Letter 1

Customizing Speckle Intensity StatisticsNICHOLAS BENDER1, HASAN YILMAZ1, YARON BROMBERG2, AND HUI CAO1,*

1Department of Applied Physics, Yale University, New Haven CT 06520, USA2Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel*Corresponding author: hui.cao@yale.edu

Compiled December 1, 2017

We develop a general method for customizing theintensity statistics of speckle patterns. By judi-ciously modulating the phase-front of a monochro-matic laser beam, we experimentally generatespeckle patterns with arbitrarily-tailored inten-sity probability-density functions (PDF). Basedon applying a local intensity transformation to aRayleigh speckle pattern, our method allows fortopological changes in the customized speckleswhile preserving their granularity. In addition totailoring the functional form of the intensity dis-tribution, we can separately control the intensityrange of the PDF and thereby tune the speckle con-trast.OCIS codes: (030.0030) Coherence and statistical optics;(030.6140) Speckle; (030.6600) Statistical optics.

Speckle formation is a phenomenon inherent to both classicaland quantum waves. Characterized by a random grainy struc-ture, speckle patterns arise when a coherent wave undergoes adisorder-inducing scattering process. The statistical propertiesof a speckle pattern are generally universal, known as Rayleighstatistics, featuring a negative-exponential for the intensity prob-ability density function [1–3]. In recent years there has been agreat deal of interest in generating speckles with non-Rayleighstatistics [4–9]. Such patterns are useful for structured illumina-tion in imaging applications, for example: dynamic speckle illu-mination microscopy [10–12], super-resolution imaging [13–17],and pseudo-thermal light sources for high-order ghost imaging[18–20]. Furthermore, a general method for customizing boththe statistics and structure of laser speckle patterns would be avaluable tool for synthesizing optical potentials for cold atoms[21–24], colloidal particles [25–28], and active media [29–31].

Recently a simple method was developed to create non-Rayleigh speckle patterns with a phase-only spatial light modu-lator (SLM) [5]. High-order correlations were encoded into thefield by the SLM, leading to a redistribution of the light intensityamong the speckle grains in the far-field. In this method, thespeckle contrast could either be greater than or less than that ofa Rayleigh speckle with the corresponding PDF decaying slower

(super-Rayleigh) or faster (sub-Rayleigh) than an exponentialfunction (Rayleigh). However it was not known if it was possi-ble to have an intensity PDF of any functional form, while stillmaintaining the random granularity of a speckle pattern.

In this letter, we present a general method for customizingthe intensity statistics of speckle patterns by modulating thephase front of a laser beam with a SLM. To maintain the char-acteristic granularity of a speckle pattern in this process, wegenerate a Rayleigh speckle pattern and numerically perform alocal intensity transformation to obtain a new speckle patternwhich is governed by the target PDF. Subsequently this patternis experimentally generated in the far field of the SLM, wherethe requisite phase modulation is determined numerically viaa non-linear optimization algorithm. In this way we can createspeckle patterns governed by distinct statistical distributions,e.g., the intensity PDF that remains constant within a predefinedintensity range of interest, or increases linearly with intensity, orfeatures either a single peak or double peaks at specified intensi-ties. The customized speckle patterns exhibit distinct topologiesrelative to Rayleigh speckles, yet retain their granularity.

Here we describe how to synthesize a target speckle patterngoverned by an arbitrary PDF. Experimentally a phase-onlyreflective SLM, illuminated by a linearly polarized laser beam ofwavelength λ = 642 nm, is placed at the front focal plane of alens and displays a random phase pattern. The correspondingintensity pattern at the back focal plane is recorded by a CCDcamera. The recorded intensity pattern is a Rayleigh specklepattern, like Fig. 1(a), because the phases of the SLM elementsare independent random variables. We numerically perform alocal intensity transformation on the recorded speckle patternwhich converts it to a speckle pattern governed by the desiredPDF, F( I). The intensity PDF of the Rayleigh speckle pattern,P(I) = exp[−I/〈I〉]/〈I〉, can be related to the target PDF F( I)by:

