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1Department of Physics of Complex Systems, Weizmann Institute of Science,Rehovot 7610001, Israel. 2Department of Physics, Indian Institute of TechnologyRopar, Rupnagar 140001, Punjab, India.*Corresponding author. Email: chene.tradonsky@weizmann.ac.il

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Rapid laser solver for the phase retrieval problemC. Tradonsky1*, I. Gershenzon1, V. Pal1,2, R. Chriki1, A. A. Friesem1, O. Raz1, N. Davidson1

Tailored physical systems were recently exploited to rapidly solve hard computational challenges, such as spinsimulators, combinatorial optimization, and focusing through scattering media. Here, we address the phaseretrieval problem where an object is reconstructed from its scattered intensity distribution. This is a key probleminmany applications, ranging from x-ray imaging to astrophysics, and currently, it lacks efficient direct reconstructionmethods: The widely used indirect iterative algorithms are inherently slow.We present an optical approach basedon a digital degenerate cavity laser, whose most probable lasing mode rapidly and efficiently reconstructs theobject. Our experimental results suggest that the gain competition between the many lasing modes acts as ahighly parallel computer that could rapidly solve the phase retrieval problem. Our approach applies to mosttwo-dimensional objects with known compact support, including complex-valued objects, and can be generalizedto imaging through scattering media and other hard computational tasks.

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INTRODUCTIONCalculating the intensity distribution of light scattered far from a knownobject is relatively easy: It is the square of the absolute value of the object’sFourier transform (1). However, reconstructing an object from itsscattered intensity distribution is generally an ill-posed problembecausethe phase information is lost and different choices of phase distributionsresult in different reconstructions. Fortunately, in many applications,additional prior information—e.g., the object’s shape, positivity, spatialsymmetry, or sparsity—can be exploited to remove extraneous phasedistributions and, hence, retrieve the original phase distribution andenable object reconstruction. Examples for applications can be foundin astronomy (2), short-pulse characterization (3), x-ray diffraction(4), radar detection (5), speech recognition (6), and imaging throughturbid media (7, 8).

For objects with a finite extent (compact support), a unique solutionto the phase retrieval problem almost always exists (up to trivial ambi-guities), provided that the scattered intensity is sampled at a sufficientlyhigh resolution (9). During the last decades, several algorithms forsolving the phase retrieval problem have been developed. These includethe Gerchberg-Saxton (GS) error reduction algorithm (10), hybridinput-output algorithm (11), relaxed averaged alternating reflections(RAAR) algorithm (12), difference map algorithm (13), and shrink-wrap algorithm (14) [see (15–17) for a modern review]. Unfortunately,these algorithms are based on iterative projections and hence are rela-tively slow even with high-performance computers (16).

An alternative approach to address computational challenges, suchas phase retrieval and other optimization problems, is through specif-ically tailored physical systems. These systems are not universal Turingmachines, namely, they cannot perform arbitrary calculations, but theycould, potentially, solve a specific class of problems very efficiently.Prominent examples for these systems are the d-wave machine thatfinds the ground state of a complicated Hamiltonian through quantumannealing (18–20), coupled lasers, degenerate optical parametric oscil-lators (OPO), and coupled polariton systems for solving various difficultoptimization problems (21–28). Solving hard problems with thesesystems could, potentially, prove advantageous in computation timeand resources over conventional computers (21, 29).

Here, we present and experimentally demonstrate a novel opticalsystem that can rapidly solve phase retrieval problems. It is based on adigital degenerate cavity laser (DDCL) (30, 31), in which two constraints,the Fourier magnitudes of the scattered light from an object and thecompact support, are incorporated. The nonlinear lasing process in thiscavity results in a self-consistent solution that best satisfies both con-straints. An upper bound of 100 ns was measured for the time neededby the DDCL to converge to a stable solution.

The physical mechanism that generates the nonlinear lasing processinDDCLs is similar to that of theOPO spin simulators (23, 24, 26). Boththe OPO simulators and the DDCLs have distinct advantages inperforming optimization, including extremely fast operation (21, 26),ability to avoid local minima when the pumping is increased slowlyenough (23, 25), and having a non-Gaussian wave packet (32). Thetwo fundamental physical processes in the system, which allow theDDCL system to find the optimal solution, are the phase locking ofthe lasers and the nonlinear mode competition in the cavity (33–35).The compact support aperture inside the cavity ensures that differentconfigurations of laser phases result in different losses and the con-figuration with theminimal losses wins the mode competition; thereby,the solution for the phase problem is found.

