We present an extension of the direct-binary-search algorithm for designing high-efficiency multi-wavelength diffractive optics that reconstruct in the Fresnel domain. A fast computation method for solving the optimization problem is proposed. Examples of three-wavelength diffractive optics with over 90% diffraction efficiency are presented. These diffractive optical elements reconstruct three distinct image patterns when probed using the design wavelengths. Detailed parametric and sensitivity studies are conducted, which provide insight into the diffractive optic’s performance when subject to different design conditions as well as common systematic and fabrication errors.
© 2012 OSA
Diffractive optics offer significant advantages over conventional refractive optics due to their versatility, and their lightweight and planar geometries . Recent advances in microfabrication technologies have made possible the fabrication of diffractive optics in mass scale at low cost . This has enabled their adoption in a variety of application such as imaging [3,4], optical communications , and lithography [6–8]. In many applications, the diffractive optical element (DOE) is operated using monochromatic, spatially coherent light. However, other applications require the DOE to be operated using broadband light. The design of broadband DOEs is more challenging as they suffer from strong chromatic aberrations and material dispersion that needs to be compensated numerically. In addition, the complexity of the corresponding optimization problem is higher compared to the monochromatic case due to the added non-linear constraints incorporated as some form of wavelength multiplexing.
A variety of systems and methods for designing multi-wavelength DOEs have been proposed in the past. Hybrid geometries composed of refractive and diffractive lenses  as well as arrays of tandem diffractive lenses of complementary materials  have been proposed for reducing the chromatic aberrations in imaging applications. Other methods rely on thick DOEs in which the required phase distribution at a particular wavelength is matched to that of a different wavelength by adding multiples of 2π . An iterative method based on the Gerchberg-Saxton algorithm was proposed for the design of multi-wavelength DOEs for beam shaping applications . Direct binary search (DBS) methods were also proposed for the optimization of monochromatic  and broadband DOEs  showing superior performance compared to error reduction, error diffusion and projection methods. Previous work on the DBS algorithm for multi-wavelength DOEs focused on one-dimensional, far-field diffractive optics used as fan-out gratings.
In this paper, we extend the DBS algorithm for the design and optimization of two-dimensional multi-wavelength DOEs for general patterns that reconstruct in the Fresnel domain. In contrast to previous work in which the phase of the DOE’s transmission function,, is directly perturbed, we perturb the actual height profile of the DOE, , where is the operating wavelength and is the DOE’s material refractive index. Operating directly on the DOE’s height profile is beneficial as fabrication constraints such as maximum height of the designed profile, number of quantization levels, as well as proximity effects, shrinkage, or other fabrication related phenomena can be incorporated into the optimization model. We present an example of a wavelength multiplexed DOE designed to reconstruct different patterns when probed with light of different wavelengths with potential security applications. A careful parametric analysis reveals useful insights regarding the design procedure. Finally, a sensitivity analysis is conducted to elucidate the tolerance of such optics to common fabrication and systemic errors.
2. Optimization problem
The geometry of the optimization problem is shown in Fig. 1(a) . In order to maximize diffraction efficiency only pure phase DOEs are considered. The DOE is discretized by pixels of size and , along the x and y-directions respectively. The DOE’s height profile is given by
The corresponding transmission function of the DOE isEq. (2), a unit amplitude boundary condition is assumed. This corresponds to patterning the DOE on a clear, non-absorbing substrate such as glass. Only the light that falls inside the DOE’s active area is diffracted. Light that falls outside this area propagates though unaltered.
For a given operating wavelength, the diffracted field at the reconstruction plane is given by the Fresnel transformationEq. (3), an on-axis, unit amplitude illumination wave was assumed:.
In order to calculate the diffracted field in a computationally efficient manner, Eq. (3) is rewritten asEq. (4) is computationally efficient as only two point-wise matrix multiplications and one Fourier transform are required: . This is in contrast to the traditional Fourier-based method, , where H is the Fresnel transfer function:
In order to find the optimum height profile that results in the highest average diffraction efficiency for the given number of operating wavelengths, , the following optimization problem is solved:
To solve this nonlinear optimization problem, a variation of the DBS algorithm is implemented. The optimization algorithm begins by calculating an initial condition for the starting height profile. Since this is a local search optimization method, the algorithm is sensitive to the choice of initial condition. If a poor initial condition is chosen, the optimization might suffer from premature convergence or it will take longer to arrive to satisfactory optimum state. Two methods were evaluated as a choice for the initial condition: random height profile and pre-optimized height profile using the Modified Error Reduction (MER) algorithm for a set reference wavelength as described in .
