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DiffRayve: Differentiable Ray-Wave Method for Polarized and Unpolarized Diffractive-Refractive Optical Systems
Samuel Audia, Shrey Patel, Dinesh Manocha, Matthias Zwicker
TL;DR
Existing optical models struggle to balance computational efficiency and physical accuracy for complex diffractive and polarized systems. DiffRayve uses differentiable SBR to model geometric and physical optics together, and its multiple-DOE wide-field lens demonstration halves chromatic aberration.
Problem
Existing optical models trade physical accuracy for speed, and current methods have limitations for wave effects in compound diffractive and polarized systems.
Source: §1 · p. 1Method
DiffRayve uses a fully differentiable SBR propagation model to jointly capture geometric- and physical-optics behavior across refractive and diffractive elements.
Source: §1 · p. 2Results
DiffRayve’s multiple-DOE wide-field system halves chromatic aberration, and analytical-pattern comparisons show 10 to 30 dB higher peak signal-to-noise ratios than other baselines.
Source: §1 · p. 2Takeaways & Limitations
The method is demonstrated for wide-field achromatic multiple-DOE lens design and for end-to-end lens and classifier codesign using polarized scattering.
Source: §6 · p. 8SBR’s surface-current approximations cannot characterize discontinuities on optical elements that are on the order of a wavelength.
Source: §6 · p. 8
Abstract
from arXiv · showhide
Current deep optics simulations struggle to balance high fidelity with computational efficiency, particularly for complex polarized and diffractive systems. To address this, we introduce a fully differentiable Shooting and Bouncing Ray (SBR) algorithm that simultaneously models geometric and physical optics fields. Unlike standard wave optics methods, our approach enables accurate, gradient-based optimization of polarized compound refractive-diffractive systems without the computational cost of full-wave simulations. We validate our method against analytical Fraunhofer diffraction patterns and state-of-the-art Fourier optics and wave optics methods. We showcase the practical utility of our differentiable engine through one application: designing a wide field-of-view achromatic lens with multiple Diffractive Optical Elements (DOEs).
1. Introduction
Existing optical models trade physical accuracy against speed and struggle with wave effects in demanding lens systems. The paper introduces an SBR-based differentiable method for optimizing compound refractive-diffractive systems with polarized or unpolarized light.
1. Introduction
- The method targets a gap in which existing models increasingly fail to capture wave effects needed by demanding lens systems.Geometric, physical, and Fourier optics are common approaches, but existing models simplify propagation for speed or have limitations for desired systems.Source: §1 · p. 1
- Compound refractive-diffractive design matters for miniaturization and broadband performance, while polarization-aware lens design has been underexplored because of solver limitations.DOEs can be designed for negative dispersion to improve chromatic correction while reducing system weight and thickness.Source: §1 · p. 1
- DiffRayve traces geometric-optics rays, calculates equivalent currents at intersections, propagates currents to the sensor, and optimizes DOE height maps.The illustrated system is not a restriction: the approach allows any combination of refractive and diffractive elements.Source: Figure 1 · p. 2
Main Results
- SBR combines geometric and physical optics using surface equivalence to capture wave effects without FDTD’s prohibitive computational cost.The authors report that SBR had not previously been used for optimization-based optical engineering.Source: §1 · p. 2
- DiffRayve’s multiple-DOE wide-field system halves chromatic aberration, while its analytical-pattern comparisons achieve 10 to 30 dB higher peak signal-to-noise ratios than other baselines.The chromatic-aberration measure is the weighted mean and variance of the point spread function.Source: §1 · p. 2
2. Related Work
Differentiable optics supports gradient-based design, but existing geometric, Fourier, and hybrid approaches have distinct limitations for wave effects, wide fields of view, polarization, and DOE placement.
2. Related Work
- First-order optimization uses simulation gradients for fast convergence in high-dimensional designs, unlike zero-order approaches that become infeasible for DOE phase masks with thousands of parameters.The passage gives stochastic gradient descent as an example of a first-order optimizer.Source: §2 · p. 2
- Differentiable geometric optics neglects wave effects, while differentiable Fourier optics struggles with the paraxial approximations required for large fields of view.The passage describes a resulting shift toward hybrid models of ray and wave behavior.Source: §2 · p. 2
- Earlier hybrids either use a neural surrogate for metalens propagation with ray-conversion costs and accuracy limits [38], or restrict diffractive modeling to a phase mask at the ray-to-wave transition [12] [35].The passage also notes that Ho et al. [12] ignore DOEs and are limited to refractive optics.Source: §2 · pp. 2–3
- Prior hybrids restrict DOE placement to the final element and cannot model polarized light, whereas Fourier optics is limited at wide fields and geometric optics ignores diffraction.The cited comparison is summarized in Table 1.Source: §2 · p. 3, Table 1 · p. 3
3. Shooting and Bouncing Ray Method
The SBR method approximates the electric field through geometric-optics propagation, surface-current computation, and scattering those currents to the sensor.
3. Shooting and Bouncing Ray Method
- SBR iterates three steps: propagate geometric-optics waves, compute physical-optics currents at surfaces, then scatter the currents to the sensor using the EFIE.The method combines techniques to efficiently approximate the electric field.Source: §3 · p. 3
- Incident plane-wave fields are discretized as geometric-optics rays that track phase and polarization as they traverse changing refractive indices and Fresnel coefficients.The initial field intensity and polarization, propagation direction, and spatial coordinate describe the rays.Source: §3 · p. 3
- Equivalent electric and magnetic currents are computed at optical-element boundaries, and the EFIE gives the scattered field at a point from those currents.Unlike other hybrid methods, the currents are computed along all optical-element boundaries, allowing multiple DOEs to be modeled.Source: §3 · p. 3
4. DiffRayve: Differentiable Ray-Wave
DiffRayve adapts SBR with near-field scattering and automatic differentiation of Monte Carlo-accelerated propagation. Its pipeline alternates ray tracing, equivalent-current calculation, and field accumulation at sensor pixels until rays leave the system.
4. DiffRayve: Differentiable Ray-Wave
- DiffRayve combines near-field scattering and automatic differentiation with an SBR pipeline that repeats ray tracing, surface-current calculation, and sensor-field accumulation.The repeated stages continue until rays have left the system.Source: §4 · p. 3, Figure 2 · p. 4
4.1. SBR Simplification of EFIE
- Lens systems require the radiative near-field form of the scattering equation because the far-field plane-wave approximation used by most SBR implementations is insufficient.The method adapts the near-field integral for Monte Carlo integration over ray paths.Source: §4.1 · p. 3, §4.2 · p. 4
4.2. Monte Carlo Near-Field
- Monte Carlo integration estimates the scattered field from sampled ray paths, with path probabilities reflecting the probabilities of their ray samples.Rays use stratified sampling, with a wavelength-sized stratum balancing performance and accuracy; reflection or refraction choices are weighted by average Fresnel reflection coefficients, and paths are truncated at twice the number of optical elements. The cited SBR work makes computational and memory complexity linear rather than exponential in the number of optical elements.Source: §4.2 · p. 4
4.3. Differentiable Shooting and Bouncing Ray
- Gradients are computed by automatic differentiation of Equation 5, which the authors report works well for lens design despite known limitations.They attribute this performance to low-dimensional light paths and limited discontinuities when optimizing DOEs.Source: §4.3 · p. 4
4.4. Polarized and Unpolarized Light
- The unpolarized-light intensity combines the scattered fields from two assumed-incoherent orthogonal incident polarizations, V and H.The resulting intensity is calculated from the sum of the squared field magnitudes for the two polarization states.Source: §4.4 · p. 5
4.5. Diffractive-Refractive Element Models
- Refractive and diffractive elements use 3D triangular meshes, with diffractive surfaces modeled as pixel height maps or binary phase masks.Pixel height maps offset mesh vertices using a 2D texture; binary phase masks use learnable coefficients in a radial polynomial representation.Source: §4.5 · p. 5
4.6. Sensor Model
- The sensor model places pixels on a rectangular grid specified by width, height, pixel counts, and sensor orientation.The text states that other sensors could be included by differentiable post-processing of the calculated electric field.Source: §4.6 · p. 5
5. Results
Section 5 validates SBR against optical references, then applies it to a multi-DOE wide-field-of-view lens. The demonstrations cover scalar and polarized scattering and lens optimization.