P(I)dI = F( I)dI (1)

This expression determines the local intensity transformationI = f (I), which must be applied to the intensity values of theRayleigh speckle pattern in order to create a new speckle patternwith the desired PDF. To solve for the intensity transformation,we write Eq. 1 in integral form with 〈I〉 = 1:∫ I

0e−I ′dI′ =

∫ I

Imin

F( I′)dI′ (2)

In addition to altering the intensity PDF, such a transformation

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Fig. 1. A Rayleigh speckle pattern (a) and customized speckle patterns with distinct intensity statistics (b-e). In the top row, eachpattern has the size of 504 µm by 504 µm, and the maximum intensity is normalized to 1. The associated PDF, shown in the lowerrow, is constant (b), linearly increasing (c), peaked at a non-zero intensity (d), bimodal (e), within a predefined range of intensity.The red solid curves are experimental data, the blue dashed curves are from numerically generated target speckle patterns. Both areaveraged over 50 independent realizations.

provides the freedom to regulate the maximum and/or mini-mum intensity values of the transformed pattern. We may arbi-trarily set Imax or Imin, as long as the following normalization

conditions hold:∫ Imax

IminF( I′)dI′ = 1, and 〈 I〉 =

∫ ImaxImin

I′F( I′)dI′ =〈I〉. Such regulatory ability is useful for practical applicationssuch as speckle illumination, where without altering the totalpower of the beam the maximal intensity value can be set belowthe damage threshold of a sample and the minimum intensityvalue can be set to exceed the noise floor.

Experimentally the phase-modulating region of the SLM ispartitioned into a square array of N × N macro-pixels. Sincethe laser beam incident upon the SLM is linearly polarized andthe scattering angles from the SLM are too small to introduceradial polarization, the light reaching the camera retains thelinear polarization and can be modeled as a scalar wave. To agood approximation, the field pattern at the camera plane is theFourier transform of the SLM phase pattern. Thus the SLM arraycomposes the spatial frequency spectrum of the speckle patternon the camera. Numerically a digital Fourier transform of theN × N SLM elements gives an N × N array, each representinga speckle grain. To avert the effects of aliasing and uniquelydefine the spatial profile measured by the camera, it is necessaryto sample the speckle pattern at or above the Nyquist limit, i.e.,each speckle grain is sampled at least twice along each spatialaxis, and the N × N speckles are sampled by 2N × 2N points.However, it is generally impossible to control all 4N2 intensityvalues in the camera array by N2 independent phases of the SLM,thus we only focus on tailoring the speckle statistics within thecentral quarter region of the speckle pattern, denoted the targetregion, which has an equivalent number of intensity values N2.

Given the constraints imposed by the Nyquist limit, we needto revisit the original intensity transformation in Eq. 2. Such atransformation produces spatial frequency components in thetransformed pattern that are higher than those present in theoriginal pattern, and therefore outside the range covered by theSLM. Nevertheless, these components can be removed from theintensity pattern by applying a digital low-pass Fourier filter,however the resulting pattern will have an intensity PDF F(I)

slightly deviating from the target one F( I). Such deviations canbe eliminated by applying an additional intensity transforma-

tion I = f (I) that is obtained from∫ I

IminF(I′)dI′ =

∫ IImin

F( I′)dI′.The process of performing a local intensity transformation, andsubsequently applying a digital low-pass Fourier filter, can beiteratively repeated as a Gerchberg-Saxton-type algorithm un-til the target PDF is obtained for a speckle pattern obeying thespatial frequency restrictions. Repetition of this operation withdifferent initial Rayleigh speckles creates a unique set of speck-led intensity patterns which possess the same PDFs.

