Validating Pseudo-Free-Space Conditions in a Planar Waveguide Using Phase Retrieval from Fresnel Diffraction Patterns

Abstract

In this study, we address the question of whether a waveguide with absorbing sidewalls can be considered pseudo free space and if the free-space transfer function is valid in such a medium. We test this hypothesis by applying a phase retrieval algorithm based on the free-space transfer function. First, optical measurements are carried out to measure the optical properties of a stack of thin films and select the parameters of simulations. Next, the propagation of light in a waveguide was simulated in COMSOL, and the phase of a wave was retrieved in MATLAB. Analysis was performed both for free-space conditions, and for a waveguide with absorbing sidewalls. The cross-correlation between the distributions of intensity under both conditions was about 0.40. The RMS error of the wave retrieved under free-space conditions was 0.378 rad, while that in the case of absorbing sidewalls was 0.323 rad, indicating successful retrieval. The successfully recovered phase of the input wave suggests that a waveguide with absorbing sidewalls can be approximated as pseudo free space and the free-space transfer function may be valid. These results may be used in future studies on how to shorten the phase retrieval of two-dimensional objects.

1. Introduction

In many applications dealing with the propagation of a wave along a waveguide, the complex field either at the input, or the output of a waveguide must be known. There are several kinds of modes supported by a waveguide, and each mode has its own distribution of amplitude and phase. Modes can be calculated by solving Maxwell’s equations subject to the boundary conditions at the interface between the core and cladding [1]. There is a limited number of guided modes supported by the core and decaying outside the core [1,2]. There is also an unlimited number of radiation modes. Few of these modes become trapped in the cladding, giving rise to the cladding modes [3]. There are also leaky modes radiating power out of the core and losing their power as they propagate along a waveguide [4].

Various methods to calculate the phase of a wave propagating along a waveguide have been proposed. In the study by Shechtman [5], the input wave exciting an array of waveguides was algorithmically calculated from the intensity measurements at the output. Recovering information about the phase of a wave from intensity measurements is known as phase retrieval [6] and has gained popularity in computational optics. Doughan et al. used a customized wavefront folding interferometer to retrieve the phase of a wave [1], while Pelliccia et al. used iterative correlation analysis to retrieve the complex X-ray field at the end of a waveguide [7]. The phase of the output wave must also be known to measure the impulse response in photonic integrated circuits [8]. This requires an additional delay path, making the phase retrieval complex.

In free space, the propagation of a wave is given by the free-space transfer function. If a complex wave is known at one plane, the free-space transfer function can be conveniently applied to calculate the wave at any other plane based on the propagation of the angular spectrum [9]. As many phase retrieval algorithms are often based on the propagation of a wave in free space, waveguides with the optical properties of free space despite being physically confined spaces are desired because of the benefits provided to numerous applications. Examples of these applications include but are not limited to waveguide arrays [1,2], optical communications [10], integrated photonic circuits [11], creation of free-space modes [12], shortening the time needed for two-dimensional phase retrieval [13], etc.

Fabricating a waveguide imitating free space seems a challenging task as it would constitute a complex study involving both the development of mathematical theory and materials science; however, it must be possible by modulating the amplitude and phase of the wave reflected from the interface between the core and cladding. There have been studies on how to create metasurfaces on the outside of a waveguide [12,14,15]; however, as far as the authors of this study know, the question of how to make the core of a waveguide optically equal to free space has not been addressed previously. In this paper, we address the question of whether a planar waveguide, i.e., a physically confined space along one dimension, can be considered a pseudo free space by making two of its sidewalls perfect absorbers over a wide range of the angle of incidence and at a given wavelength. We chose planar waveguides for our study, as they are used extensively in photonic crystals [16,17,18], logic gates [19], telecommunications [20], and many other applications. We verify our hypothesis by applying a phase retrieval algorithm based on the free-space transfer function.

2. Materials and Methods

2.1. Fabrication of Samples and Measurements of Optical Properties

First, a silicon wafer was cleaned in oxygen plasma. The parameters of the recipe were as follows: flowrate 1000 sccm, power 800 W, pressure 0.133 mbar, and duration 10 min. Next, a stack of Cr/Au/Ge thin films was deposited on the silicon wafer, as a stack such as this is a highly efficient absorber for various combinations of the type of polarization, the angle of incidence, and the wavelength of light [21]. Chromium and gold were thermally deposited using a thermal evaporator BOC Edwards Auto 306 (manufacturer: Edwards; Burgess Hill, UK). The thickness of chromium thin film was 10 nm, and it served to enhance the adhesion of the gold thin film deposited above it. The thickness of the gold thin film was 50 nm, and it was optically almost opaque in the visible range of the electromagnetic spectrum. Germanium thin film was thermally deposited using a magnetron at rate from 0.1 to 0.4 Å/s and at a pressure of 7.6·10⁻⁴ Pa. The thickness of the germanium thin film was 7 nm. SiO₂ was deposited using plasma enhanced chemical vapor deposition (PECVD). The PECVD deposition parameters were as follows: process gases N₂O (710 sccm) and 2% SiH₄/Ar (500 sccm), power 20 W, temperature 200 °C, pressure 1 Torr, and deposition time 4 min 10 s. The optical properties of the Cr/Au/Ge/SiO₂ stack were measured using a spectrophotometer Cary 7000 (manufacturer: Agilent; Santa Clara, CA, USA). The reflectance of s- and p-polarized light was measured at wavelengths ranging from 380 nm to 760 nm and angles of incidence ranging from 6 degrees to 85 degrees. Next, the range of wavelengths at which reflectance was low across the whole range of angles of incidence were identified, and the largest wavelength was selected so that the mesh in COMSOL has the lowest density, thereby reducing the computational load.

2.2. Simulations of the Propagation of Light and Phase Retrieval

Simulations were carried out in COMSOL Multiphysics (version 6.2) and MATLAB R2020a. The propagation of a plane wave was simulated both in free space and in a planar waveguide with absorbing sidewalls. In both cases, a block of SiO₂ was used as the domain through which light was propagated; however, the scattering boundary condition was used for simulating free space while perfectly matched layers (PMLs) were applied to the top and bottom sidewalls of the waveguide to simulate a planar waveguide with absorbing sidewalls. The planar waveguide simulated in this study was equivalent to a strip waveguide [22,23] buried in a hole etched in a silicon wafer (substrate), with its top surface lying in the same plane as the top surface of the substrate and having a top silicon wafer bonded to the substrate. Simulations were carried out for p-polarized light (directed along Z axis) at vacuum wavelength of 500 nm with low reflectance at all angles of incidence. The 500 nm wavelength was selected because this is the largest wavelength at which reflectance is low for all angles of incidence and requires the least dense mesh, thereby reducing the computational requirements in COMSOL. The dimensions of the simulation domain were as follows: width 5 µm, depth 3.2 µm, and height 2.1 µm. The height of the core was 2 µm, while that of the PMLs was 50 nm. The input power was about 9.5 pW, as if it were illuminated by a collimated top-hat beam of diameter 25.4 mm and power 5 mW. The type of physics selected for the simulations was “Electromagnetic Waves, Frequency Domain”. The minimum size of the element of the mesh was equal to one fifth of the wavelength in the media. The study was computed in the frequency domain, and the distributions of intensity at various planes of the waveguide were calculated.

The similarity between the distributions of intensity at each distance from the input of the waveguide was assessed based on the cross-correlation. The two distributions of intensity were shifted relative to one another, and the value of cross-correlation vs. the displacement was plotted for all distances of propagation. Cross-correlation has also been used previously to compare atomic pair distribution functions [24].

The applicability of the free-space transfer function was verified in a way described in more detail below. A plane wave was propagated from the input port to five planes at various distances from the input port. The distance of propagation ranged from one micrometer to five micrometers in steps of one micrometer. Next, the multiple-plane Gerchberg–Saxton (MPGS) algorithm was applied in MATLAB to the distributions of intensity at these planes of the waveguide. The MPGS algorithm is based on the alternating projections and the free-space transfer function, and it iteratively updates the phase of a wave as it is propagated from one plane to another while the distributions of intensity are preserved [13]. The complex wave at the middle plane located three micrometers from the input port was retrieved and then propagated backwards using the free-space transfer function to calculate the wave at the input of the waveguide.

3. Results

3.1. Reflectance Maps and the Distribution of Intensity

Figure 1 shows the reflectance spectra of the stack of Cr/Au/Ge/SiO₂ thin films for both s polarization (top) and p polarization (bottom). Interference fringes can be clearly seen superimposed on the maps. For p-polarized light, reflectance is low for short wavelengths across the whole range of the angles of incidence, and the interference fringes are not very expressed; however, this is not the case for s polarization. Based on the reflectance maps, p-polarized light at the wavelength 500 nm was selected for simulations. While the reflectance at some angles of incidence may still be around 25%, these reflections can be minimized by multiple reflections as the wave propagates along the waveguide.

3.2. Intensity Distribution and Phase Retrieval

Figure 2 shows the distributions of intensity obtained when simulating the propagation of a plane wave in free space, while those obtained when propagating the wave in the SiO₂ core of the waveguide with PMLs applied to its sidewalls are shown in Figure 3. The propagation distance increases from one micrometer to five micrometers in steps of one micrometer from left to right and from top to bottom. In both cases, the distributions follow typical Fresnel diffraction patterns despite the differences in the shape at some distances of propagation. The biggest difference between both distributions is observed for the 3 μm distance of propagation—without PMLs applied to the sidewalls, a valley is observed at the center, while a peak is observed when PMLs are applied.

For each distance of propagation, the cross-correlation between the two distributions versus the displacement between those distributions is shown in Figure 4. The maximum cross-correlation between the two distributions varied with the propagation distance within a narrow range from low (0.38) to medium (0.45).

