduction to the Theoryof Random Signal and Noise (McGraw-Hill, 1958), Chaps. 12–13, pp. 255–311
12. J. Van de Vegte, Fundamentals of Digital Signal Processing(Prentice-Hall, 2002), Chap. 15, pp. 659–660.
13. J. L. Horner and M. A. Flavin, “Average amplitude matchedfilter paper,” Opt. Eng. 29, 31–37 (1990).
14. B. Haji-saeed, “Development of novel device assemblies andtechniques for improving adaptive optics imaging systems,”Ph.D. dissertation (University of Massachusetts at Lowell,2006).
15. R. E. Hufnagel and N. R. Stanley, “Modulation transfer func-tion associated with image transmission through turbulentmedia,” J. Opt. Soc. Am. 54, 52–61 (1964).
16. R. C. Gonzalez and R. E. Woods, Digital Image Processing(Prentice-Hall, 2002).
17. G. Asimellis, J. Khoury, and C.Woods, “Effects of saturation onthe nonlinear incoherent-erasure joint-transform correlator,”J. Opt. Soc. Am. A 13, 1345–1356 (1996).
18. J. Khoury, G. Asimellis, P. D. Gianino, and C. L. Woods“Nonlinear compansive noise reduction in joint transformcorrelators,” Opt. Eng. 37, 66–74 (1998).
19. J. Khoury, P. D. Gianino, and C. L.Woods “Engineering aspectsof the two-beam coupling correlator,” Opt. Eng. 39, 1177–1183 (2000).