5.1.1. Scalar Wave Optics Scattering
- For scalar scattering, SBR strongly matches the analytical Fraunhofer solution; Figure 3 reports PSNR comparisons with Chromatix and DeepLens, while GO predicts no diffraction.The validation covers a circular lens and a sinusoidal phase grating.Source: §5.1 · p. 5, Figure 3 · p. 6
5.1. Validation
- The validation compares normalized 550 nm point-spread functions with Fourier optics, a ray-wave method, geometric optics, and the analytical solution.The analytical solution serves as ground truth.Source: §5.1 · p. 5
5.1.2. Polarized Wave Optics Scattering
- For a polarized field through a circular plano-convex lens, DiffRayve captures the field structure, although Chromatix has higher PSNR values in Figure 4.Both methods qualitatively match the analytical solution.Source: §5.1.2 · p. 5, Figure 4 · p. 6
5.2. Multiple Diffractive Element Optimization
- The optimization uses a binary DOE, a plano-convex refractive lens, and a pixel-mask DOE to focus light on the sensor.The binary DOE has 5 learnable parameters and the pixel mask has 214.Source: §5.2 · p. 6, Figure 5 · p. 7
- Error nearly halved for 8° PSFs and decreased 66% at 4°, while overall system error fell from 0.4 to 0.25.After optimization, the PSFs condensed toward a white dot where the three color channels align.Source: §5.2 · pp. 6–7, Figure 6 · p. 7
- The optimized pixel phase mask resembles multiple Kinoform lenses centered at angles of interest, and gradients quickly achieve the optimization objective.The reported improved PSFs also hold at angles outside the training dataset, and Appendix E demonstrates vector-field optimization.Source: §5.2 · pp. 7–8
6. Conclusion, Limitations, and Future Work
DiffRayve applies differentiable SBR to optical lens design, modeling refractive, diffractive, and polarized light behavior. Its scope is bounded by computational costs for unpolarized systems and difficulty characterizing wavelength-scale discontinuities.
6. Conclusion, Limitations, and Future Work
- DiffRayve combines geometric and physical optics in a differentiable method for optical lens design and demonstrates optimization of a multiple-DOE wide-field system.The work also demonstrates end-to-end codesign of a lens and classification network using polarized scattering.Source: §6 · p. 8
- Unpolarized systems currently require two ray traces per wavelength, while wavelength-scale discontinuities may require higher-fidelity methods.The authors note that simpler refractive systems may be better served by existing solvers until ray-trace optimizations are realized.Source: §6 · p. 8
A. DiffRayve Details
- The appendix reports that implementation choices do not influence the main-paper results and says the authors plan to pursue open-sourcing the simulation upon acceptance.Source: A. DiffRayve Details · p. 10
A.1. Near Field EFIE
- For lens design, the near-field derivation retains the inverse-distance term because it exceeds the inverse-square and inverse-cubic terms by 3 and 6 orders of magnitude at 550 nm and 100 µm.The retained term is used in the radiative near-field integral for propagation to the sensor.Source: Figure 7 · p. 10, §A.1 · p. 10
A.2. Scaling
- Scaling the system and lowering frequency by the same factor preserves constructive and destructive interference; PSF normalization removes the resulting field-scale factor.The scaling addresses loss of computed-field detail for a 3 mm lens at 550 nm.Source: §A.2 · p. 10
A.3. Monte Carlo Sampling Probabilities
- Near-normal incidence weights Monte Carlo ray decisions toward refraction because transmission coefficients exceed reflection coefficients.Source: §A.3 · p. 10
A.4. Bounce Count
For normalized PSFs, the sensor field is well characterized once rays can reach the lens system’s final surface; additional bounces increase runtime and may be needed for throughput.
A.4. Bounce Count
- Once rays can reach the final lens surface, the sensor field is well characterized; additional bounces did not greatly influence the result.The comparison considered maximum path lengths of 1, 2, 4, 6, and 8.Source: §A.4 · p. 10, Figure 9 · p. 11
- Computing light throughput without normalizing the PSF may require more bounces than characterizing a normalized PSF.Source: §A.4 · p. 11
- For normalized PSFs, the authors recommend setting the maximum bounce count to twice the number of optical elements so all surfaces are intersected without unnecessary runtime.Additional bounces increase runtime.Source: §A.4 · p. 11
A.5. Sampling Density
Sampling density controls incident-plane-wave ray launches: higher density can improve sampling, but runtime rises quadratically, and one sample per wavelength is the paper’s chosen balance.
A.5. Sampling Density
- Fewer samples are sufficient for smoothly varying surfaces under near-normal incidence, but the required density is difficult to know in advance.Density is set along one dimension, so the sampled area scales as its square.Source: §A.5 · p. 11
- A density of 1 sample per wavelength is the chosen balance between performance and adequate sampling across geometric surfaces.The comparison reports that the circular-lens PSF can be resolved with very few samples.Source: Table 3 · p. 11, §A.5 · p. 11, Figure 10 · p. 11
- Ray count and runtime increase quadratically with sampling density, so higher density quickly raises simulation cost.Source: Table 3 · p. 11, §A.5 · p. 11
B. Multiple DOE Training Details
The multiple-DOE optimization minimizes PSF spread across incident angles, three wavelengths, and polarizations using normalized sensor-plane PSFs and gradient updates.
