Enhanced phase retrieval using nonlinear dynamics

Jen-Tang Lu; Chien-Hung Lu; Jason W. Fleischer

doi:10.1364/OE.24.025091

1. Introduction

The phase of complex waves plays an important role in a variety of fields, ranging from x-ray crystallography [1] and astronomy [2] to wavefront sensing [3] and biomedical imaging [4]. Indeed, it is well known that phase contains most of the information of a signal [5], and that phase (frequency) modulation is more versatile and noise-resistant for signal transmission [6]. Consequently, there have been many methods devised to measure the phase of a wave. As electronic devices record only time-averaged intensity, and most illumination sources are not coherent, noninterferometric methods have become increasingly popular. These approaches typically take two (or more) intensity measurements and use a numerical algorithm to reconstruct the phase. The first such method was the Gerchberg–Saxton (GS) algorithm [7], where phase is iteratively reconstructed with two measured intensity distributions at two different measurement planes (typically near field and far field). However, systems with multiple measurement planes require either a beam splitter or mechanical translational stages, which increase the bulk of the setup and introduce potential alignment issues. In order to achieve phase retrieval with only one measurement plane, other forms of constraint or parameter diversity have been used, including knowledge of the pupil function [8], compact support of the field [9], multiple object orientations [10], transverse displacement [11], wavelength changes [12], binary objects [13], atomicity constraints [14], and differing polarization [15]. However, all of these methods considered only linear propagation or linear systems.

Phase algorithms using linear propagation have several limitations. Chief among these is the issue of convergence, as iteration is a “black box” method that does not provide an independent measure of the phase being retrieved. This makes linear methods susceptible to noise, conjugate solutions, and local minima in phase space [16]. There are also no guidelines for the boundary conditions, as initial phase guesses are random and no phase reference is available for stopping the iterations.

Here, we show that phase retrieval algorithms using nonlinear propagation can overcome these issues. In general, considering nonlinearity introduces a new degree of freedom, with the potential to surpass linear limits of imaging [17]. For signal transmission, nonlinear systems can be much more sensitive to phase inputs, as intensity-induced index changes upon propagation also contribute to phase evolution [17,18]. At the same time, noise can be reduced, as phase-matching conditions (conservation of wave energy and momentum) effectively act as a nonlinear filter [19,20]. In this paper, we show that nonlinear feedback leads to a remarkable convergence property in the iteration algorithm, and that the reconstructed phase error can be minimized by tuning the nonlinearity.

2. Phase retrieval algorithm

2.1 Nonlinear field propagation

We consider the propagation of continuous-wave beams in a nonlinear medium, such as a photorefractive crystal. To a good approximation, beam evolution can be described by the nonlinear Schrödinger equation:

\frac{\partial ψ}{\partial z} = [i \frac{1}{2 k} \nabla_{⊥}^{2} + i Δ n ({| ψ |}^{2})] ψ \equiv [D + N ({| ψ |}^{2})] ψ

where ψ is the slowly varying electric field, z is the optical axis, k = 2π/λ, Δn(|ѱ|²) is the nonlinear index change, and D and N are the linear and nonlinear operators, respectively. In this work, the nonlinear operator is chosen as the Kerr nonlinearity, with Δn = γ|ѱ|², where γ is a tunable nonlinear coefficient. With an input or output intensity and phase, one can calculate the propagation or reverse propagation of ѱ(x,y,z), e.g. via the Fourier split-step method, where the linear and nonlinear operators act individually for each increment of propagation distance [21]. In this work, a symmetrized split-step Fourier method is used:

ψ (z + d z) = e^{\frac{d z}{2} . D} e^{d z . N} e^{\frac{d z}{2} . D} ψ (z)

For backward propagation, we take dz → -dz: the complex field at location z_i, where z_i<z_f, can thus be computed as:

ψ (z_{i}) = e^{- \frac{d z}{2} . D} e^{- d z . N} e^{- \frac{d z}{2} . D} ψ (z_{f} = z_{i} + d z)

This reverse algorithm uses the output amplitude as the starting point for an initial value problem, which may be sensitive to the initial conditions but always gives a unique final solution.

We note that this system is in some senses foundational and can be generalized in several ways. For beams, paraxial propagation is assumed but not required [22], as more accurate evolution equations can be used as well, e.g [23]. For pulses, time evolution and pulse dispersion must be taken into account. Indeed, in the latter case, nonlinearity has been fundamental for phase retrieval, as nonlinearity enables the spectral beating of different pulse components [24] and the reconstruction of complex chirp [24,25]. Here, we use nonlinear mixing not only for self-reference but for amplification of phase/amplitude effects and bandwidth extrapolation [17,26] as well.