Our DDCL system has several particular attractive and importantfeatures. These include high parallelism that simultaneously providesmillions of parallel realizations; short round trip times (~20 ns), leadingto fast convergence time; and inherent selection of the mode withminimal loss as a result of mode competition.

RESULTSExperimental arrangementThe basic DDCL arrangement for rapidly solving the phase retrievalproblem is schematically presented in Fig. 1. It consists of a ring degene-rate cavity laser that includes a gain medium, two 4f telescopes, anamplitude spatial light modulator (SLM), an intracavity aperture, threehigh-reflectivity mirrors, and an output coupler. The left 4f telescope( f1 and f2) images the center of the gain medium onto the SLM, wherethe transmittance at each pixel is controlled independently (30). Theintracavity aperture, together with the SLM, serves to control and formthe output lasing intensity distribution (36). In the absence of the intra-cavity aperture, the right 4f telescope simply reimages the SLM backonto the gain medium, so all phase distributions can lase with equalprobability, i.e., the amplification and losses are phase independent.

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However, when an intracavity aperture (compact support mask) isplaced at the Fourier plane between the two lenses, each phase dis-tribution has a different level of loss. In this case, the phase distributionthat experiences theminimal loss is themost probable lasingmode, as aresult of mode competition.

Theoretically, two constraints are required to solve phase retrievalproblems with our system: (i) an SLM transmittance pattern that isdetermined by the intensity distribution of the Fourier transform ofthe object and (ii) an intracavity aperture having the size and shapeof the compact support. The transmittance of each SLM pixel is set toa local function of the Fourier intensity described in the theory section(Eq. 3). Because of laser cavity imperfections, we set the transmittance ofthe SLM using an iterative scheme outlined in section S1. The intracav-ity aperture size and shape, enforcing the compact support constraint,were predetermined in accordance to the known object size and shape.With both constraints of the phase retrieval problem implemented inthe cavity, the most probable lasing mode corresponds to the optimalsolution of the phase retrieval problem. The field distribution of anyother lasing mode spreads beyond the boundaries of the intracavityaperture and therefore suffers additional loss. The field distributionof the resulting lasing mode, i.e., of the reconstructed object, is formedwithin the intracavity aperture and is imaged through the output cou-pler onto the camera. In a typical application, the Fourier intensitydistribution and the compact support would both be provided as exter-nal inputs, for example, those obtained by x-ray diffraction experiments.

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All the experimental results reported in this work were obtained withsynthetic inputs generated in the following manner. Starting from adigitized object field, the Fourier intensity distributionwas calculated bya discrete Fourier transformation and then used to set the SLM trans-mittance. The compact support size and shape, commonly obtained bya low-resolution imaging of the object, were determined from the digi-tized object. The actual and detailed experimental arrangement that wasused in our experiments is presented in fig. S1.

Experimental resultsWe conducted extensive investigations on the performance of experi-mental system shown in fig. S1. We considered two figures of merit toquantify the quality of our system: solution fidelity and the computationtime. The fidelity of a solution intensity distribution is calculated as

Fidelity ¼ 1� 12‖IdetðxÞ � IorigðxÞ‖

2

2 ð1Þ

where Idet is the aligned normalized intensity of the detected recon-structed object (or Fourier distribution), Iorig is the normalized intensityof the original object (or calculated Fourier distribution), and ‖I(x)‖2 isthe L2 norm of an intensity pattern I(x). The use of the intensity-basedfidelity metric in Eq. 1 is justified by the uniqueness property of thephase retrieval problem solution (9). Specifically, the uniqueness prop-erty guarantees that the object phase distribution can be easily obtained

Fig. 1. Basic DDCL arrangement for rapid phase retrieval. (A) Calculated scattered intensity distribution from the object (essentially the Fourier intensitydistribution) is applied onto an SLM, which is incorporated into a ring degenerate cavity laser that can support up to 100,000 degenerate transverse modes. A maskshaped as the object boundaries (compact support) at the Fourier plane filters out extraneous modes that do not match the compact support. With this laser arrange-ment, the lasing process yields a self-consistent solution that satisfies both the scattered intensity distribution shown in (B) and the compact support constraint. (C) Thereconstructed object intensity appears at the compact support mask and is imaged onto the camera. a.u., arbitrary units.