Once the initial condition is set, the algorithm starts by addressing each pixel in the DOE and performing a perturbation by increasing the height by . Next, the corresponding diffracted field is calculated and the cost function of Eq. (5) is evaluated. If the new cost value is lower than the original one, the perturbation is accepted. Otherwise, the perturbation is rejected and the pixel returns to its original height value. This process continues by addressing every pixel in a prescribed order, such as lexicographic or random. The algorithm continues with the pixel perturbation until no pixel yields to a lower cost upon completing a full loop. Other termination conditions can be incorporated, such as performing an early termination if the cost doesn’t change within a tolerance for a set number of iteration or if a target diffraction efficiency value is reached.
In order to calculate the perturbed diffracted field in a computationally efficient manner, an analytical expression for the required perturbation is derived. After perturbing the pixel, the new DOE’s transmission function is given by
To speed up the calculation of a given perturbation step, Eq. (9) can be computed at the beginning of the optimization and cached. Similarly, for the case of square DOEs (i.e. ), can be sped up by precomputing and caching a matrix B with rows equal to , for . Then, for a given perturbation state the following calculation is carried out: , where is the transpose operator, and represents the vector flip or reverse operator.
3. Design example
We now apply the optimization algorithm described in the previous section for designing a multi-wavelength DOE used to reconstruct three different target binary image intensity distributions as shown in Fig. 1(b). Each of the three images is reconstructed at the image plane when the DOE is probed by the corresponding illumination wavelength. The design parameters are: ; ; ; ; and Also, it is assumed that the DOE’s height profile is fabricated on a Shipley S1813 photoresist on top of a glass substrate. The propagation distance was calculated using the grating equation for the shortest wavelength: , where the paraxial approximation is assumed. This guarantees that the sampling requirements are satisfied given the DOE’s pixel size and all the high spatial frequencies of interest are properly encoded. The initial condition was chosen to be the pre-optimized solution of the MER algorithm as discussed previously.
The resulting optimized height profile map of the DOE is shown in Fig. 2(a) . The wavelength-multiplexed information is encoded as a set of fringes similar to those found in conventional holography. The reconstructed absolute amplitude distributions for the three operating wavelengths are shown in Figs. 2(c) and 2(d). The individual diffraction efficiencies are also indicated. The final average efficiency of the DOE is 79.8% (losses due to Fresnel reflections and material absorption are ignored). The speckle present in the reconstructed images is due to the choice of cost function and the tradeoff between diffraction efficiency and uniformity. If a higher uniformity is required, the cost function of Eq. (5) can be replaced by a mean-square-error (MSE) metric based on the difference between the reconstructed and target intensity distributions. In this paper we focus on maximizing the reconstruction efficiency in order to minimize the crosstalk between different target image patterns.
The convergence of the optimization algorithm is shown in Fig. 3 as the evolution of the average efficiency and number of perturbed pixels per iteration. Each iteration refers to one complete pass through all available pixels. As can be seen, the algorithm rapidly converges to a local minimum within 32 iterations despite the large number of degrees of freedom.
4. Parametric analysis
In the following sections, we analyze the effect of the design parameters on the optical performance of the DOE.
4.1 Number of height quantization levels
The topography of the DOE determines the phase shift applied to each illuminating wavelength. To simplify fabrication using, for example, conventional lithographic methods, the height profile is quantized in number of levels. Reducing the number of quantization levels decreases the maximum diffraction efficiency that can be achieved by the DOE. This corresponds to reducing the number of degrees-of-freedom of the system that results in phase-shift errors as the available discrete heights cannot properly match the required phase shift for all wavelengths at a given location. Figure 4(a) shows this dependence for the design example described in the previous section. As expected, the optical efficiency increases as the number of quantization levels increases. However, the optical efficiency starts to saturate at large number of pixel-height levels. This is also expected since the large number of height levels begins to closely approximate the continuous phase profile, which would represent the efficiency upper bound. In addition, as shown in Fig. 4(a), highly efficient multi-wavelength DOEs can be designed with the proposed optimization algorithm. Over 90% diffraction efficiency can be attained for three-wavelength DOEs with 64 height levels. DOEs with such number of levels could be fabricated using, for example, a pattern generator or diamond turning.
4.2 Reconstruction distance
The distance between the DOE and the image plane is an important choice in the design process. We need to ensure that the shortest wavelength, which diffracts by the smallest angle, has sufficient distance to propagate to the image plane. However, if the reconstruction distance is too large, then longest wavelength, which has the largest diffraction angle, could potentially miss the image plane resulting in low diffraction efficiency. As expected, the highest efficiency is achieved near the reconstruction distance calculated using the grating equation as shown in Fig. 4(b), where all the other simulation parameters were kept the same as those from the previous section.
4.3 Total number of pixels
Similar to the number of quantized height levels, the total number of pixels determines the number of degrees-of-freedom in the system. From an optics point-of-view, increasing the total number of pixels while fixing the DOE’s size improves the ability to encode higher spatial frequency signals and higher diffraction efficiencies can be reached. However, as the number of pixels increases, the total number of optimization variables and problem complexity also increases. As a result, it is harder for the optimization algorithm to converge to a good solution and may suffer premature convergence. This is an intrinsic scalability property of the algorithm due to its local-based search nature. To overcome this problem, a perturbation strategy to escape from prematurely converged local minima may be adopted, such as that based on random perturbations or a simulated annealing scheme . Figure 4(c) shows this dependence for the design example of the previous section.
4.4 Number of design wavelengths
Another important parameter for the algorithm is the number of design wavelengths, . As the DOE has finite number of degrees-of-freedom, increasing results in a degradation in optical performance. Also, the complexity of the optimization problem increases as new constraints are added and the algorithm may again suffer from premature convergence. This is illustrated in Fig. 4(d), where the design example of the previous section was extended to reconstruct the following letters: U (351nm), B (405nm), G (532nm), R (633nm), I (720nm) and K (850nm). In this case, the optimization started with a design for B and G. Then, more wavelengths were added slowly. For each set of wavelengths, the algorithm was run until it converged. The corresponding reconstructed amplitude distributions are shown in Fig. 5 . As the number of design wavelengths increases, the average diffraction efficiency decreases and crosstalk between different target images appears.
5. Sensitivity analysis
In the following sections, a sensitivity analysis is performed to study the effect of different operating parameters and fabrication errors.
5.1 Chromatic effects
When an optimized DOE is illuminated by a wavelength other than the design wavelength, an intermediate intensity distribution is reconstructed. This intermediate intensity distribution has lower performance as it is dominated by crosstalk between the two closest design wavelengths as shown in Fig. 6 . The amount of chromatic effects is proportional to the wavelength shift. Chromatic effects can be minimized by increasing the number of design wavelengths and reducing their difference, ; however, as the number of wavelengths increases, the average diffraction efficiency decreases as explained in the previous section.
4.2 Effect of defocus
Defocus occurs when the reconstructed intensity distribution for a given operating wavelength is evaluated at a plane different than the one considered during the optimization. The effect of defocus is analogous to that of chromatic aberration due to the wavelength-distance dependence present in the Fresnel propagation transfer function: , where u and v are the spatial frequency coordinates. Hence, fixing the operating wavelength and varying the propagation distance is analogous to fixing the propagation distance and varying the illumination wavelength. Figure 7 shows the resulting average diffraction efficiency as a function of propagation distance, d. The DOE was optimized for .
4.3 Effect of pixel-height error
One of the common fabrication errors is the discrepancy between the designed pixel height and the fabricated pixel height. In order to simulate this effect, two types of errors were simulated: random pixel-height errors and a uniform height error. The first error can be caused by, for example, dose errors during the laser writing procedure for DOEs fabricated using a pattern generator. To simulate this error, randomly generated pixel-heights drawn from a normal distribution with zero mean and varying standard deviations were added to the original DOE. Figure 8(a) shows the resulting average diffraction efficiency as a function of standard deviation. The second error can be caused by, for example, photoresist thickness errors during the spin process. The effect of this error is shown in Fig. 8(b). The insets in both cases show the corresponding reconstructed amplitude distributions at the two extremes.
6. Summary and conclusions
An extension to the direct binary search (DBS) algorithm is presented for the design of high-efficiency two-dimensional multi-wavelength DOEs of that reconstruct general patterns in the Fresnel regime. An analytical formulation of the DOE’s transmission function as well as its corresponding diffracted field is provided. In order to speed up the numerical computation of the diffracted field, an alternate method is proposed that has a reduced number of operations compared to the standard Fourier-based approach. Similarly, an analytical formulation of the perturbation required by the DBS algorithm is derived and a fast method for its computation is proposed. In contrast to previous work in which the phase of the DOE’s transmission function is directly perturbed, we perturb the actual height profile, which allows accounting for different fabrication constraints and errors. The numerical optimization is conducted for maximizing the average diffraction efficiency for a multi-wavelength DOE designed to reconstruct different target image patterns. Design examples that showcase the robustness and performance of the proposed optimization algorithm are provided. High optical performance three-wavelength DOEs are presented with average diffraction efficiencies over 90%. The fast convergence behavior of the optimization algorithm is shown. A parametric analysis is conducted to study the effect of different design parameters on the optical performance. The following design parameters are analyzed: number of height quantization levels, reconstruction distance, total number of pixels, and number of design wavelengths. A sensitivity analysis is conducted to study the tolerance of the multi-wavelength DOE to common fabrication and systematic errors. Finally, chromatic aberration, defocus and pixel height errors are considered. We conclude that the described optimization method is highly robust for designing multi-wavelength DOEs for various applications, such as chromatically encrypted security tags.
This work was partially funded by the Utah Science Technology and Research (USTAR) initiative and a Utah Technology Commercialization Grant. G.K. was partially supported by an Undergraduate Research Opportunities Project. We thank Mark Ogden and Stewart Brock for computer support.
References and links
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