2.2 Pseudocode

A pseudocode for the algorithm is shown in Fig. 1. As only the light intensity exiting the nonlinear crystal is accessible, to reconstruct an unknown input phase we first measure two amplitude distributions at the exit plane. While any difference in nonlinearity will work [18], we consider the simplest case of one linear and one nonlinear output intensity: I_L = |A_L|² (N = 0) and I_NL = |A_NL|² (N≠0), respectively. With an initial phase estimate ϕ₀ = 0 (other phase estimates are also acceptable [18]), the algorithm proceeds as follows:

Fig. 1 Pseudocode for phase retrieval with nonlinear propagation.

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(1) A linear output field with amplitude A_L is back-propagated from the output (measured) face to the input face using D.
(2) The simulated field is then forward-propagated from the input face to the output face using D and N, yielding a simulated nonlinear output field with amplitude $A_{N L}^{'}$ .
(3) The simulated amplitude $A_{N L}^{'}$ is replaced by the measured nonlinear amplitude A_NL.
(4) The new field is back-propagated again to the input plane using D and N.
(5) The field is then forward-propagated from the input plane to the output plane using D, yielding a simulated linear output field with amplitude $A_{L}^{'}$ .
(6) The simulated amplitude $A_{L}^{'}$ is replaced by the measured linear amplitude A_L. At this point, the algorithm restarts from step 1.
(7) As the iterations stop, the reconstructed phase image is extracted at the input plane.

As a representative example, we consider a self-defocusing Kerr nonlinearity, with Δn/n₀ = -|γ|I, where n₀ = 2.3 is the base refractive index of the nonlinear crystal and I is light intensity. For each iteration, the error between computed amplitudes and measured amplitudes is determined by E_n = 0.5 * $\sum_{i j} (| A_{L} - A_{L}^{'} | / A_{L} + | A_{N L} - A_{N L}^{'} | / A_{N L})$ , where i, j denote the pixel position and $A_{L}^{'}$ and $A_{N L}^{'}$ represent the simulated amplitudes at the n^th iteration. Between two successive iterations, the convergence rate is defined as C = E_n-1 - E_n, while the error of reconstruction is defined using the L1 norm:

E R = \frac{\sum_{r \in S} | | g (r) | - | f (r) | |}{\sum_{r \in S} | f (r) |}

where g(r), f(r), and S represent the reconstructed image, the exact solution, and the image space, respectively.

2.3 Simulation

Simulations of phase retrieval, based on Eq. (1) and nonlinear propagation in a 1cm-long photorefractive Kerr-type crystal, are shown in Fig. 2. For input, we choose the amplitude and phase shown in Figs. 2(a) and 2(b): 0.72 mm $\times$ 0.72 mm models of a golf ball and Princeton Tiger, respectively. Figure 2(c) shows the simulated output amplitude after linear propagation and Fig. 2(d) shows nonlinear output for a defocusing Kerr nonlinearity Δn/n₀ = −1.1 * 10⁻⁵.For the propagation algorithm, a step dz = 200 μm was used (no significant variation was seen for step size dz = 100 μm, indicating numerical convergence and stability). As shown in Fig. 2(f), nonlinear phase reconstruction successfully retrieves most features of the Tiger and gives a reconstruction error ER = 0.43. In order to have a fair comparison, we kept the same propagation distance (1 cm) and performed the GS algorithm with the input amplitudes Fig. 2(a) and the linear output amplitudes Fig. 2(c). As shown in Fig. 2(e), the GS reconstruction is less sharp and displays more high-frequency noise. Quantitatively, the linear method gives a reconstruction error ER = 0.85, which is two times worse than the nonlinear reconstruction.

Fig. 2 Numerical demonstration of nonlinear phase retrieval. (a) Input amplitude image (golf ball). (b) Input phase image (Princeton Tiger). (c) Amplitude of simulated linear output (d) Amplitude of simulated nonlinear output with ∆n/n₀ = −1.1 * 10⁻⁵. (e,f) Reconstructed phase image with the GS algorithm for (e) linear and (f) nonlinear propagation.

