OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 3 — Jan. 30, 2012
  • pp: 2814–2823
« Show journal navigation

Design and analysis of multi-wavelength diffractive optics

Ganghun Kim, José A. Domínguez-Caballero, and Rajesh Menon  »View Author Affiliations


Optics Express, Vol. 20, Issue 3, pp. 2814-2823 (2012)
http://dx.doi.org/10.1364/OE.20.002814


View Full Text Article

Acrobat PDF (12174 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

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

1. Introduction

Diffractive optics offer significant advantages over conventional refractive optics due to their versatility, and their lightweight and planar geometries [1

1. B. Kress and P. Meyrueis, Digital Diffractive Optics: An Introduction to Planar Diffractive Optics and Related Technology (John Wiley, 2000).

]. Recent advances in microfabrication technologies have made possible the fabrication of diffractive optics in mass scale at low cost [2

2. C. Dankwart, C. Falldorf, R. Gläbe, A. Meier, C. V. Kopylow, and R. B. Bergmann, “Design of diamond-turned holograms incorporating properties of the fabrication process,” Appl. Opt. 49(20), 3949–3955 (2010). [CrossRef] [PubMed]

]. This has enabled their adoption in a variety of application such as imaging [3

3. D. Faklis and G. M. Morris, “Spectral properties of multiorder diffractive lenses,” Appl. Opt. 34(14), 2462–2468 (1995). [CrossRef] [PubMed]

,4

4. D. Faklis and G. M. Morris, “Polychromatic diffractive lenses,” U.S. patent 5,589,982 (31 December 1996).

], optical communications [5

5. D. Prongué, H. P. Herzig, R. Dändliker, and M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. 31(26), 5706–5711 (1992). [CrossRef] [PubMed]

], and lithography [6

6. J. A. Domínguez-Caballero, S. Takahashi, G. Barbastathis, and S. J. Lee, “Design and sensitivity analysis of Fresnel domain computer generated holograms,” Int. J. Nanomanufacturing 6(1/2/3/4), 207 (2010). [CrossRef]

8

8. H.-Y. Tsai, H. I. Smith, and R. Menon, “Reduction of focal-spot size using dichromats in absorbance modulation,” Opt. Lett. 33(24), 2916–2918 (2008). [CrossRef] [PubMed]

]. 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.

2. Optimization problem

The geometry of the optimization problem is shown in Fig. 1(a)
Fig. 1 (a) Schematic of the optimization problem geometry. (b) Target binary images used for the design example containing 3 letters to be reconstructed by 3 distinct wavelengths.
. In order to maximize diffraction efficiency only pure phase DOEs are considered. The DOE is discretized by (M+1)×(N+1) pixels of size Δx and Δy, along the x and y-directions respectively. The DOE’s height profile is given by
h(x,y)=mnΔhpm,nrect(xmΔxΔx)rect(ynΔyΔy),
(1)
and Δh=hmax/Nlevels, is the height step; hmaxis the maximum height of the profile; Nlevelsis the total number of quantization levels; pm,nis a positive integer within the interval [0,Nlevels]; rect() is the rectangle function; and the indices m[M/2,M/2] and n[N/2,N/2].

The corresponding transmission function of the DOE is
T(x,y;λ)=eiϕ(x,y;λ)=1+mnrect(xmΔxΔx)rect(ynΔyΔy)(eia(λ)pm,n1),
(2)
where a(λ)=kΔh[n(λ)1], and k=2π/λ is the wave number. In deriving Eq. (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 transformation
U(x',y';λ)=eikdiλdgillum(x,y;λ)T(x,y;λ)eik2d[(x'x)2+(y'y)2]dxdy=eikd[1+ΔxΔyeik2d(x'2+y'2)mnΚm,n(λ)(eikd(x'mΔx+y'nΔy)sinc(x'Δxλd)sinc(y'Δyλd)eik2d(x'2+y'2))],
(3)
where d is the propagation distance, and Κm,n(λ)=eia(λ)pm,n1. In deriving Eq. (3), an on-axis, unit amplitude illumination wave was assumed:gillum=1.

In order to calculate the diffracted field in a computationally efficient manner, Eq. (3) is rewritten as
U(x',y';λ)=1+ΔxΔyQ(x',y'){Tc(x,y)Q(x,y)},
(4)
with Q(x',y')=eik2d(x'2+y'2) is a quadratic phase factor that can be cached; Tc(x,y;λ)=Κm,n(λ)rect(xmΔx/Δx)rect(ynΔy/Δy), is a complex transmission function; the spatial frequencies are u=x'/λd and v=y'/λd; and the constant term, eikd, was dropped. Calculating the diffracted field using Eq. (4) is computationally efficient as only two point-wise matrix multiplications and one Fourier transform are required: O(2N2)+O(N2logN). This is in contrast to the traditional Fourier-based method, U=1{{T}H}, where H is the Fresnel transfer function: O(N2)+O(2N2logN).