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from the reconstructed intensity (10). This is verified in section S5 andfig. S6. We also show, in section S5 and fig. S6, that the Fourier phasedistribution can be directly measured in our system.

Representative experimental results, including the fidelity figures ofmerit, for different objects are presented in Figs. 2 to 5. In these figures,columns A show the intensity and/or phase distributions of the actualobjects; columns B (and columnC in Fig. 5) show the detected intensitydistributions of the reconstructed objects along with the compact sup-port outlines; columns C (columnD in Figs. 4 and 5) show the intensitydistributions at the SLM (calculated Fourier intensity distributions ofthe object after modification by the SLM properties), which controlledthe transmission of the SLM (see section S1); and column C in Fig. 4shows examples of calculated Fourier intensity distributions beforemodification by the SLM properties.

Representative results for centrosymmetric objects with uniformphase distribution (i.e., real-valued objects) and circular compact sup-port are presented in Fig. 2. As evident, there is very good agreementbetween the intensity distributions of the actual objects and those of thereconstructed objects. Figure 3 shows the results for objects with cen-trosymmetric intensity distributions and various complex phase dis-tributions and circular compact support. The first row shows the

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results for an object with uniform phase distribution, where thecorresponding Fourier intensity distribution has a 12-fold symmetry.The second row shows an object with centrosymmetric phasedistribution, so both the object and the corresponding Fourier intensitydistribution are centrosymmetric. Note that our system correctly recon-structs the actual objects, although the Fourier intensity distributions forthe first and second rows are notably different. The high-quality recon-structions (reconstruction fidelities of 0.81 to 0.91) indicate that our ap-proach is also valid for complex-valued objects, which are generallyharder to solve computationally (37).

A known ambiguity in phase retrieval emerges when the object isnoncentrosymmetric, but the assumed compact support is centro-symmetric. An example for such a case is shown in the third row ofFig. 3, where the object has a random asymmetric phase distributionso the corresponding Fourier intensity distribution is also asymmetric.Here, the reconstruction fidelity is only 0.81, and the blurred recon-structed object differs from the actual object because of interferencesbetween two degenerate solutions (one is the image of the object andthe other the inverted phase-conjugated image). Applying a properlydesigned, noncentrosymmetric compact support (for a noncentrosym-metric object) removes this degeneracy, so as to obtain a high-quality

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Fig. 2. Experimental results for real-valued centrosymmetric objects. Column (A) Intensity distributions of the actual objects. Column (B) Detected intensitydistribution of the reconstructed objects, using a circular aperture as compact support. Column (C) Fourier intensity distributions at the SLM.

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reconstruction with a fidelity of 0.91, as shown in the fourth row ofFig. 3. Note that the design of the noncentrosymmetric compactsupport requires knowledge about the asymmetry of the object.For example, in x-ray applications, this can usually be obtained bylow-resolution imaging of the object.

The effect of object complexity on the reconstruction fidelity wasmeasured and is presented in Fig. 4. The original objects consisted ofan array with an even number of spots and alternating phases of 0

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and p, arranged in a ring geometry. The number of spots ranged from4 to 30, while the size of all spots was kept constant. Each data pointin Fig. 4 is made up of the average fidelities of the input Fourier andreconstructed object intensity distributions obtained in 48 repetitionsfor each number of spots. Representative intensity distributions forobjects with 4, 16, and 30 spots are presented in the top part of Fig. 4.These indicate that higher-complexity objects (with more spots) have ahigher-complexity Fourier intensity distribution with smaller details

Fig. 3. Experimental results for complex-valued objects. Column (A) Intensity (brightness) and phase (hue) distributions of the actual objects. Column (B) Detectedintensity distribution of the reconstructed objects, using mainly a circular aperture as compact support. Column (C) Fourier intensity distributions at the SLM. The firstrow shows an object with uniform phase distribution with a reconstruction fidelity of 0.91. The second (third) row shows the same object with arbitrary centro-symmetric (asymmetric) phase distribution with a reconstruction fidelity of 0.89 (0.81). The fourth row shows a noncentrosymmetric object with random asymmetricphase distribution and noncircular compact support with a reconstruction fidelity of 0.91.