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It is important to emphasize that both algorithms are inherently nonlinear, independent of the type of propagation, since the change in amplitude upon substitution at iteration n depends on the magnitude of the amplitude at iteration n-1. The substitution of measured amplitudes for computed ones enforces the experimental boundary conditions; in turn, these values become initial conditions for the subsequent propagation (reverse propagation) step. It is this stage of the algorithm that depends sensitively on whether or not the propagation itself is nonlinear. Indeed, the introduction of nonlinearity into the propagation equation is known to be a singular perturbation, enabling solutions that can be distinctly different than those available in linear systems. Consequently, the behavior of phase retrieval can be distinctly different as well.

In addition to better final performance, the advantage of nonlinear dynamics can be seen by following the algorithms as each is iterated. Figure 3 shows the convergence characteristics for linear (GS) and nonlinear propagation. While both algorithms display a monotonically decaying convergence, there is a subtle but significant difference in the profiles. In the linear case Fig. 3(a), C(n) asymptotically approaches zero; in the nonlinear case Fig. 3(b), C(n) crosses zero at a finite iteration number and continues to decay (indicating an unstable oscillation in amplitude, for at least some pixels). In a more traditional, positive-definite measure of amplitude error, |C(n)|, the singular nature of the nonlinearity is more pronounced (Fig. 3(d)).

Fig. 3 Convergence characteristics of phase retrieval. (a-d) Convergence of the phase retrieval algorithm for (a,c) linear and (b,d) nonlinear propagation. (a,b) are in a linear scale and the red dashed lines represent zeros, while (c,d) show the absolute value of the convergence in a logarithmic scale. (e,f) Reconstructed phase error of the GS algorithm for (e) linear and (f) nonlinear propagation. (g) Phase reconstruction error vs. noise for linear (blue) and nonlinear (red) propagation.

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The corresponding behavior of the reconstructed phase error ER(n) is shown in Figs. 3(e) and 3(f). For linear propagation (Fig. 3(e)), ER also decays monotonically [16], with an error that does not change significantly after iteration number 20 (though there is no way of knowing this in a real system with unknown phase; that is, it is difficult to determine where to stop the iterations to minimize the phase error and avoid extended computation with diminishing returns). In contrast, the ER profile for nonlinear propagation, shown in Fig. 3(f), is distinctly non-monotonic. The convex profile tracks the dip of |C(n)|, with the phase error reaching a minimum (0.43) exactly at the zero crossing of C.

Nonlinear dynamics give imaging methods that are both more sensitive to signal and less sensitive to noise [17,19,20]. To demonstrate this for the phase retrieval here, we examine the algorithms under the condition of random salt-and-pepper noise added to the input amplitude. As shown in Fig. 3(g), nonlinear propagation gives a reconstructed phase error that is both lower and less susceptible (gentler slope) than the corresponding algorithm with linear propagation.

The robustness of nonlinear propagation to noise arises primarily from the defocusing nature of the response. While cross-phase interaction can lead to modulation [27] and snake [28] instabilities, their appearance and growth is normally less volatile than modulation instability in the self-focusing case. Indeed, while self-focusing nonlinearity can enhance phase retrieval as well (Figs. 4(a) and 4(c)), the algorithm can react uncontrollably from just numerical discretization alone (Figs. 4(b) and 4(d)).

Fig. 4 Convergence (a,b) and reconstructed phase error (c,d) of the nonlinear algorithm with self-focusing nonlinearity of (a,c) ∆n/n₀ = 2.5*10⁻⁶ and (b,d) 3.5*10⁻⁶.

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In general, the nonlinear advantages are optimized for intermediate propagation distances, where the GS algorithm is less efficient [29] (it works best in the far field) and the expense of extra nonlinear computation is not too burdensome. To see this, one must compare not only the number of iterations to achieve a certain convergence level but also the number of computations required per iteration. For example, to achieve the convergence dip (10^-4.5) in Fig. 3(d), the nonlinear algorithm takes only 16 iterations, while the GS algorithm needs about 200 iterations to reach the same value of C. Both require a Fourier (or fractional Fourier) transform to calculate diffraction, but nonlinear propagation requires extra steps to accumulate the intensity-dependent phase. In turn, these extra calculations depend on the length of the crystal and the numerical step size. In this case (1-cm long crystal and 200 micron step size), there are 50 steps in each iteration for the nonlinear algorithm, and it takes ~4 times longer to include nonlinear propagation. It is up to the user to decide if the improvement in accuracy and robustness to noise is worth the extra processing time.