In order to find the optimum height profile that results in the highest average diffraction efficiency for the given number of operating wavelengths, Nλ, the following optimization problem is solved:
minC(pm,n);C=1η¯(pm,n)subjecttopm,nΖ,pm,n[0,Nlevels]
(5)
where the integers pm,nare the optimization variables, and
η¯(pm,n)=λ[mnIT(λ)|U(pm,n)|2Pin(λ)](W(λ)Nλ);
(6)
Pin(λ) is the input power of the illuminating light at a given wavelength; W(λ) is a weight assigned to add flexibility in controlling relative diffraction efficiencies between different wavelengths; IT(λ) is the designed target binary image pattern that may be different for each wavelength.

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 Δh. 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 (m',n') pixel, the new DOE’s transmission function is given by
Tnew(x,y;λ)=T(x,y;λ)eia(λ)rect(xm'ΔxΔx)rect(yn'ΔyΔy)=T(x,y;λ)+[Ψm',n'(λ)rect(xm'ΔxΔx)rect(yn'ΔyΔy)],
(7)
where Ψm',n'(λ)=ΚΔh(Κm',n'(λ)+1), and ΚΔh=(eia(λ)+1). The corresponding Fresnel diffracted field is Unew(x',y';λ)=U(x',y';λ)+Upert(x',y';λ), where the perturbation field is given by
Upert(x',y';λ)=Ζm',n'(x',y';λ)Ω(x',y';λ),
(8)
with Ζm',n'(x',y';λ)=Ψm',n'(λ)eikd(m'Δxx'+n'Δyy'), and

Ω(x',y';λ)=ΔxΔyeikdeik2d(x'2+y'2)[sinc(x'Δxλd)sinc(y'Δyλd)eik2d(x'2+y'2)].
(9)

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. M=N),Ζm',n'(x',y';λ) can be sped up by precomputing and caching a matrix B with rows equal to br,:=eikd(rΔxx'), for r[M/2,M/2]. Then, for a given perturbation state the following calculation is carried out: Ζm',n'=Ψm',n'(λ)[(b¯m',:)Tbn',:], where ()T is the transpose operator, and b¯ represents the vector flip or reverse operator.

3. Design example

The resulting optimized height profile map of the DOE is shown in Fig. 2(a)
Fig. 2 Multi-wavelength DOE designed for 3 discrete wavelengths and target images. (a) DOE’s optimized height profile map. (b) Magnified view of a 13 X 13 pixels region outlined by the white square in (a). Reconstructed image amplitude distributions at (c) λ = 405nm, (d) 532nm and (e) 633nm. The refractive indices at these wavelengths were 1.6894 (405nm), 1.6482 (532nm) and 1.6347 (633nm). The corresponding optical efficiencies are indicated on top of each figure. The average diffraction efficiency is 79.8%.
. 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
Fig. 3 Convergence plots. Evolution of (a) average efficiency and (b) number of perturbed pixels with number of iterations.
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

4.2 Reconstruction distance

4.3 Total number of pixels

4.4 Number of design wavelengths

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

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: H(u,v;λ,d)=eiπλd(u2+v2), 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
Fig. 7 Effect of defocus as a function of average diffraction efficiency for (a) λ = 405nm, (b) 532nm and (c) 633nm.
shows the resulting average diffraction efficiency as a function of propagation distance, d. The DOE was optimized for d=5cm.

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)
Fig. 8 Effect of pixel-height error on DOE’s average diffraction efficiency. (a) Random pixel-height error; (b) Uniform pixel height error.
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

Acknowledgments

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

1.

B. Kress and P. Meyrueis, Digital Diffractive Optics: An Introduction to Planar Diffractive Optics and Related Technology (John Wiley, 2000).

2.

C. Dankwart, C. Falldorf, R. Gläbe, A. Meier, C. V. Kopylow, and R. B. Bergmann, “Design of diamond-turned holograms incorporating properties of the fabrication process,” Appl. Opt. 49(20), 3949–3955 (2010). [CrossRef] [PubMed]

3.

D. Faklis and G. M. Morris, “Spectral properties of multiorder diffractive lenses,” Appl. Opt. 34(14), 2462–2468 (1995). [CrossRef] [PubMed]

4.

D. Faklis and G. M. Morris, “Polychromatic diffractive lenses,” U.S. patent 5,589,982 (31 December 1996).

5.

D. Prongué, H. P. Herzig, R. Dändliker, and M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. 31(26), 5706–5711 (1992). [CrossRef] [PubMed]

6.

J. A. Domínguez-Caballero, S. Takahashi, G. Barbastathis, and S. J. Lee, “Design and sensitivity analysis of Fresnel domain computer generated holograms,” Int. J. Nanomanufacturing 6(1/2/3/4), 207 (2010). [CrossRef]

7.

R. Menon, P. Rogge, and H.-Y. Tsai, “Design of diffractive lenses that generate optical nulls without phase singularities,” J. Opt. Soc. Am. A 26(2), 297–304 (2009). [CrossRef] [PubMed]

8.