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Fig. 4. Experimental and quantitative results for fidelity as a function of object complexity. Top: Representative intensity distributions of objects with 4, 16, and30 spots. Column (A) Intensity (brightness) and phase (hue) distributions of the actual objects. Column (B) Detected intensity distribution of the reconstructed objects,using a circular aperture as compact support. Column (C) Calculated Fourier intensity distributions applied to control the SLM. Column (D) Detected correspondingFourier intensity distributions after modifications by SLM properties. Bottom: Quantitative fidelity values of the Fourier intensity distributions (blue) and the recon-structed object intensity distributions (red) as a function of the number of spots in the object (4 to 30). Inset: Fidelity values of the reconstructed object intensitydistributions as a function of the fidelity values of the Fourier intensity distributions for all the measurements.

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that cannot be properly resolved by our system. The quantitative ex-perimental results for the fidelity as a function of object complexity(number of spots) are shown in the bottom part of Fig. 4. As expected,both the input and reconstruction fidelities decrease as the object com-plexity increases. In addition, these fidelities are highly correlated (Pearsoncorrelation value of 0.76; see inset of Fig. 4). The fidelity correlationresults suggest that improving the fidelity between the calculated andmeasured input Fourier intensity distributions will improve the fidelityof the reconstructed object. Note that both fidelities exhibited substan-tial statistical spreading, quantified by error bars, around their meanvalues. These fluctuations are caused by technical noise arising fromthe laser pumping method (flash lamp), which could be alleviated byresorting to diode pumping.

The experimental results on the qualitative effect of tightness andsymmetry of the compact support on the reconstruction quality arepresented in Fig. 5. Column A shows the intensity distributions ofthe actual objects; columns B and C show the detected intensity distri-butions of the reconstructed objects along with the compact supportoutlines, and column D shows the intensity distributions at the SLM,which controlled the transmission of the SLM. The results in the firstrow demonstrate that a tight compact support (square rather than a cir-cular aperture) substantially improves the quality of reconstructedsquare object. However, since both the object and the support are cen-trosymmetric, even the nontight circular aperture leads to a reasonablereconstruction. The results in the second row demonstrate the impor-tance of centrosymmetry, as also shown in Fig. 3.

The quantitative effect of tightness of the size of the compact supporton the reconstruction quality and fidelity is presented in Fig. 6. It showsthe reconstruction fidelity as a function of the radius of the compactsupport aperture normalized by the object size and representative inten-sity distributions in insets A to C. As expected, when the object size isbigger than the compact support, there is a rapid decay in the recon-

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struction fidelity since the laser cannot support the correct object shape.When the object size is smaller than the compact support aperture,there is a slower decay in the fidelity due to overlap ofmultiple solutionsthat are supported by the aperture. Specifically, when the compactsupport is larger than the reconstructed object size, many translated so-lutions, known to be “trivial ambiguities,” are valid. Accordingly, thereconstruction fidelity is reduced when the camera averages overmultiple realizations, as in our system.

Typically, the resolution of the reconstructed objects was rela-tively low (about 20 × 20, corresponding to a spot size of 500 mm).As indicated by numerical simulations (see section S3 and fig. S4), theresolution is limited mainly because of phase aberrations in the lasercavity.We believe that reducing these aberrations, e.g., by proper com-pensation with the intracavity SLM, could improve the resolution.Complete elimination of phase aberrations should substantially en-hance the resolution to that of the SLM, which, in our system, wouldbe 500 × 500 (corresponding to a spot size of 20 mm). An additionallimiting factor on the reconstruction fidelity is the dynamic range ofthe input Fourier intensity distribution, which is often very high, e.g.,in x-ray applications. In the current system, the dynamic range is quitelimited. As already visible in Fig. 2, objects exhibiting a bright centralpeak in the Fourier distribution (the two upper panels in Fig. 2) tend tohave poorer reconstruction fidelity compared to objects where thecentral peak is dimmer (bottom panels in Fig. 2). The dynamic rangecould be improved by controlling the spatial distribution of the pumpintensity (end pumping).

Last, we analyze the time to solution of the system. In our system,two different processes contribute to the overall computation time. Thefirst process includes the overhead durations for forming the specificintensity pattern on the SLMand for detecting the solutionwith a camera.These durations are dictated by the SLM response time and the com-plementary metal-oxide semiconductor camera readout time and,

Fig. 5. Experimental results demonstrating the qualitative effect of tightness and asymmetry of compact supports. Column (A) Intensity distribution of theactual objects. Column (B) Detected intensity distribution of the reconstructed objects, using a circular aperture as compact support. Column (C) Detected intensitydistribution of the reconstructed objects, using a square aperture as tight compact support (top row) and a circular aperture with a wedge as asymmetric compactsupport (bottom row). Column (D) Fourier intensity distributions at the SLM.