In Figure 5, the top row shows the retrieved phase under free-space conditions. The panel on the left side shows the wrapped phase, while the phase after applying a phase unwrapping algorithm is shown on the right side. As the input wave was planar, there should be no wrapping points. However, there is still one wrapping point successfully unwrapped after applying a phase unwrapping algorithm. The value of the RMS error of the retrieved phase is 0.378 rad when compared with an absolutely planar wave. The bottom row shows the phase retrieved from the diffraction patterns formed by the waveguide with PMLs applied to its sidewalls. Despite the medium cross-correlation between the intensity distributions at each distance of propagation, the value for the RMS error was again very low, i.e., 0.323 rad. While the Strehl ratio is commonly used in astronomy for two-dimensional objects, it may still be applied to one-dimensional objects, for example, cylindrical wavefronts propagating behind cylindrical optical elements. According to the Marechal criterion, the Strehl ratio corresponding to the RMS values mentioned previously are 0.867 and 0.901, respectively.

4. Discussion

The values of the RMS errors obtained under free-space conditions (0.378 rad) and when using absorbing sidewalls (0.323 rad) were similar and only about 5.5% of a full wave despite the moderate cross-correlation (around 0.4) between the intensity distributions in both cases. The RMS error reported in our previous study on the applicability of digital signal processors for phase retrieval with MPGS was about 1 rad or 15.9% of a full wave. These results suggest that a waveguide with absorbing sidewalls may serve as a good approximation of free space despite the high angular frequencies lost due to the absorption by the sidewalls. The MPGS algorithm applied for phase retrieval itself is based on the free-space transfer function, and the low values of RMS errors suggest that the distributions of intensity are accurate Fresnel diffraction patterns related to each other by the free-space transfer function. As noted by Pelliccia et al. [7], in a multimode fiber, there is very good agreement between the true and retrieved amplitude but not the phase. The differences between the true phase and the retrieved phase may be attributed to multimodal propagation.

Due to the limited computational resources available, it was impossible to calculate the intensity distribution for larger models, as this would require cluster computing not available to the authors of this study. Due to the small size of the domain, the current study was performed in the near field; however, it can be expected that a waveguide with absorbing sidewalls would also act as a pseudo free space in the far-field. In practice, waveguides with large dimensions would have a low throughput of light, thereby limiting the maximum permissible length of the waveguide. If the input was irradiated by a circular beam with a diameter of 25.4 mm and a total power 5 mW, then the peak output irradiance would be around 0.3 µW/cm². Under low-light conditions, the Poisson noise is often the dominant noise, and if the mean and variance of the Poisson distribution becomes about 1 × 10⁻⁴ of the maximum of the diffraction pattern, then the phase of a wave can no longer be retrieved reliably [25]. The problem may be solved by using fiber collimators with a collimating ball lens with high numerical aperture and thereby providing a sufficient amount of light. This effect can also be minimized by using phase retrieval algorithms that are tolerant to low signal-to-noise ratios and can retrieve the phase from sparse data [6,26]. If sparse data is acceptable, the bit depth can be made low, thereby zeroing the lowest intensity levels corrupted by noise. It must also be taken into account that the fabrication of the proposed waveguide would have very strict requirements for the flatness of the surfaces, as any inhomogeneities could have devastating effects on the quality of the diffraction patterns.

The stack of thin films investigated has a relatively simple structure and could therefore be applicable for building the sidewalls of a waveguide. As germanium is a highly lossy dielectric, the thickness of this material required for destructive interference can be much smaller than a quarter wavelength, thereby saving on the material needed for fabricating the waveguide [21]. One of the drawbacks of the studied structure is the dependence of reflectance on the state of polarization. In the current study, broader regions of low reflectance were observed for p-polarized light, as in the study by Kats et al. [21]. This problem could be mitigated by using absorbers based on plasmonic resonance [27]. It is also known that the refractive index of thin Ge films depends on the thickness of the film [28], imposing additional difficulties regarding the fabrication of the waveguide. It must also be noted that the reflectance does not fall to zero at any point, making the model somewhat inaccurate. The lowest reflectance of the sidewalls is actually about 12.7%. However, the light hitting the sidewalls may become attenuated through multiple reflections, thereby minimizing the impact on the output.

The results are promising to the various fields mentioned previously that involve the need to recover and control the phase of a wave travelling along a waveguide. In future, we plan to extend this study to the phase retrieval of two-dimensional objects. A two-dimensional object could be split into individual vectors, each serving as an input to its own planar waveguide, as all the waveguides together would for a stack. Processing the diffraction patterns formed by these waveguides in parallel could shorten the time needed for two-dimensional phase retrieval. In combination with computationally efficient embedded systems [29], this would make the ultrafast phase retrieval of two-dimensional objects feasible.

5. Conclusions

The results suggest that an optical media with absorbing sidewalls is optically close to free space, as shown by the moderate cross-correlation between the distributions of intensity under free-space conditions and in the case of a waveguide with absorbing sidewalls. This conclusion is also supported by the small RMS error values of the retrieved waves. However, there is still a question regarding whether the similarity is preserved until the far-field. Applicability of the free-space transfer function to optical media with absorbing sidewalls could facilitate the development of various applications where information about the phase of a wave is important.