H.-Y. Tsai, H. I. Smith, and R. Menon, “Reduction of focal-spot size using dichromats in absorbance modulation,” Opt. Lett. 33(24), 2916–2918 (2008). [CrossRef] [PubMed]

9.

T. Stone and N. George, “Hybrid diffractive-refractive lenses and achromats,” Appl. Opt. 27(14), 2960–2971 (1988). [CrossRef] [PubMed]

10.

Y. Arieli, S. Noach, S. Ozeri, and N. Eisenberg, “Design of diffractive optical elements for multiple wavelengths,” Appl. Opt. 37(26), 6174–6177 (1998). [CrossRef] [PubMed]

11.

D. W. Sweeney and G. E. Sommargren, “Harmonic diffractive lenses,” Appl. Opt. 34(14), 2469–2475 (1995). [CrossRef] [PubMed]

12.

S. Noach, A. Lewis, Y. Arieli, and N. Eisenberg, “Integrated diffractive andrefractive elements for spectrum shaping,” Appl. Opt. 35(19), 3635–3639 (1996). [CrossRef] [PubMed]

13.

M. A. Seldowitz, J. P. Allebach, and D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26(14), 2788–2798 (1987). [CrossRef] [PubMed]

14.

T. R. M. Sales and D. H. Raguin, “Multiwavelength operation with thin diffractive elements,” Appl. Opt. 38(14), 3012–3018 (1999). [CrossRef] [PubMed]

OCIS Codes
(050.1970) Diffraction and gratings : Diffractive optics
(090.1995) Holography : Digital holography

History
Original Manuscript: November 23, 2011
Revised Manuscript: January 11, 2012
Manuscript Accepted: January 12, 2012
Published: January 23, 2012

Citation
Ganghun Kim, José A. Domínguez-Caballero, and Rajesh Menon, "Design and analysis of multi-wavelength diffractive optics," Opt. Express 20, 2814-2823 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-2814


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. B. Kress and P. Meyrueis, Digital Diffractive Optics: An Introduction to Planar Diffractive Optics and Related Technology (John Wiley, 2000).
  2. C. Dankwart, C. Falldorf, R. Gläbe, A. Meier, C. V. Kopylow, and R. B. Bergmann, “Design of diamond-turned holograms incorporating properties of the fabrication process,” Appl. Opt.49(20), 3949–3955 (2010). [CrossRef] [PubMed]
  3. D. Faklis and G. M. Morris, “Spectral properties of multiorder diffractive lenses,” Appl. Opt.34(14), 2462–2468 (1995). [CrossRef] [PubMed]
  4. D. Faklis and G. M. Morris, “Polychromatic diffractive lenses,” U.S. patent 5,589,982 (31 December 1996).
  5. D. Prongué, H. P. Herzig, R. Dändliker, and M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt.31(26), 5706–5711 (1992). [CrossRef] [PubMed]
  6. J. A. Domínguez-Caballero, S. Takahashi, G. Barbastathis, and S. J. Lee, “Design and sensitivity analysis of Fresnel domain computer generated holograms,” Int. J. Nanomanufacturing6(1/2/3/4), 207 (2010). [CrossRef]
  7. R. Menon, P. Rogge, and H.-Y. Tsai, “Design of diffractive lenses that generate optical nulls without phase singularities,” J. Opt. Soc. Am. A26(2), 297–304 (2009). [CrossRef] [PubMed]
  8. H.-Y. Tsai, H. I. Smith, and R. Menon, “Reduction of focal-spot size using dichromats in absorbance modulation,” Opt. Lett.33(24), 2916–2918 (2008). [CrossRef] [PubMed]
  9. T. Stone and N. George, “Hybrid diffractive-refractive lenses and achromats,” Appl. Opt.27(14), 2960–2971 (1988). [CrossRef] [PubMed]
  10. Y. Arieli, S. Noach, S. Ozeri, and N. Eisenberg, “Design of diffractive optical elements for multiple wavelengths,” Appl. Opt.37(26), 6174–6177 (1998). [CrossRef] [PubMed]
  11. D. W. Sweeney and G. E. Sommargren, “Harmonic diffractive lenses,” Appl. Opt.34(14), 2469–2475 (1995). [CrossRef] [PubMed]
  12. S. Noach, A. Lewis, Y. Arieli, and N. Eisenberg, “Integrated diffractive andrefractive elements for spectrum shaping,” Appl. Opt.35(19), 3635–3639 (1996). [CrossRef] [PubMed]
  13. M. A. Seldowitz, J. P. Allebach, and D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt.26(14), 2788–2798 (1987). [CrossRef] [PubMed]
  14. T. R. M. Sales and D. H. Raguin, “Multiwavelength operation with thin diffractive elements,” Appl. Opt.38(14), 3012–3018 (1999). [CrossRef] [PubMed]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited