Diffraction light analysis method for a diffraction grating imaging lens Takamasa Ando,1,2,3,* Tsuguhiro Korenaga,2 Masa-aki Suzuki,2 and Jun Tanida3 1
2
New Products & Applications Department, Panasonic Photo & Lighting Co., Ltd./1-1 Saiwai-cho, Takatsuki City, Osaka 569-1193, Japan
Device Solutions Center, Panasonic Corporation/3-1-1 Yagumo-nakamachi, Moriguchi City, Osaka 570-8501, Japan 3
Department of Information and Physical Sciences, Graduate School of Information Science and Technology, Osaka University, 1-5 Yamadaoka, Suita, Osaka 565-0871, Japan *Corresponding author: [email protected] Received 14 January 2014; revised 8 March 2014; accepted 12 March 2014; posted 13 March 2014 (Doc. ID 204498); published 10 April 2014
We have developed a new method to analyze the amount and distribution of diffraction light for a diffraction grating lens. We have found that diffraction light includes each-order diffraction light and striped diffraction light. In this paper, we describe characteristics of striped diffraction light and suggest a way to analyze diffraction light. Our analysis method, which considers the structure of diffraction grating steps, can simulate the aberrations of an optical system, each-order diffraction light, and striped diffraction light simultaneously with high accuracy. A comparison between the simulation and experimental results is presented, and we also show how our analysis method can be used to optimize a diffraction grating lens with low flare light. © 2014 Optical Society of America OCIS codes: (050.1965) Diffractive lenses; (220.3620) Lens system design. http://dx.doi.org/10.1364/AO.53.002532
1. Introduction
Diffraction gratings have been widely used for optical pickup lenses of CDs, DVDs, and BDs for multiple-focus operation [1] and aberration correction. Furthermore, they have been used to develop diffractive bifocal contact lenses and intraocular lenses suitable for both myopia and hyperopia [2,3]. Recently, diffraction gratings have also attracted attention in lens design for optical imaging, since they effectively correct lens aberrations, such as chromatic aberration and field curvature [4]. These diffraction grating lenses are generally designed by a high refractive index method or a phase function method [5]. However, such methods are based only on ray tracing [6,7], and thus they are not capable of estimating the intensity of each 1559-128X/14/112532-07$15.00/0 © 2014 Optical Society of America 2532
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diffraction order generated by the diffraction grating lens. Scalar diffraction theory can calculate diffraction efficiency [8], but this theory has difficulty estimating it precisely with an optical imaging system because the incident light on the diffraction grating surface arrives at several angles. Moreover, it is impossible for the theory to calculate the striped diffraction light generated through Fraunhofer diffraction [9]. Electromagnetic field analyses, such as finitedifference time-domain (FDTD) [10] and rigorous coupled-wave analysis (RCWA) [11] can be used to analyze the amount of each-order diffraction light. But the FDTD method is impractical to use with an optical imaging system because the analysis can consider only a small area of diffraction grating surface due to the analysis volume, and the RCWA method is inefficient because it supposes periodic structure. As the number of diffraction grating rings
increases, estimates of the amount and distribution of diffraction light will become less accurate, as seen through a comparison of analysis and experimental results. Therefore, the development of an analysis method that can calculate diffraction light very precisely in a short time is essential to manufacturing diffraction grating lenses effectively with a minimum of trial-and-error procedures. In this paper, we suggest a novel analysis method for diffraction gratings that can estimate the amount and distribution of diffraction light accurately and quickly. This analysis is appropriate for not only flare simulation but also overall simulation of diffraction gratings, such as superposition analysis of the multidiffraction light of pickup lenses and the convoluted images of bifocal or multifocal lenses, and the applications of our method can be even more widely extended.
Fig. 1. Schematic of optical imaging system with diffraction grating lens.
2. Striped Diffraction Light
Unnecessary-order diffraction light, such as zerothand second-order diffraction light, is well known to be given off by a diffraction grating lens. However, we have discovered another kind of diffraction light that is created when a grating lens is irradiated with intense light, and we have elucidated the generation principle of this light [12]. Diffraction light produces a striped pattern near the intended focal point as it passes through minute slits formed by diffraction grating rings. Since the striped diffraction light is generated based on Fraunhofer diffraction, the width of the rings correlates closely with the interval of the striped diffraction patterns [12]. This work uses an optical imaging system with a diffraction grating lens (Table 1, Fig. 1). We positioned a stop in the optical system at the diffraction grating surface so that chromatic aberration could be corrected effectively. Our optical system consisted of two lenses with a front element and a rear element, where the rear element’s surface has a diffraction grating. To evaluate the dependence on the width of the rings, we changed the rear element and investigated four types, each with a different number of grating rings. In order to keep the focal length constant when we change the width of the grating rings, we changed the form of aspherical surface together. The width of the rings becomes narrow in proportion to the number of rings, since the stop diameters are roughly the same in each lens. Figure 2 shows images captured by these lenses sequentially. The point light source was set at 60° of the field of view. This result indicates that the wider the width of the rings becomes, the narrower the striped interval appears Table 1.