B. Multiple DOE Training Details
- The optimization shrinks PSFs for 350 nm, 550 nm, and 750 nm to reduce chromatic aberration, using SGD with a 1e-3 learning rate for 9 updates.The learning rate is halved after 7 updates, and momentum 0.9 gave the best results.Source: B. Multiple DOE Training Details · p. 11
- Each gradient step covers 16 incident angles, three color channels, and orthogonal polarizations, requiring 96 SBR forward simulations and backpropagations.At 1 ray per wavelength, each simulation propagates an average of 4 million geometric-optics rays.Source: B. Multiple DOE Training Details · p. 11
- The loss weights normalized pixel intensity by radial distance and adds a variance term to further reduce PSF spread.The PSFs are evaluated on a 10 × 10 grid with 1.6 µm pixel pitch.Source: B. Multiple DOE Training Details · pp. 11–12
C. Out of Distribution Angles
Tests at angles excluded from optimization show improved, more condensed PSFs, with stronger performance inside the trained field of view. The section also details analytical references and the Chromatix baselines used for comparison.
C. Out of Distribution Angles
- Unoptimized angles also show improved, more condensed PSFs, with stronger gains within the trained field of view and only slight improvement at wider angles.The authors attribute weaker gains at wider angles to less geometric overlap with the training set.Source: C. Out of Distribution Angles · p. 12, Figure 11 · p. 12
D. Baseline Details
- The baseline-details section describes Chromatix and DeepLens simulations, including small code modifications to make some inputs equivalent.The supplied passage introduces these baseline details but does not specify the modifications.Source: D. Baseline Details · p. 12
D.1. Fraunhofer Diffraction Equations
- The analytical references include an Airy-disk solution for a circular lens and a known Fraunhofer solution for a sinusoidal phase grating.The circular-lens setup uses a 1mm diameter, 10mm focal length, refractive index 1.5, and 550nm incident wavelength.Source: §D.1 · p. 12
D.2.1. Circular Lens
- Chromatix circular-lens simulations use an x-polarized 550nm wavefront on a 128 × 128 grid, with a circular pupil and high-NA lens model.The lens is 1mm in diameter, has a 10mm focal length and refractive index 1.5; the sensor is about 27µm with 62 × 64 resolution.Source: §D.2.1 · p. 13
D.2.2. Sinusoidal Grating
- The Chromatix grating is modeled as a linear operator and propagated by convolutional Fresnel propagation to a sensor 100µm away.Its half-width is 5µm, its frequency is 0.5µm−1, and the incident wavefront uses an 8400 × 8400 grid.Source: §D.2.2 · p. 13
D.3.1. Circular Lens
The DeepLens circular-lens baseline uses a hybrid ray-wave configuration because geometric optics alone does not produce the characteristic sensor-plane PSF. Its grating baseline instead propagates a phase-modified wavefront to the sensor.
D.3.1. Circular Lens
- For the circular-lens experiment, DeepLens uses a 50mm plano-convex lens with a 500mm curvature radius and a 64 × 64 sensor at the 1m focal length.These parameters are scaled up by a factor of 100 from the Diffrayve and Chromatix setup.Source: §D.3.1 · p. 14
- DeepLens geometric optics sends rays to one sensor point rather than producing the characteristic PSF, so the circular-lens baseline adds an identity-phase DOE and switches from rays to a wavefront.Monte-Carlo integration forms the complex wavefront at the DOE, which is then propagated to the sensor using the angular spectrum method.Source: §D.3.1 · p. 14
D.3.2. Sinusoidal Grating
- The sinusoidal grating uses a phase transformation on a DOE and Fresnel propagation to a sensor 0.1mm away.The incident wavefront and sensor are 30 times the DOE size, using a 4800 × 4800 grid to capture the output pattern.Source: §D.3.2 · p. 14
E. Vector Diffractive Optical Element Optimization
A pixel-mask DOE is optimized to make the y-component of a polarized field match its x-component. The loss falls sharply, although unconstrained x- and z-components are also affected.
E. Vector Diffractive Optical Element Optimization
- The optimization figure plots the loss curve and DOE phase mask, while the PSF figure shows the resulting three-dimensional vector field at the sensor.The phase mask is plotted as per-pixel phase offsets intended to transform the y-component to match the x-component.Source: Figure 17 · p. 15, Figure 18 · p. 15
- The experiment uses a single 550nm wavelength, paraxial angles, and 200 stochastic-gradient-descent steps for a pixel-mask DOE.Optimization takes about 1 minute and uses less than 1 GB of memory.Source: E. Vector Diffractive Optical Element… · p. 15
- The polarized-field optimization reduces loss from 0.1 to 0.0005 while matching the y-component to the x-component, but also affects the x- and z-components.The authors attribute those changes to the loss function's lack of constraints on those components.Source: E. Vector Diffractive Optical Element… · p. 15, Figure 17 · p. 15, Figure 18 · p. 15