To experimentally generate target speckle patterns with cus-tomized intensity PDFs, we need to determine the requiredphase modulation for the SLM to apply upon the incident laserbeam. Since the Fourier relation between the field reflected offthe SLM and the camera plane is only approximate, we charac-terize the precise relation by measuring the field transmissionmatrix (T-matrix). In addition to encapsulating the experimentalimperfections induced by optical misalignment, lens aberra-tions, nonuniform laser illumination of the SLM, employing thetransmission matrix provides a general formalism which canbe adapted to other setups (e.g., holographic optical tweezers[32]) or to tailor the speckle statistics at a different plane thanthe Fourier plane. In this work, the T-matrix is measured witha common path interference method akin to those in [33–35].For more details, see the supplementary material. Using themeasured T-matrix, we find the requisite phase pattern on theSLM with a nonlinear-optimization algorithm. Numerically weminimize the difference between the target speckle intensity pat-tern and the intensity pattern obtained by applying the T-matrixto the SLM phase array, using the low-storage BFGS algorithmin the NLopt software library [36, 37]. The algorithm finds anappropriate solution for the SLM phase array which generatesthe target speckle pattern with a customized intensity PDF onthe camera plane.

Examples of experimentally generated speckle patterns withcustomized intensity statistics are shown in Fig. 1 (b-e). In (b)the speckle pattern has a uniform intensity PDF, over a prede-fined intensity range. This example illustrates that it is possible

Letter 3

Fig. 2. Evolution of customized speckle patterns upon axial propagation. The intensity PDF at the Fourier plane of the SLM (z = 0)is linearly increasing (a) and bimodal (e). The distance from the Fourier plane is Rl/5 (b, f), (2/5)Rl (c, g), and (10/7)Rl (d, h).

to create speckle patterns with non-decaying PDFs in addition toconfining the speckle intensities within a finite range. Taking thisone step further in (c), we first make the PDF increase linearlywith intensity, then have it drop rapidly to zero above the speci-fied threshold. To demonstrate that our method is not restrictedto monotonic functions, in (d) we create a speckle pattern whosePDF is peaked in the middle of Imin and Imax. To further increasethe complexity of the speckle statistics, (e) shows an exampleof a bimodal PDF. Since the customized speckle patterns in Fig.1(b-e) have a similar range of intensities, their distinct PDFs leadto different speckle contrasts. However, the speckle contrastis not directly related to the functional form for the intensityPDF. We can tune the speckle contrast by changing the inten-sity range while keeping the functional form of PDF (see thesupplementary material for more information).

Additionally Fig. 1 illustrates how the topology of the cus-tomized speckle patterns changes in accordance with the PDF.The spatial intensity profile of a Rayleigh speckle pattern in (a)can be characterized as a random interconnected web of darkchannels surrounding bright islands. Conversely for speckleswith a linearly increasing PDF in (c), the spatial intensity profileis an interwoven web of bright channels with randomly dis-persed dark islands. Similarly the spatial structure of speckleswith a bimodal PDF in (e) consists of interlaced bright and dimchannels. The topological changes in the customized speckles re-sult from the local intensity transformation and digital low-passFourier filtering (see the supplementary material for details).

The continuous network of high intensity in the customizedspeckle pattern, which is absent in the Rayleigh speckle pattern,will be useful for controlling the transport of trapped atoms orparticles in optical potentials.

In all cases, the experimentally generated speckle patternshave PDFs which follow the target functional form over the in-tensity regions of interest and converge to zero, quickly, outside.Small deviations between the experimental PDFs and the tar-get ones are caused by error in the T-matrix measurement andtemporal decorrelation of the experimental setup. We modelthese effects and numerically reproduce the deviations (see thesupplementary material for a full description of our model). Ourmodel explains why the deviations are stronger at higher in-tensity values or where the PDF varies rapidly with intensity.Moreover, certain PDFs cannot be generated because of the finiterange of spatial frequencies available on the SLM plane.

Given the drastic alterations made to the speckle PDFs, it isimportant to verify that the patterns generated by our methodmaintain the characteristic of speckle fields. The generatedspeckle fields in the target region possess a uniform phase-distribution over a range of 2π. Furthermore, the spatial fieldcorrelation function and the spatial intensity correlation functionare both identical to those of the Rayleigh speckles subjected tothe same digital low-pass filtering as the customized speckles(see the supplementary material for details). Hence, we can ma-nipulate the intensity statistics without introducing additionalspatial correlations in the speckle pattern.

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Finally we study how the tailored speckle patterns evolve, asthey axially propagate away from the Fourier plane along thez-axis. The previous method gives the target PDF for specklepatterns at the Fourier plane of the SLM. Outside of the Fourierplane however, the intensity statistics and topology may change.We know that in the case of a Rayleigh speckle pattern, thespatial pattern changes upon propagation while the intensityPDF remains a negative exponential. From this we can defineRl as the axial correlation length of a Rayleigh speckle pattern,which is equal to the longitudinal length of a single speckle.Fresnel propagation of a speckle field over a distance z can beapproximated by adding a quadratic phase to its spatial Fourierspectrum.

The top row of Fig. 2 shows the axial evolution of specklesthat has a linearly increasing PDF at the Fourier plane z = 0in (a). As the speckle pattern propagates to z = Rl/5, the PDFbecomes bell-shaped in (b). With further propagation, the maxi-mum of the PDF migrates to a smaller intensity value, as shownin (c) for z = (2/5)Rl , until it reaches I = 0. The speckles revertto Rayleigh statistics at z ≈ Rl , beyond which the PDF main-tains a negative exponential, as shown in (d) for z = (10/7)Rl .The topology of the speckle pattern evolves together with theintensity PDF: the interconnected web of bright channels firstattenuates upon propagation, then breaks into islands with darkchannels forming. In the lower row of Fig. 2, we show the axialevolution of a speckle pattern with a bimodal PDF at z = 0 in (e).As the pattern propagates to z = Rl/5 in (f), the peaks are asym-metrically eroded, with the high intensity peak diminishingfirst. Further propagation to z = (2/5)Rl results in a unimodalPDF in (g). Once the axial distance z exceeds Rl , the specklesreturn to Rayleigh statistics, as shown in (h) for z = (10/7)Rl .A corresponding change of speckle topology is seen: the brightchannels disappear first, then the dim channels fracture, whileneighboring dark islands merge to form channels. Therefore,axial propagation, within the range of Rl , alters the intensityPDF functional form and the speckle topology.

In conclusion, we have presented a general method for cus-tomizing speckle intensity statistics using a phase-only SLM.The generated speckle patterns possess radically different inten-sity statistics and topology than Rayleigh speckles, while stillmaintaining a random granularity. In addition to tailoring theprobability density function, we adjust the intensity range andtune the speckle contrast. Our method is versatile and com-patible with a broad range of optical setups. Therefore giventhe plethora of potential applications, it paves the way for newdirections in both fundamental research (interactions of complexsystems with random optical potentials of customized statistics)and applied research (computational imaging, dynamic speckleillumination microscopy, super-resolution imaging).

See Supplement 1 for supporting content.

Acknowledgment.We thank Chia Wei Hsu and Owen Miller for useful discussions.

Funding.This work is supported partly by the MURI grant no. N00014-13-1-0649 from the US Office of Naval Research.

Letter 5

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lation of non-rayleigh speckle fields and its application insuper-resolution imaging,” Laser Phys. 26, 055007 (2016).

21. J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lu-gan, D. Clément, L. Sanchez-Palencia, P. Bouyer, and A. As-pect, “Direct observation of anderson localization of matterwaves in a controlled disorder,” Nature 453, 891–894 (2008).

22. G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zac-canti, G. Modugno, M. Modugno, and M. Inguscio, “An-derson localization of a non-interacting bose-einstein con-densate,” Nature 453, 895–898 (2008).

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Supplementary Material

AbstractThis document provides supplementary information to “Cus-tomizing Speckle Intensity Statistics. We describe the detailsof the experimental setup in the first section. The second sec-tion elaborates upon our discussion of the Nyquist samplingcriterion. In the third section, we cover details pertaining tothe T-matrix measurement. We provide additional informationabout the customized PDFs in the fourth section. The fifth sec-tion is devoted to a theoretical assessment of the experimentalerror. We verify that our patterns maintain the characteristic ofspeckle fields in section six. The seventh section describes thetopological changes in the customized speckle patterns. Sectioneight presents an experimental demonstration of tailoring theintensity PDF range without altering the functional form of thePDF. In the final section, we present some details pertaining toaxial propagation of speckle fields.

1. EXPERIMENTAL SETUP

In our experimental setup, we illuminate a reflective, phase-only SLM (Hamamatsu LCoS X10468) with a linearly-polarizedmonochromatic laser beam at the wavelength λ = 642 nm (Co-herent OBIS). The beam is expanded and clipped by an iris touniformly illuminate the SLM phase modulation region. Thecentral part of the phase modulation region of the SLM is parti-tioned into a square array of 32× 32 macro-pixels, each consist-ing of 16× 16 pixels. The remaining illuminated pixels outsidethe central square diffract the laser beam via a phase grating.The SLM is placed at the front focal plane of a lens, f = 50cm, and the intensity pattern at the back focal plane is recordedby a 12 bit charge-coupled device (CCD) camera (Allied VisionProsilica GC660).

2. TARGET REGION VS. JUNKYARD

To avert the effects of aliasing and uniquely define the spatialprofile measured by the camera, it is necessary to sample thespeckle pattern at or above the Nyquist limit. This means everyspeckle grain is sampled at least twice along each spatial axis.Thus the N × N speckles generated at the camera plane must besampled by at least 2N× 2N points. To connect the N×N phasepattern on the SLM to the 2N × 2N array at the camera, via adiscrete Fourier transform, one can zero-pad the phase array toincrease its size to 2N × 2N. This is equivalent to assuming thatthe SLM phase pattern consists of 2N × 2N elements, howeverthe highest-half of the spatial frequency components – alongeach coordinate-axis – are set to zero.

Generally it is impossible to control 4N2 intensity points inthe speckle pattern using N2 independent phases of the SLM.Thus we reduce the area of control in the camera plane to thecentral quarter, denoted the target region, and neglect the re-maining three quarters, denoted the junkyard. An illustrationof the spatial partition of the target region and the junkyard isgiven in Fig. 3. Experimentally we sample the speckle patternin the target region well above the Nyquist limit with the CCDcamera: ∼ 10 camera pixels per speckle grain along each axis.

3. TRANSMISSION MATRIX MEASUREMENT

In this work the T-matrix is measured using an interferencemethod akin to those in [33–35]. The phase-modulating region

Fig. 3. An illustration of the Fourier transform between thefields on the front and back focal planes of a lens. The redframes delineate the outer edges of the discrete Fourier trans-form arrays, each consisting of 2N × 2N elements. On the frontfocal plane, the SLM modulates the phases of N × N macro-pixels in the central quarter, the rest of the elements (in thedark area) diffract light away from the lens. On the back focalplane where the camera is placed, the intensity statistics of thespeckles within the central quarter (target region, orange) thatconsists of N × N intensity values are tailored. The specklepattern in the remaining area (junkyard, black) is ignored.

of the SLM (with 32× 32 macropixels) is divided into two subre-gions, I and II. First, subregion I provides the reference signal tointerfere with a series of field patterns produced by subregion II,then subregion II serves as the reference for subregion I. One dif-ference from the previous methods is that the reference signalsare speckle patterns with sub-Rayleigh statistics. The dimensionof the T-matrix is 8100× 1024, and we average over multipleT-matrix measurements to reduce the error. The measured matri-ces deviate only slightly from an ideal T-matrix correspondingto a discrete Fourier transform.

4. INTENSITY PROBABILITY DENSITY FUNCTION

Figure 1 in the main text shows four PDFs with 〈I〉 = 1. In(a) the probability density is equal to 1/2 between I = 0 and2. In (b), the probability density increases linearly from 0 to√

2 in the intensity range of[0,√

2]. In (c), the probability

density is peaked at I = 1. The functional form is given by(π/4) sin[(π/2)I] H(2− I), where H(x) is the Heaviside stepfunction that sets the range of I to [0, 2]. In (d), the PDF is bi-modal in the intensity range of [0, 2]. The original functionalform is sin2(π I) H[2− I], making the probability vanish at I = 1.This means the intensity value of 1 is absent in the speckle pat-tern, but the intensities below and above 1 are present in thespeckle pattern. Creating such a pattern would require full con-trol of the 4N2 frequency components that the SLM array is com-posed of, but we only modulate the phases of N2 SLM macro-pixels (the rest having zero amplitude) to satisfy the Nyquistsampling criterion. Therefore, the iterative transformation al-gorithm converges to the PDF plotted in blue in Fig. 4, whichdiffers from the original one plotted in black. The blue shading

Letter 7

in this figure highlights the regions where deviations occur, pri-marily centered around I = 1. Therefore, the target PDF cannotvanish or have a discontinuity at any intensity value in betweenthe minimal and maximal intensities.

Fig. 4. Bimodal PDF for speckle intensity. The black curve isthe original PDF, and the blue curve is the numerically con-verged one. Their difference (shaded area) is due to the limitedrange of the spatial frequencies in the speckle pattern.

5. EXPERIMENTAL ERROR

In this section we perform a theoretical assessment of the experi-mental error in order to understand how the T-matrix measure-ment error and temporal decorrelation of the experimental setupaffect the intensity PDF of the tailored speckle patterns. Thedifference between the measured T-matrix Tm and the actualT-matrix Ta that produces the measured speckle pattern is:

∆T = Tm − Ta (3)

Tm is used in the numerical optimization to obtain the SLM field~Ψs to create the target speckle intensity pattern

~Id = |Tm ~Ψs|2 (4)

The measured speckle intensity pattern is

~Ie = |Ta ~Ψs|2 (5)

The difference between the two intensities, to the first order in∆T, is

~Ie −~Id = 2<[(Tm ~Ψs)(∆T~Ψs)

∗] = 2<[√

~Id ei~θd (∆T~Ψs)∗] , (6)

where Tm ~Ψs =√~Id ei~θd . Assuming the scalar elements of ∆T

are uncorrelated with those in ~Ψs, then the scalar quantities of

(~Ie −~Id)/√~Id = 2<

[(∆T~Ψs)∗

]will obey Gaussian statistics, as

verified experimentally in Fig. 5. The statistical distribution for(Ie − Id)/

√Id is:

G(Ie, Id) = A exp[−(Ie − Id)

2

2aId

](7)

where a is a coefficient that quantifies the experimental error, andA is the normalization constant given by

∫ ∞0 G(Ie, Id)dId = 1.

Therefore, the normalized expression for G(Ie, Id) is

G(Ie, Id) =exp

[−(Ie)2−(Id)2

2aId

]2IeK1[Ie/a]

(8)

where K1 is the Bessel function of the first kind.

Fig. 5. The statistical distribution of (Ie − Id)/√

Id, extractedfrom the experimental data (symbols), is fit well by the Gaus-sian distribution G(Ie, Id) in Eq. 7 (lines). The fitting parameteris given by a = 0.020 for the constant PDF (black), a = 0.019for the linearly increasing PDF (red), a = 0.037 for the PDFwith a single peak (blue) and a = 0.029 for the bimodal PDF(green).

The probability density function of the measured speckleintensity F(Ie) is equal to the target distribution F(Id) convolvedwith the error function G(Ie, Id) in Eq.(8):

F(Ie) =∫ ∞

0F(Id)G(Ie, Id)dId (9)

The convolution corresponds to an averaging of F(Id) overadjacent values of Id. Consequently, the measured PDF F(Ie)displays more discrepancy in the region where F(Id) changesrapidly. Since G(Ie, Id) is wider at larger Ie, the averaging effectis stronger, leading to a larger error at higher intensity. Theseeffects are confirmed in Fig. 6, where we plot Eq.(9) for the fourPDFs shown in Fig. 1 of the main text. For the constant andlinear increasing PDFs in (a) and (b), the abrupt drop in F(Id)at the upper boundary of the intensity range is smoothed outin F(Ie). In (c), the single-peaked PDF has relatively small error,although the deviation from the target PDF is clearly larger athigher intensity. For the bimodal PDF in (d), the peak at largerintensity is suppressed more due to stronger averaging effect,and the fine features around the dip in between the two peaksare removed by averaging.

Fig. 6. Deviation (shaded area) of the intensity PDF (blue solidline) from the target one (black dashed line) is reproducednumerically by Eq. 9. (a-d) are for the four PDFs shown in Fig.1 of the main text.

Letter 8

6. CUSTOMIZED SPECKLE PATTERN PROPERTIES

In this section, we investigate the characteristics of our cus-tomized speckle patterns presented in Figure 1 of the main text,and compare them with Rayleigh speckles. We apply the samedigital low-pass Fourier filter, used in the creation of target in-tensity patterns, to the Rayleigh patterns, when comparing theirspatial field and intensity correlation functions, so that any dif-ference between them is not caused by the Fourier filtering.

We start our investigation by looking at the phase distributionΦ(θ) of the speckle fields. To find Φ(θ), we use the measuredtransmission matrix and the associated SLM phase pattern torecover the field on the camera plane. Figure 7(a) plots Φ(θ) inthe target region for four PDFs that are demonstrated in Fig. 1of the main text, in addition to the case of a Rayleigh PDF. All ofthem have nearly constant values over [0, 2π], thus the phasesare uniformly distributed over a range of 2π.

Fig. 7. Characteristic of customized speckle patterns. (a) His-togram of phases of the speckle fields. (b) Spatial field cor-relation function CE(|∆r|). (c) Normalized spatial intensitycorrelation function CI(|∆r|). In (a-c), the four customizedspeckle patterns have constant (black), linear increasing (red),unimodal (blue), bimodal (green) PDFs, and the purple is forRayleigh speckles. (d) Comparing |CE(|∆r|)|2 (black curve) toCI(|∆r|) (blue dashed curve), both averaged over the 5 curvesin (b) and (c) respectively.

Next we compute the spatial correlation function of thespeckle field: CE(∆r) = 〈E(r)E(r + ∆r)〉/〈 I〉1/2. As shown inFig. 7(b), the customized speckles have the same field correlationfunction as a corresponding Rayleigh speckle pattern. Hence,the way we tailor the speckle statistics does not affect the fieldcorrelation.

Furthermore, we compute the spatial intensity correlationfunction CI(∆r) = (〈 I(r) I(r+∆r)〉− 〈 I〉2)/(〈 I2〉− 〈 I〉2), whichis normalized to 1 at |∆r| = 0. Figure 7(c) shows the normalizedspatial intensity correlation functions for the four customizedspeckles. They have the same line-shape as a Rayleigh specklepattern subjected to the digital low-pass Fourier filter. Therefore,we can manipulate the speckle intensity PDF without introduc-ing additional spatial correlations in the intensity.

In Fig. 7(d), we compare 〈|CE(∆r)|2〉 and 〈CI(∆r)〉, both av-eraged over the five curves in Figure 7(b,c). The comparison

demonstrates that the speckle field and intensity have the samecorrelation length, which is the average size of a speckle grain.However, 〈CI(∆r)〉 exhibits small oscillations at the tail. Theyare attributed to the low-pass Fourier filtering of the intensitypattern, which we use to remove the high spatial frequency com-ponents introduced during the nonlinear transformation of aRayleigh speckle pattern. For confirmation of this, we appliedthe digital low-pass Fourier filter to Rayleigh speckle patternsand the same oscillations appeared in the spatial intensity corre-lation function, shown in Fig. 7(c).

Despite that CI(∆r) is almost identical to |CE(∆r)|2, thespeckle fields with customized intensity PDFs are not Gaus-sian random variables. This is evident from the differences intheir high-order moments, 〈In〉 =

∫ ∞0 InP(I)dI, from those of

Rayleigh speckles, as shown in Table 1.

Table 1. Intensity moments of speckle patterns with differ-ent intensity PDFs

PDF 〈I〉 〈I2〉 〈I3〉 〈I4〉 〈I5〉 〈I6〉

Negative Exponential 1.00 2.00 6.00 24.0 120 720

Constant 1.00 1.35 2.06 3.39 5.87 10.51

Linearly Increasing 1.00 1.16 1.45 1.92 2.64 3.77

Unimodal 1.00 1.18 1.55 2.22 3.40 5.50

Bimodal 1.00 1.29 1.9 2.99 4.93 8.42

7. TOPOLOGICAL CHANGE

As shown in Fig. 1 of the main text, the topology of the cus-tomized speckle patterns is distinct from Rayleigh speckle pat-terns. The change of speckle topology is caused by the combina-tion of the local intensity transformation and the digital low-passFourier filtering. One example is given in Fig. 8. The originalRayleigh speckle pattern in (a) has a maximal probability-densityof vanishing intensity, leading to the dark channels surround-ing bright islands in the spatial profile of the speckle pattern.Application of a local intensity transformation to make the PDFincrease linearly with intensity and then rapidly converge to zeroabove a threshold results in the speckle pattern shown in (b).Due to the enhanced probability-density of high intensity andreduced probability-density of low intensity, the bright grainsare enlarged while the dark channels narrow. The application ofa digital low-pass Fourier filter severs the narrow dark lines inbetween the bright grains. After iterating the process of a localintensity transformation followed by a low-pass filter, neighbor-ing bright grains are merged to form channels that encompassdark islands.

8. SPECKLE CONTRAST

The speckle contrast is not uniquely defined by the functionalform of the intensity PDF, and it depends on the range of thespeckle intensity. Thus we can tune the speckle contrast bychanging the intensity range while keeping the functional formof the PDF. One example is given in Fig. 9. The two specklepatterns have distinct contrast, because their intensity PDFs havedifferent width. Narrowing the PDF in (a) by a multiplicativefactor and rescaling its height makes it overlap with the PDF in(b), confirming the two PDFs have an identical function form.

Letter 9

Fig. 8. A Rayleigh speckle pattern in (a) is transformed to the speckle pattern in (b) via a local intensity transformation to have alinearly increasing PDF. Application of a digital low-pass Fourier filter results in the pattern in (c). Multiple iterations of intensitytransformations and filtering result in the final pattern in (d) which obeys the desired intensity PDF and spatial frequency con-straints.

Fig. 9. (a,b) Two speckle patterns with different contrast, C= 0.42 in (a), and C = 0.20 in (b). Their intensity distributionshave similar functional form but different width. Narrowingthe PDF in (a) by a multiplicative factor of ∼ 3.5 and rescalingits maximum value, plotted by the dashed purple line in (b),coincides with the PDF of the speckle pattern in (b).

9. AXIAL EVOLUTION

An interesting property of the tailored speckles is their returnto Rayleigh statistics, as they axially propagate away from theFourier plane of the SLM. To study this, we make the Fresnelapproximation, namely, the axial propagation of a field pattern isapproximated by adding a quadratic phase to its spatial Fourierspectrum. Therefore, to measure the speckle pattern at a planeof distance z from the Fourier plane, we add a phase parabolaπz(x2 + y2)/λ f 2 to the SLM pattern and record the intensity atthe Fourier plane.

Experimentally, we validated that Fresnel diffraction equa-tion accurately described the axial evolution of the speckle fieldaway from the Fourier-plane of the SLM in our setup. First weplaced the CCD camera at the Fourier-plane of the SLM andrecorded a Rayleigh speckle pattern. Then we added a phaseparabola πz(x2 + y2)/λ f 2 to the SLM, and translated the cam-era over the corresponding distance z. The recorded imagesmatch the one taken at z = 0 without the additional phase. We

repeated this verification for different values of z within therange of ±4Rl .