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together, are about 20ms. The second important process is the actualcomputation time of lasing, which, as discussed below and in fig. S2,is less than 100 ns. Unlike most other computational devices, where theinput/output overhead duration is negligible compared to the com-putation time, in our system, it is the bottleneck in the total computationtime. Fortunately, this bottleneck can be alleviated (reducing overheaddurations to submilliseconds) by resorting to improved input/outputdevices, which are continuously becoming available. To determine anupper bound on the minimum duration needed for the laser to findthe optimal lasing mode and reconstruct the object, we resorted to aQ-switched linear degenerate cavity laser arrangement that included aPockels cell, as shown in fig. S2. The shortest pulse that we could gen-erate was 100 ns long, as shown in the pulse profile in fig. S2B, and eventhen, the laser reconstructed the object successfully. In general, theresults obtained with Q-switching were essentially the same as thosewith quasi–continuous wave (CW) lasing operation. This indicates thatthe computation time of the system is, at most, 100 ns. For comparison,we measured the median reconstruction time of the RAAR algorithmon the problem shown in the bottommost panels of Fig. 3 and found itto be about 1 s (see section S4 for additional details).

TheoryThe lasing mode in the system is a complex field at the SLM, Eð→k; tÞ,where

→k is the position at the SLM plane, mapped onto itself after pro-

pagating through the cavity. In other words, it is a stationary solution ofthe field propagation equation,which, for a cavity round trip of durationt, can be written as

Eð→k; t þ tÞ ¼ e�aTSLMð→kÞGð→k; tÞ F�1 Mðx→ÞF ½Eð→k; tÞ�h i

ð2Þ

where TSLMð→kÞ is the amplitude transmittances at the SLM; Gð→k; t Þ isthe (nonlinear) gain of the system; a is the loss coefficient per roundtrip; F is a two-dimensional Fourier transform (performed by thelenses); and Mðx→Þ represents the spatial compact support imposed by

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the intracavitymask, wherex→is the position at themask plane.Note that

the mapping ofEð→k; tÞ is nonlinear because of the nonlinear gain withsaturation (38)Gð→k; tÞ ¼ expðg0ð1þ ∣Eð→k; tÞ∣2=IsatÞ�1Þ, where g0 isthe linear gain at very low intensities set by the pumping strength andIsat is the saturation intensity.

Now, consider the electric field Esolð→kÞ, which corresponds to thesolution for the phase retrieval problem. This field passes through thecompact support without any changes,F�1½Mðx→Þ F½Esolð→kÞ�� ¼ Esolð→kÞ.Assuming that Esolð→kÞ is a stable, time-independent solution ofEq. 2, e�aTSLMð→kÞGsolð→kÞ ¼ 1. With this

TSLMð→kÞ ¼ exp a� g0ð1þ ∣Esolð→kÞ∣2=IsatÞ�1� �

ð3Þ

the solution of the phase retrieval is a possible lasingmode in the system.Note that the SLM transmittance,TSLM, is a local function of the Fourierintensity∣Esolð→kÞ∣2 and cavity parameters. Substituting the expressionsfor TSLMð→kÞ and Gð→k; tÞ into Eq. 2 yields

Eð→k ; t þ tÞ ¼

exp g01

1þ ∣Eð→k; tÞ∣2=Isat� 1

1þ ∣Esolð→kÞ∣2=Isat

! !

F�1 Mðx→ÞF ½Eð→k; tÞ�h i

ð4Þ

Equation 4 can be considered as a modified GS iterative projectionprocess, in which the fastest growingmode corresponds to the solution.To verify our approach,we performednumerical simulations of the cav-ity. The simulation results are presented in fig. S3, along with additionaldetails on the numerical simulations.

At the transition to lasing, the amplified spontaneous emission(ASE) modes with the highest energy win the mode competition overthe limited gain. In the initial growth stage of the electric field inside the

Fig. 6. Experimental quantitative results for reconstruction fidelity as a function of the compact support radius of the aperture normalized by the object size.Insets: Typical reconstructed object intensity distributions. (A) Compact support radius is 87% of the object radius. (B) Object radius is equal to compact support radius.(C) Compact support radius is 152% of the object radius.

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cavity, ∣Eð→k; tÞ∣2 of each ASE mode is extremely small. Therefore,Eq. 4 can be approximated as

Eð→k ; t þ tÞeexp g0∣Esolð→kÞ∣2=Isat

1þ ∣Esolð→kÞ∣2=Isat

! !F�1 Mðx→Þ F½Eð→k; tÞ�

h ið5Þ

Under this approximation, the round trip mapping is linear. Thefastest growingmode in this stage is hence the eigenmode of the linearmapping with the highest eigenvalue. When Isat >> ∣Esolð→kÞ∣2, Eq. 5is, to a good approximation, a projection on the compact support.Hence, allmodeswithin the support grow exponentially faster than oth-ermodes. Thus, the solutions of the phase retrieval problemare both thefastest growing modes in the initial stage and the stable lasing modes.Moreover, since the phase retrieval problem has a unique solution, itassures thatEsolð→kÞwould be the only stable lasing mode from all themodes with a certain compact support. Thus, if the solution is presentin the ASE, it is expected to be the lasing configuration of the system.

Unfortunately, there might be additional stable lasing modes in thenonlinear cavity that can lead to a wrong solution. Out of the possiblestable lasing modes, which one is chosen by the system as the actuallasing mode? Before the lasing transition, the gain operates in theincoherent ASE regime, where the number of different phase configura-tions is very large, as quantified below. Each of these configurationsevolves according to Eq. 4. Out of all these time-evolving phase config-urations, the one with the highest energy (smallest loss) wins the modecompetition over the limited gain. Therefore, the larger the number ofinitial independent configurations, the higher the probability of the sys-tem to find the correct solution, which is the unique stable configurationwith no losses on the compact support mask.

Let us therefore evaluate the number of different ASE phase realiza-tions inside the cavity. This number quantifies the independent parallel“computations” running simultaneously inside the cavity. These phaserealizations are first amplified independently, until the gain mediumapproaches saturation. The nonlinear mode competition leads to theselection of the optimal configuration. A phase configuration is definedby a set of N phases, one for each of the N spatial modes in the cavity(i.e., the phase at each pixel on the SLM). For ASE, each spatial modehas a coherence length dictated by the bandwidth. The length of the cavitydivided by this coherence length therefore dictates the number of differentphases in each spatial mode in the cavity, denoted by K, the number oflongitudinal modes. However, in ASE, the different spatial modes areindependent. For simplicity, we model the phase of each spatial modeto evolve as aPoissonprocesswith an average timebetweenphase changescorresponding to the coherence length. Under this assumption, for Nindependent spatial modes, the rate at which the phase configurationchanges isN times larger than that of a singlemode. Therefore, the overallnumber of different phase realizations in the cavity is the number of lon-gitudinal modes K times the number of spatial modes N, namely, KN.

In practice, our laser does not lase in a single longitudinal mode, al-though the number of modes decreases by a factor of 10 (39). For Nd–yttrium-aluminum-garnet (YAG) around 1064 nm, the ASE coherencelength is about 2 mm; hence, in our 5-m cavity, there are about 2500independent longitudinal modes. We estimate the number of spatialmodes by the number of pixels in the SLM to be ~106. Therefore, theoverall number of different initial conditions is ~109, and the number offinal realizations in the cavity is ~102.

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CONCLUSIONWe presented an optical system for rapid phase retrieval, using a novelDDCL. The computation time is bounded by 100 ns, orders of magni-tude faster than conventional computation systems (see, for compari-son, reconstruction timewith the RAAR algorithm in section S4). Note,however, that setting the scattered intensity distribution as the inputon the SLM could take at least a few milliseconds. A direct approach,which uses the scattered light from the unknown object as an on-axisstructured pump, could substantially speed up the process.

Several modifications to our DDCL system can potentially improvethe performance. These include overcoming phase aberrations byadaptive calibration of the intracavity SLM (40), improving the stabilityand repeatability of pumping by resorting to diode pumping instead offlash light pumping, pump ramping in time to improve conversion tothe ground state (23, 25), introducing coupling between longitudinalcavitymodes by either using an intracavity saturable absorber ormeasur-ing all different realizations inside the cavity and choosing the optimalone, and increasing the number of independent parallel realizationsusing a longer cavity and different gainmediumwith a broader spectrumof amplification (shorter correlation time in the ASE).

We believe that, in addition to rapidly finding solutions to phaseretrieval problems, our DDCL systems can be exploited for solving avariety of problems that occur in many fields, e.g., resolving imageryafter propagation through scattering media (8).