3. Experiment

We experimentally demonstrate nonlinear phase retrieval with a 2 × 5 × 10 mm³ SBN:75 (Sr_0.75Ba_0.25Nb₂O₆) photorefractive crystal. For weak illumination, the index change in this crystal can be approximated as a Kerr nonlinearity: ∆n = (1/2)n₀³γ₃₃E_appI, where n₀ and γ₃₃ are the base (linear) index and extraordinary electro-optic coefficient, respectively, E_app is an electric field applied across the crystalline c-axis, and I is light intensity normalized to background (dark current) intensity [30,31]. The experimental setup is shown in Fig. 5. An input phase was created by modulating an extraordinarily polarized plane wave with a spatial light modulator (Holoeye HEO 1080P), where a grating profile was produced and the first-order diffracted beam was imaged onto the input plane of the crystal via a 4-f system. For simplicity, we considered a phase-only input of the Princeton Tiger pattern (1.35 mm × 1.35 mm). An imaging lens with 1.25 magnification and a CCD camera were used to record the intensity at the crystal output with and without an applied field. To use the latter measurement in the conventional GS algorithm, for comparison, we experimentally recorded the input amplitude as well.

Fig. 5 Experimental setup for nonlinear phase retrieval.

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The linear output, shown in Fig. 6(a), was obtained without applying a voltage across the crystal.The nonlinear output, shown in Fig. 6(b), was obtained with an electric field of −350V/cm (corresponding to a maximum index change of Δn/n₀ = −8.7 × 10⁻⁶). The nonlinear output is more diverged than the linear one, because of the self-defocusing effect, and shows extra interference fringes, due to nonlinear wave mixing (esp. noticeable for the eyes and nose/mouth). Figures 6(c) and 6(d) show the linear and nonlinear reconstruction, respectively. As in the simulations, the nonlinear reconstruction is sharper, with more successful reconstructions of the small features of the tiger. It also has noise that has both lower intensity and lower spatial frequency than the linear case. Quantitatively, the nonlinear method gives a 17% improvement in L1 error (ER = 1.25 vs. 1.50).

Fig. 6 Reconstruction of a Princeton Tiger phase image from experimental measurements. (a) Amplitude of measured linear output (b) Amplitude of measured nonlinear output of −350V/cm (c,d) Reconstructed phase image for (c) linear and (d) nonlinear propagation.

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Figure 7 shows the convergence characteristics of linear vs. nonlinear phase reconstruction. As shown in Figs. 7(a) and 7(c), the GS algorithm still possesses a monotonically decaying convergence (asymptotically approaching zero) between the measured and numerically calculated amplitude. Interestingly, the phase error, shown in Fig. 7(e), experiences a minimum at iteration 5, after which it increases monotonically. For a real-world (vs. test) system, this behavior would be inaccessible, without any indication of where to stop the iterations. In contrast, the nonlinear reconstruction, shown in Fig. 7(b), has a zero-crossing convergence profile, which leads to a convergence dip in the L1 measure log(|C(n)|) (Fig. 7(d)). In this case, the dip in convergence occurs near, but not at, the minimum in reconstructed phase error (as shown in Fig. 7(f), iteration 20 vs. 16, corresponding to ER 1.2525 vs. 1.2461, a difference of 0.5%).

Fig. 7 Experimental convergence characteristics. (a-d) show the experimental convergence for (a,c) linear and (b,d) nonlinear propagation. (a,b) are in a linear scale and the red dashed lines represent zeros, while (c,d) show the absolute value of the convergence in a logarithmic scale. (e,f) Corresponding reconstructed phase error using (e) linear and (f) nonlinear propagation.

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4. Discussion

To explain the discrepancy between minima in Fig. 7 (also apparent in the self-focusing case of Fig. 4), we must consider the dynamics of nonlinear propagation in more detail. In nonlinear media, the intensity changes the refractive index of the medium and thus affects the phase of the wavefront. In turn, the modified phase gives a change in intensity upon propagation (from linear diffraction). The feedback gives a resonance between amplitude and phase, determined physically by phase matching between the mixing waves. Mathematically, the interplay can be seen from the two operators in Eq. (1), which is codified in the split-step method used to advance the linear and nonlinear terms. For the problem here of reconstructing an unknown phase, the resonance gives a sharp dip in intensity error when agreement occurs between the measured and calculated amplitudes in the iteration loop.