MATERIALS AND METHODSIn our experiments, we actually used a reflective phase-only SLM, ratherthan a transmissive SLM, as depicted in the experimental arrangementshown in Fig. 1. The reflective SLM had a relatively high light efficiencyand a high damage threshold. Accordingly, the laser arrangement wasmodified to retain the same operation functionality while accom-modating the reflective SLM. The detailed experimental arrangementthat includes the reflective SLM is schematically presented in fig. S1,along with an explanation on how a phase-only SLM together withthe intracavity aperture can control the amplitude transmittance of eacheffective pixel in the SLM (30).

In our experimental arrangement, the laser gain medium was a1.1% doped Nd-YAG rod of 10-mm diameter and 11-cm length. Forquasi-CWoperation, the gainmediumwas pumped high above thresh-old by a 100-ms pulsed xenon flash lamp operating at 1500 V and arepetition rate of 1 Hz to avoid thermal lensing. Each 4f telescope con-sists of two plano-convex lenses, with diameters of 50.8 mm and focallengths of f1 = 750 mm and f2 = 500 mm at the lasing wavelength of1064 nm. The SLMwasHamamatsu [liquid crystal on silicon (LCOS)–SLM X13138-03] with a high reflectivity of about 98% at 1064-nmwavelength, an area of 15.9 mm by 12.8 mm, 1272 × 1024 resolution,12.5-mm pixel size, and a high damage threshold. Compact supportaperture diameters ranged between 2 and 10 mm, depending on theobject size. For Q-switch operation, a Pockels cell was incorporatedinto a linear degenerate cavity laser of the same gain and pump asthe quasi-CWoperation, and the focal lengths of two lenses in the tele-scope were 250 mm. Additional details are presented in section S2.

SUPPLEMENTARY MATERIALSSupplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/10/eaax4530/DC1Section S1. Detailed experimental arrangement

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Section S2. Convergence time to reach a solutionSection S3. Simulation resultsSection S4. Runtime comparison to the RAAR phase retrieval algorithmSection S5. Phase measurement and reconstructionFig. S1. Detailed experimental digital ring degenerate cavity laser arrangement.Fig. S2. Experimental arrangement with corresponding results for demonstratingrapid solutions.Fig. S3. Representative simulation results of inverse problem solutions with a DDCL.Fig. S4. Representative simulation results with phase aberrations.Fig. S5. Reconstruction fidelity in the RAAR algorithm.Fig. S6. Phase measurement methods.References (41–44)

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Acknowledgments: We would like to thank Y. Eliezer for useful comments and discussions.Funding: This work was supported by the Israel Science Foundation (grant number: 1881/17)and the United States–Israel Binational Science Foundation (grant number 2105509). O.R. isthe incumbent of the Shlomo andMichla Tomarin career development chair and is supported bya research grant from Mr. and Mrs. Dan Kane and the Abramson Family Center for YoungScientists at WIS. Author contributions: All authors (C. T., I. G., V. P., R. C., A.A. F., O. R., and N. D.)contributed to writing the manuscript. C.T. designed the experimental arrangement and thenumerical simulations with contributions and input from V. P. and R. C. C.T. and I.G. performed

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the experiments and the data analysis with contributions and inputs from all authors. O.R. andI.G. developed the theory. Competing interests: N.D., A.A.F., O.R., and C.T. are inventors ona patent application related to this work assigned to Yeda Research and Development Co. Ltd.(application no. PCT/IL2019/050473). The authors declare no other competing interests.Data and materials availability: All data needed to evaluate the conclusions in the paperare present in the paper and/or the Supplementary Materials. Additional data related to thispaper may be requested from the authors.

Tradonsky et al., Sci. Adv. 2019;5 : eaax4530 4 October 2019

Submitted 25 March 2019Accepted 9 September 2019Published 4 October 201910.1126/sciadv.aax4530

Citation: C. Tradonsky, I. Gershenzon, V. Pal, R. Chriki, A. A. Friesem, O. Raz, N. Davidson, Rapidlaser solver for the phase retrieval problem. Sci. Adv. 5, eaax4530 (2019).

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Rapid laser solver for the phase retrieval problemC. Tradonsky, I. Gershenzon, V. Pal, R. Chriki, A. A. Friesem, O. Raz and N. Davidson

DOI: 10.1126/sciadv.aax4530 (10), eaax4530.5Sci Adv 

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