Since different nonlinearities lead to different output amplitude distributions, both the type and strength of the nonlinearity for phase retrieval play a key role in the reconstruction process. To examine this further, we keep the same Kerr-like response as above and investigate the reconstruction as a function of varying strength. As a baseline, we use the linear output amplitude A and consider the deviation obtained for differing nonlinear outputs B. For a concrete metric, we use the Pearson product-moment correlation coefficient [32]:

r = \frac{\sum_{i j} (A_{i j} - \overset{}{\bar{A}}) (B_{i j} - \overset{}{\bar{B}})}{\sqrt{\sum_{i j} {(A_{i j} - \bar{A})}^{2}} \sqrt{\sum_{i j} {(B_{i j} - \bar{B})}^{2}}} .

For the Princeton Tiger phase-only input, we numerically investigate the convergence characteristics for different correlations: r = −0.2 ~0.7. Figures 8(a) and 8(b) show the convergence and reconstructed error for r = −0.2, 0, and 0.7 (obtained using Δn/n₀ = −1.5 × 10⁻⁵, −1.0 × 10⁻⁵, and −4.0 × 10⁻⁶). For each case, there is a distinct dip in the convergence profile that occurs at or near the minimum in phase error. As shown in Fig. 8(c), the lowest reconstructed phase error is ER = 0.55, obtained when the correlation r = 0.1, compared to ER = 0.90 and 0.84 for r = −0.2 and 0.7, respectively. In other words, the optimum value occurs when there is nearly no overlap between the linear and nonlinear wavefunctions, corresponding to near-total information diversity between the linear and nonlinear outputs (some overlap must remain to keep the outputs correlated). A double resonance is therefore possible, allowing the minimum in amplitude error and the minimum in phase error to align at the same iteration number. This simultaneously gives a rule to stop iteration and a method of optimizing the reconstructed phase.

Fig. 8 Numerical simulations of phase retrieval as a function of nonlinear coupling strength. (a) Convergence and (b) Reconstructed phase error of the nonlinear algorithm for r =0.7 (blue), 0 (red), and −0.2 (green) (c) Relation between reconstructed phase error and correlation r.

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The region around each phase minimum is broad and flat. There is thus a significant margin for improvement beyond reconstruction with linear propagation, even if the improvement is sub-optimal. This range, and the generally flat response to noise (Fig. 3(g)) implies that the system is also robust to (spatial) inhomogeneity in nonlinearity as well. From another perspective, the ability to reconstruct phases for known objects is a sensitive way to characterize materials and their responses [21,31,33]. In the Kerr case, at least, the broad convergence curves suggest that the same reconstruction parameters are applicable to objects that are similar to each other. That is, the system provides a means of grouping (phase) objects into related classes, and the possibility of using a fixed system, without tuning, to recover these classes.

Likewise, it is clear that nonlinear propagation should benefit other methods of phase retrieval and diffraction imaging as well, such as ptychography [34], difference-map algorithms [35], and RAAR [36]. It is expected that the algorithms proposed here should work in an ever-increasing variety of fields, as more and more media become available with high sensitivity and/or wideband nonlinear response [37].

5. Conclusion

We have proposed a phase retrieval algorithm based on nonlinear propagation. Intensity-dependent changes to the index of refraction link amplitude with phase, making the method more sensitive than those using linear propagation. At the same time, a resonance condition arising from phase matching (energy and momentum conservation) makes the method less susceptible to noise. Further, nonlinearity creates a large amplitude response, altering the steady, asymptotic convergence typical of linear propagation. The result is a zero crossing in the convergence profile of the observable amplitudes (or cusp when absolute values are taken). With suitable nonlinear tuning, the minimum in amplitude variation and the minimum in phase error occur at the same iteration number. The proposed algorithm generalizes the conventional Gerchberg-Saxton algorithm to include nonlinear propagation, provides a clear stopping point for iterations, enhances reconstruction of phase, and is robust to noise and aberrations.

Funding

This work was supported by DARPA and the AFOSR.

Acknowledgments

The authors thank Prof. J. Nathan Kutz and Dr. Christopher Barsi for valuable discussion.

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Enhanced phase retrieval using nonlinear dynamics

Abstract

1. Introduction

2. Phase retrieval algorithm

2.1 Nonlinear field propagation

2.2 Pseudocode

2.3 Simulation

3. Experiment

4. Discussion

5. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (8)

Equations (5)

Optics Express