Fig. 2. Images of a point light source’s light captured by a diffraction grating lens.
(Fig. 3). This corresponds to the Fraunhofer diffraction theory. As the striped intervals on the image plane become narrow, the diffraction light shrinks near to the central main light and becomes unnoticeable (Fig. 2). However, the intensity and clarity of the striped diffraction light are not necessarily related to the width of the rings, especially on a diffraction grating lens that has plural grating rings, since the striped diffraction light is formed as an integration of the light from each diffraction grating ring. As shown in Fig. 2, a grating lens with 18 rings provides higher and clearer intensity than does one with 24 rings, despite having fewer rings. This implies that the phase function of the diffraction gratings should be designed to weaken the amplitude of the stripe pattern when the striped light from each grating ring is
Specifications of Optical Imaging System
Number of lenses Total optical track length Angle of view Resolution Stop position Grating rings
2 8.5 mm 180° 1.2 megapixels At diffraction grating surface 0–24 rings (positive power)
Fig. 3. Experimental data of 3rd–4th striped interval from central main light. 10 April 2014 / Vol. 53, No. 11 / APPLIED OPTICS
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superimposed with the other rings’ light. In the previous paper [12], we proposed an original design method for minimizing such light by controlling the phase function and finding a suitable combination of widths of the diffraction grating rings. 3. Diffraction Light Analysis Method Considering Grating Steps Structure
The design method in the previous paper [12] involved analysis of striped diffraction light that considered only one surface of a diffraction grating, and aberration was not incorporated in the results. To ensure that the effect of aberration is reflected in the results, the analysis should be carried out using all of the lenses in the optical system. Figure 4(a) shows the conventional ray tracing method for a diffraction grating surface. The diffraction grating is expressed by the phase function ϕ and the ray tracing on the diffraction grating surface is executed by vector analysis with Eq. (1): n1 S1 × E n0 S0 × E m
λ ∇ψr × E; λ0
(1)
wherein ψr a1 · r2 a2 · r4 a3 · r6 a4 · r8 a5 · r10 ; (2) ϕr
2π ψr: λ0
(3)
Here, n0 is the refractive index of the medium before passing through the surface, n1 is the refractive index of the medium after passing through the
Fig. 4. Principle of real-shape analysis. 2534
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surface, m is the diffraction order, λ1 is the wavelength of the light, λ2 is the construction wavelength, E is the unit normal vector on the surface, S0 is the unit vector of the incident ray, S1 is the unit vector of the exit ray, ψ is the optical path difference function, r is the distance from the optical axis, and a1 ; a2 ; a3 ; a4 ; a5 ; …, are coefficients of the optical path difference function. Equation (1) means that the exit direction after passing through the surface is calculated by adding the diffraction component of the phase function to the refractive component of the aspherical surface. However, in this case, the wavefront passing through the diffraction grating surface is seamless, thus making the first-order diffraction light 100%; the unnecessary-order diffraction light does not appear on the image plane, although it can be used to calculate aberration by analyzing all lenses in the optical system. We have developed a new analysis method that takes the unnecessary-order diffraction light into account. First, we represented the diffraction grating surface with a real grating shape having diffraction grating steps instead of a phase function. Unlike a phase function, this structure can reflect the optical path difference of the finite thickness of diffraction grating steps and generate slight wavefront gaps between neighboring rings [Fig. 4(b)]. This state of the wavefront resembles that of an actual object. This analysis is also applicable in the case of a white grating lens [13], which we have suggested reduces unnecessary-order diffraction light by controlling the refractive index of the film covering the diffraction grating surface. When the refractive index of the film is higher than that of the diffraction grating lens, the direction of the diffraction grating is inverted (Fig. 5). Second, we executed ray tracing of the optical system on each surface from the object plane to the image plane by Eq. (4):
Fig. 5. Schematic of white grating lens.
n1 S1 × E n0 S0 × E:
(4)
Compared to Eq. (1), the calculation of the phase function component can be omitted because the diffraction component is taken into consideration by the real grating shape. Note the ray should be traced with not only exit direction but also the optical path length so that wavefront gaps on the diffraction grating surface are acquired. After reaching the image plane, the ray follows the path back with the optical path length to the exit pupil position keeping the ray proceeding angle constant. By comparing the optical path length of each ray on the exit pupil with ideal spherical wave, the wavefront aberration distribution Wx; y ∈ RNx×Ny on the exit pupil can be calculated. Here, x; y are coordinates on the exit pupil, NxNy denote the number of elements along the x and y axis on the exit pupil, respectively. Figure 6 illustrates wavefront aberration on the exit pupil of the white grating lens. The number of elements Nx, Ny are 2048, respectively. In this case (Fig. 1), the exit pupil exists in the stop position. Compared to conventional analysis [Fig. 6(a), diffraction gratings with phase function], our original analysis with a real shape [Fig. 6(b)] disconnects the wavefront at the diffraction grating rings. Finally, we propagated the wavefront aberration distribution Wx; y on the exit pupil to the image plane, and thus point spread function (PSF) Pu; v could be constituted. Here, u; v are coordinates on the image plane. Propagation by Fraunhofer diffraction can be used to calculate the PSF Pu; v. Fraunhofer diffraction is solved by discrete Fourier transform: Pu; v
Ny Nx X X
ux
vy
Wx; ye−2πiNxNy
x1 y1
u 1; …; Nx v 1; …; Ny
:
(5)
The diffraction theory of Rayleigh–Sommerfeld or Fresnel diffraction can also be used to propagate the wavefront. Since the wavefront aberration considers the wave gap generated by diffraction rings, the PSF includes unnecessary-order diffraction light. Furthermore, this analysis enables us to simulate
Fig. 6. Wavefront aberration on the exit pupil (white grating lens).
Fig. 7. PSF simulated by real-shape analysis.
not only the unnecessary-order diffraction light but also the striped diffraction light, since the wavefront passing through a real grating shape is intermittent. Our analysis is able to simultaneously calculate the aberrations of an optical system, unnecessary-order diffraction light, and striped diffraction light. Moreover, since this method is based on ray tracing and wavefront propagation of exit pupil aberration, it can calculate them quickly. Besides, our method is applicable to the range of scalar field: more than several micrometer order of the width of rings. Figure 7 shows the results of PSF simulation. We simulated an optical imaging system with the white grating lens in Fig. 1 (24 rings) at 60° of the field of view. Conventional analysis, using diffraction gratings with a phase function [Fig. 7(a)], can simulate only aberrations, leaving unnecessary-order diffraction light and striped diffraction light unobserved. Meanwhile, our analysis using diffraction gratings with real shape can express the flare light around the central main light, corresponding to the experimental results in Fig. 2 (24 rings). This shows that our analysis can be used to effectively optimize the diffraction grating height of diffraction grating steps related to unnecessary-order diffraction light [14]. Figure 8(a) shows the dependence of diffraction-light flare distribution on diffraction grating height. The integrated brightness of flare light normalized by total brightness is shown in Fig. 8(b). This result indicates that the best diffraction grating height for reducing flare light can be calculated and easily determined by our analysis method. 4. Comparison of Analysis and Experiment
We compared the data of simulation and experimental results using the optical system with a white grating lens (Fig. 9). The diffraction grating is before optimization, the diffraction grating height is 15 μm, and the field of view is 60°. The wavelengths by analysis are 640, 550, and 440 nm, and the captured images of a halogen point light source were taken as experimental PSF data by a white grating lens through a dichroic filter; wavelengths were 610–650 nm (with IR cut filter), 505–575 nm, and 400–495 nm. Figure 9 shows the cross section of the PSF intensity distribution, with the position in the image plane on the abscissa and the intensity of the PSF on the 10 April 2014 / Vol. 53, No. 11 / APPLIED OPTICS
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Fig. 8. Diffraction grating height dependence of PSF distribution.
ordinate. The PSF value on the ordinate was normalized as 100 by integrated total brightness. Since the max. value on the ordinate is shown as 0.03% against integrated total brightness, the peak of PSF is saturated, and the bottom distribution of PSF can be observed in detail on the graph. The diffraction-light flare has been observed near the saturated central main lights, and these distributions of the analysis results [Fig. 9(a)] closely correspond to those of the experiment [Fig. 9(b)] at each wavelength from the viewpoints of the generated amount and direction of the diffraction-light flare. This experimental data show that the shorter the wavelength, the larger the amount of diffraction-light flare. This result also corresponds to that of the analysis.
We used our analysis method to optimize the diffraction grating surface of the optical system in Fig. 1. Figure 10(a) is the PSF intensity of the optimized diffraction grating. We adjusted the interval of the grating rings and the diffraction grating height (13.8 μm). Compared to Fig. 9(a), it is obvious that the diffraction-light flare decreases in the simulation. The reduction of diffraction-light flare, especially in the wavelength range of blue, seems to have the effect of lowering the diffraction grating height. Based on this analysis, we have successfully created an effective diffraction grating lens for reducing the diffraction-light flare. The very close correspondence of the amount and distribution of diffraction light between our analysis and the experiment is due to our method’s ability to consider the aberration characteristics and each incident angle of rays to the diffraction grating surface. This has never been achieved by conventional methods. 5. Applied Analysis
The measured modulation transfer function (MTF) of the fabricated diffraction grating lens incorporates the characteristics of the unnecessary-order diffraction light and striped diffraction light. Therefore, the simulated MTF should also be calculated by considering those types of diffraction light. Since MTF can be calculated by the wavefront aberration on the exit pupil [15], it is possible to calculate the MTF of diffraction grating lenses using our analysis. Figure 11 shows the results of MTF simulation, with defocus position on the abscissa and the MTF value on the ordinate. Figure 11(a) is a conventional MTF of a diffraction grating calculated by a phase function.
Fig. 9. Comparison of PSF intensity distribution between simulation and experiment (unoptimized diffraction grating). 2536
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Fig. 10. Comparison of PSF intensity distribution between simulation and experiment (optimized diffraction grating).
As shown in Fig. 11(b), the analysis with a real shape can calculate MTF as well as the conventional approach. Specifically, compared to a phase function, the peak values of MTF analysis with a real shape are lower by only a few percentage points. This is
caused by diffraction-light flare, and it indicates that our analysis method is appropriate for simulating the MTF of diffraction grating lenses precisely. Our analysis can also calculate light lost through the diffraction grating’s wall by tracing rays while considering refraction and reflection there. Light loss is divided into three classes: one-time refraction passing directly through the diffraction wall, threetime refraction passing through the diffraction wall
Fig. 11. MTF (simulation).
Fig. 12. Light loss simulation. 10 April 2014 / Vol. 53, No. 11 / APPLIED OPTICS
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once and the diffraction surfaces twice, and total reflection at the diffraction wall (Fig. 12). These are categorized by considering the positions of rays, the directions of diffraction gratings, the incident angles of rays, and the refractive index of the lens materials. We have used this method to successfully analyze the distribution of these types of light loss on the image plane, and this analysis is thus suitable for determining the number of diffraction grating rings from the amount of light loss. 6. Conclusion
We have developed a new method to analyze the amount and distribution of diffraction light that affects the image quality of an imaging system. This analysis has achieved high-accuracy and simultaneous simulation of the aberrations in an optical system, the unnecessary-order diffraction light, and the striped diffraction light. Consequently, by using the proposed method to clarify these types of diffraction light, even before fabrication, we can reduce the number of trial manufacturing runs and more efficiently identify the optimum structure of the diffraction grating lens. We have explained our analysis method by using the example of flare light. However, this analysis can be widely expanded to the overall simulation of diffraction gratings, such as superposition analysis of the multidiffraction light of pickup lenses and the convoluted images of bifocal lenses. We would like to thank Y. Okada, A. Murata, T. Suenaga, and N. Imamura for co-development of the white grating lens. References 1. M. H. Freeman, “Multifocal contact lenses utilizing diffraction and refraction,” U.S. patent 4,637,697 (20 January, 1987).
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2. S. Klein, “Understanding the diffractive bifocal contact lens,” Optom. Vis. Sci. 70, 439–460 (1993). 3. B. Roberto, “Multifocal intraocular lenses,” Curr. Opin. Ophthalmol. 16, 33–37 (2005). 4. T. Nakai and H. Ogawa, “Development of 3-layer diffractive optical elements employed for wide incident angle,” in Technical Digest of ICO’04 International Conference Optics and Photonics in Technology Frontier, Tokyo (2004). 5. W. C. Sweatt, “Describing holographic optical elements as lenses,” J. Opt. Soc. Am. 67, 803–808 (1977). 6. G. H. Spencer and M. V. R. K. Murty, “General ray-tracing procedure,” J. Opt. Soc. Am. 52, 672–676 (1962). 7. M. Muranaka, S. Kawai, T. Akiyama, I. Oki, T. Takamasa, Y. Hayashi, K. Akazu, K. Sakai, N. Miyatake, T. Kawamoto, H. Qi, M. Tanizu, K. Nishizawa, M. Fukuda, T. Fukuda, S. Yamaguchi, and N. Ohya, “Shin Kougaku Lens Gijutsu (Japanese),” Sci. Tech., 71–87 (2013). 8. D. A. Pommet, M. G. Moharam, and E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1834 (1994). 9. T. Ando, T. Korenaga, and M. Suzuki, “Explication of diffraction lights on an optical imaging system from a Fraunhofer diffraction perspective,” Proc. SPIE 8429, 842916 (2012). 10. H. Ichikawa, “Electromagnetic analysis of diffraction gratings by the finite-difference time-domain method,” J. Opt. Soc. Am. A 15, 152–157 (1998). 11. K. Oka, N. Ebizuka, and K. Kodate, “Optimal design of the grating with reflective plate of comb type for astronomical observation using RCWA,” Proc. SPIE 5290, 168 (2004). 12. T. Ando, T. Korenaga, and M. Suzuki, “Phase function design of a diffraction grating lens for an optical imaging system from a Fraunhofer diffraction perspective,” Appl. Opt. 52, 6543–6548 (2013). 13. Y. Okada, A. Murata, T. Ando, T. Suenaga, T. Korenaga, and M. Suzuki, “Development of a diffraction lens for color imaging using a nanocomposite material with high refractive index and high Abbe number (Japanese),” Kobunshi Ronbunshu 67, 390–396 (2010). 14. T. Korenaga, T. Ando, S. Moriguchi, F. Takami, Y. Takasu, S. Nishiwaki, M. Suzuki, and M. Nagashima, “Diffraction grating lens array,” Proc. SPIE 6501, 65010T (2007). 15. G. D. Boreman, Modulation Transfer Function in Optical and Electro-Optical Systems (SPIE, 2001).
2
New Products & Applications Department, Panasonic Photo & Lighting Co., Ltd./1-1 Saiwai-cho, Takatsuki City, Osaka 569-1193, Japan
Device Solutions Center, Panasonic Corporation/3-1-1 Yagumo-nakamachi, Moriguchi City, Osaka 570-8501, Japan 3
Department of Information and Physical Sciences, Graduate School of Information Science and Technology, Osaka University, 1-5 Yamadaoka, Suita, Osaka 565-0871, Japan *Corresponding author: [email protected] Received 14 January 2014; revised 8 March 2014; accepted 12 March 2014; posted 13 March 2014 (Doc. ID 204498); published 10 April 2014
We have developed a new method to analyze the amount and distribution of diffraction light for a diffraction grating lens. We have found that diffraction light includes each-order diffraction light and striped diffraction light. In this paper, we describe characteristics of striped diffraction light and suggest a way to analyze diffraction light. Our analysis method, which considers the structure of diffraction grating steps, can simulate the aberrations of an optical system, each-order diffraction light, and striped diffraction light simultaneously with high accuracy. A comparison between the simulation and experimental results is presented, and we also show how our analysis method can be used to optimize a diffraction grating lens with low flare light. © 2014 Optical Society of America OCIS codes: (050.1965) Diffractive lenses; (220.3620) Lens system design. http://dx.doi.org/10.1364/AO.53.002532
1. Introduction
Diffraction gratings have been widely used for optical pickup lenses of CDs, DVDs, and BDs for multiple-focus operation [1] and aberration correction. Furthermore, they have been used to develop diffractive bifocal contact lenses and intraocular lenses suitable for both myopia and hyperopia [2,3]. Recently, diffraction gratings have also attracted attention in lens design for optical imaging, since they effectively correct lens aberrations, such as chromatic aberration and field curvature [4]. These diffraction grating lenses are generally designed by a high refractive index method or a phase function method [5]. However, such methods are based only on ray tracing [6,7], and thus they are not capable of estimating the intensity of each 1559-128X/14/112532-07$15.00/0 © 2014 Optical Society of America 2532
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diffraction order generated by the diffraction grating lens. Scalar diffraction theory can calculate diffraction efficiency [8], but this theory has difficulty estimating it precisely with an optical imaging system because the incident light on the diffraction grating surface arrives at several angles. Moreover, it is impossible for the theory to calculate the striped diffraction light generated through Fraunhofer diffraction [9]. Electromagnetic field analyses, such as finitedifference time-domain (FDTD) [10] and rigorous coupled-wave analysis (RCWA) [11] can be used to analyze the amount of each-order diffraction light. But the FDTD method is impractical to use with an optical imaging system because the analysis can consider only a small area of diffraction grating surface due to the analysis volume, and the RCWA method is inefficient because it supposes periodic structure. As the number of diffraction grating rings
increases, estimates of the amount and distribution of diffraction light will become less accurate, as seen through a comparison of analysis and experimental results. Therefore, the development of an analysis method that can calculate diffraction light very precisely in a short time is essential to manufacturing diffraction grating lenses effectively with a minimum of trial-and-error procedures. In this paper, we suggest a novel analysis method for diffraction gratings that can estimate the amount and distribution of diffraction light accurately and quickly. This analysis is appropriate for not only flare simulation but also overall simulation of diffraction gratings, such as superposition analysis of the multidiffraction light of pickup lenses and the convoluted images of bifocal or multifocal lenses, and the applications of our method can be even more widely extended.
Fig. 1. Schematic of optical imaging system with diffraction grating lens.
2. Striped Diffraction Light
Unnecessary-order diffraction light, such as zerothand second-order diffraction light, is well known to be given off by a diffraction grating lens. However, we have discovered another kind of diffraction light that is created when a grating lens is irradiated with intense light, and we have elucidated the generation principle of this light [12]. Diffraction light produces a striped pattern near the intended focal point as it passes through minute slits formed by diffraction grating rings. Since the striped diffraction light is generated based on Fraunhofer diffraction, the width of the rings correlates closely with the interval of the striped diffraction patterns [12]. This work uses an optical imaging system with a diffraction grating lens (Table 1, Fig. 1). We positioned a stop in the optical system at the diffraction grating surface so that chromatic aberration could be corrected effectively. Our optical system consisted of two lenses with a front element and a rear element, where the rear element’s surface has a diffraction grating. To evaluate the dependence on the width of the rings, we changed the rear element and investigated four types, each with a different number of grating rings. In order to keep the focal length constant when we change the width of the grating rings, we changed the form of aspherical surface together. The width of the rings becomes narrow in proportion to the number of rings, since the stop diameters are roughly the same in each lens. Figure 2 shows images captured by these lenses sequentially. The point light source was set at 60° of the field of view. This result indicates that the wider the width of the rings becomes, the narrower the striped interval appears Table 1.
Fig. 2. Images of a point light source’s light captured by a diffraction grating lens.
(Fig. 3). This corresponds to the Fraunhofer diffraction theory. As the striped intervals on the image plane become narrow, the diffraction light shrinks near to the central main light and becomes unnoticeable (Fig. 2). However, the intensity and clarity of the striped diffraction light are not necessarily related to the width of the rings, especially on a diffraction grating lens that has plural grating rings, since the striped diffraction light is formed as an integration of the light from each diffraction grating ring. As shown in Fig. 2, a grating lens with 18 rings provides higher and clearer intensity than does one with 24 rings, despite having fewer rings. This implies that the phase function of the diffraction gratings should be designed to weaken the amplitude of the stripe pattern when the striped light from each grating ring is
Specifications of Optical Imaging System
Number of lenses Total optical track length Angle of view Resolution Stop position Grating rings
2 8.5 mm 180° 1.2 megapixels At diffraction grating surface 0–24 rings (positive power)
Fig. 3. Experimental data of 3rd–4th striped interval from central main light. 10 April 2014 / Vol. 53, No. 11 / APPLIED OPTICS
2533
superimposed with the other rings’ light. In the previous paper [12], we proposed an original design method for minimizing such light by controlling the phase function and finding a suitable combination of widths of the diffraction grating rings. 3. Diffraction Light Analysis Method Considering Grating Steps Structure
The design method in the previous paper [12] involved analysis of striped diffraction light that considered only one surface of a diffraction grating, and aberration was not incorporated in the results. To ensure that the effect of aberration is reflected in the results, the analysis should be carried out using all of the lenses in the optical system. Figure 4(a) shows the conventional ray tracing method for a diffraction grating surface. The diffraction grating is expressed by the phase function ϕ and the ray tracing on the diffraction grating surface is executed by vector analysis with Eq. (1): n1 S1 × E n0 S0 × E m
λ ∇ψr × E; λ0
(1)
wherein ψr a1 · r2 a2 · r4 a3 · r6 a4 · r8 a5 · r10 ; (2) ϕr
2π ψr: λ0
(3)
Here, n0 is the refractive index of the medium before passing through the surface, n1 is the refractive index of the medium after passing through the
Fig. 4. Principle of real-shape analysis. 2534
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surface, m is the diffraction order, λ1 is the wavelength of the light, λ2 is the construction wavelength, E is the unit normal vector on the surface, S0 is the unit vector of the incident ray, S1 is the unit vector of the exit ray, ψ is the optical path difference function, r is the distance from the optical axis, and a1 ; a2 ; a3 ; a4 ; a5 ; …, are coefficients of the optical path difference function. Equation (1) means that the exit direction after passing through the surface is calculated by adding the diffraction component of the phase function to the refractive component of the aspherical surface. However, in this case, the wavefront passing through the diffraction grating surface is seamless, thus making the first-order diffraction light 100%; the unnecessary-order diffraction light does not appear on the image plane, although it can be used to calculate aberration by analyzing all lenses in the optical system. We have developed a new analysis method that takes the unnecessary-order diffraction light into account. First, we represented the diffraction grating surface with a real grating shape having diffraction grating steps instead of a phase function. Unlike a phase function, this structure can reflect the optical path difference of the finite thickness of diffraction grating steps and generate slight wavefront gaps between neighboring rings [Fig. 4(b)]. This state of the wavefront resembles that of an actual object. This analysis is also applicable in the case of a white grating lens [13], which we have suggested reduces unnecessary-order diffraction light by controlling the refractive index of the film covering the diffraction grating surface. When the refractive index of the film is higher than that of the diffraction grating lens, the direction of the diffraction grating is inverted (Fig. 5). Second, we executed ray tracing of the optical system on each surface from the object plane to the image plane by Eq. (4):
Fig. 5. Schematic of white grating lens.
n1 S1 × E n0 S0 × E:
(4)
Compared to Eq. (1), the calculation of the phase function component can be omitted because the diffraction component is taken into consideration by the real grating shape. Note the ray should be traced with not only exit direction but also the optical path length so that wavefront gaps on the diffraction grating surface are acquired. After reaching the image plane, the ray follows the path back with the optical path length to the exit pupil position keeping the ray proceeding angle constant. By comparing the optical path length of each ray on the exit pupil with ideal spherical wave, the wavefront aberration distribution Wx; y ∈ RNx×Ny on the exit pupil can be calculated. Here, x; y are coordinates on the exit pupil, NxNy denote the number of elements along the x and y axis on the exit pupil, respectively. Figure 6 illustrates wavefront aberration on the exit pupil of the white grating lens. The number of elements Nx, Ny are 2048, respectively. In this case (Fig. 1), the exit pupil exists in the stop position. Compared to conventional analysis [Fig. 6(a), diffraction gratings with phase function], our original analysis with a real shape [Fig. 6(b)] disconnects the wavefront at the diffraction grating rings. Finally, we propagated the wavefront aberration distribution Wx; y on the exit pupil to the image plane, and thus point spread function (PSF) Pu; v could be constituted. Here, u; v are coordinates on the image plane. Propagation by Fraunhofer diffraction can be used to calculate the PSF Pu; v. Fraunhofer diffraction is solved by discrete Fourier transform: Pu; v
Ny Nx X X
ux
vy
Wx; ye−2πiNxNy
x1 y1
u 1; …; Nx v 1; …; Ny
:
(5)
The diffraction theory of Rayleigh–Sommerfeld or Fresnel diffraction can also be used to propagate the wavefront. Since the wavefront aberration considers the wave gap generated by diffraction rings, the PSF includes unnecessary-order diffraction light. Furthermore, this analysis enables us to simulate
Fig. 6. Wavefront aberration on the exit pupil (white grating lens).
Fig. 7. PSF simulated by real-shape analysis.
not only the unnecessary-order diffraction light but also the striped diffraction light, since the wavefront passing through a real grating shape is intermittent. Our analysis is able to simultaneously calculate the aberrations of an optical system, unnecessary-order diffraction light, and striped diffraction light. Moreover, since this method is based on ray tracing and wavefront propagation of exit pupil aberration, it can calculate them quickly. Besides, our method is applicable to the range of scalar field: more than several micrometer order of the width of rings. Figure 7 shows the results of PSF simulation. We simulated an optical imaging system with the white grating lens in Fig. 1 (24 rings) at 60° of the field of view. Conventional analysis, using diffraction gratings with a phase function [Fig. 7(a)], can simulate only aberrations, leaving unnecessary-order diffraction light and striped diffraction light unobserved. Meanwhile, our analysis using diffraction gratings with real shape can express the flare light around the central main light, corresponding to the experimental results in Fig. 2 (24 rings). This shows that our analysis can be used to effectively optimize the diffraction grating height of diffraction grating steps related to unnecessary-order diffraction light [14]. Figure 8(a) shows the dependence of diffraction-light flare distribution on diffraction grating height. The integrated brightness of flare light normalized by total brightness is shown in Fig. 8(b). This result indicates that the best diffraction grating height for reducing flare light can be calculated and easily determined by our analysis method. 4. Comparison of Analysis and Experiment
We compared the data of simulation and experimental results using the optical system with a white grating lens (Fig. 9). The diffraction grating is before optimization, the diffraction grating height is 15 μm, and the field of view is 60°. The wavelengths by analysis are 640, 550, and 440 nm, and the captured images of a halogen point light source were taken as experimental PSF data by a white grating lens through a dichroic filter; wavelengths were 610–650 nm (with IR cut filter), 505–575 nm, and 400–495 nm. Figure 9 shows the cross section of the PSF intensity distribution, with the position in the image plane on the abscissa and the intensity of the PSF on the 10 April 2014 / Vol. 53, No. 11 / APPLIED OPTICS
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Fig. 8. Diffraction grating height dependence of PSF distribution.
ordinate. The PSF value on the ordinate was normalized as 100 by integrated total brightness. Since the max. value on the ordinate is shown as 0.03% against integrated total brightness, the peak of PSF is saturated, and the bottom distribution of PSF can be observed in detail on the graph. The diffraction-light flare has been observed near the saturated central main lights, and these distributions of the analysis results [Fig. 9(a)] closely correspond to those of the experiment [Fig. 9(b)] at each wavelength from the viewpoints of the generated amount and direction of the diffraction-light flare. This experimental data show that the shorter the wavelength, the larger the amount of diffraction-light flare. This result also corresponds to that of the analysis.
We used our analysis method to optimize the diffraction grating surface of the optical system in Fig. 1. Figure 10(a) is the PSF intensity of the optimized diffraction grating. We adjusted the interval of the grating rings and the diffraction grating height (13.8 μm). Compared to Fig. 9(a), it is obvious that the diffraction-light flare decreases in the simulation. The reduction of diffraction-light flare, especially in the wavelength range of blue, seems to have the effect of lowering the diffraction grating height. Based on this analysis, we have successfully created an effective diffraction grating lens for reducing the diffraction-light flare. The very close correspondence of the amount and distribution of diffraction light between our analysis and the experiment is due to our method’s ability to consider the aberration characteristics and each incident angle of rays to the diffraction grating surface. This has never been achieved by conventional methods. 5. Applied Analysis
The measured modulation transfer function (MTF) of the fabricated diffraction grating lens incorporates the characteristics of the unnecessary-order diffraction light and striped diffraction light. Therefore, the simulated MTF should also be calculated by considering those types of diffraction light. Since MTF can be calculated by the wavefront aberration on the exit pupil [15], it is possible to calculate the MTF of diffraction grating lenses using our analysis. Figure 11 shows the results of MTF simulation, with defocus position on the abscissa and the MTF value on the ordinate. Figure 11(a) is a conventional MTF of a diffraction grating calculated by a phase function.
Fig. 9. Comparison of PSF intensity distribution between simulation and experiment (unoptimized diffraction grating). 2536
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Fig. 10. Comparison of PSF intensity distribution between simulation and experiment (optimized diffraction grating).
As shown in Fig. 11(b), the analysis with a real shape can calculate MTF as well as the conventional approach. Specifically, compared to a phase function, the peak values of MTF analysis with a real shape are lower by only a few percentage points. This is
caused by diffraction-light flare, and it indicates that our analysis method is appropriate for simulating the MTF of diffraction grating lenses precisely. Our analysis can also calculate light lost through the diffraction grating’s wall by tracing rays while considering refraction and reflection there. Light loss is divided into three classes: one-time refraction passing directly through the diffraction wall, threetime refraction passing through the diffraction wall
Fig. 11. MTF (simulation).
Fig. 12. Light loss simulation. 10 April 2014 / Vol. 53, No. 11 / APPLIED OPTICS
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once and the diffraction surfaces twice, and total reflection at the diffraction wall (Fig. 12). These are categorized by considering the positions of rays, the directions of diffraction gratings, the incident angles of rays, and the refractive index of the lens materials. We have used this method to successfully analyze the distribution of these types of light loss on the image plane, and this analysis is thus suitable for determining the number of diffraction grating rings from the amount of light loss. 6. Conclusion
We have developed a new method to analyze the amount and distribution of diffraction light that affects the image quality of an imaging system. This analysis has achieved high-accuracy and simultaneous simulation of the aberrations in an optical system, the unnecessary-order diffraction light, and the striped diffraction light. Consequently, by using the proposed method to clarify these types of diffraction light, even before fabrication, we can reduce the number of trial manufacturing runs and more efficiently identify the optimum structure of the diffraction grating lens. We have explained our analysis method by using the example of flare light. However, this analysis can be widely expanded to the overall simulation of diffraction gratings, such as superposition analysis of the multidiffraction light of pickup lenses and the convoluted images of bifocal lenses. We would like to thank Y. Okada, A. Murata, T. Suenaga, and N. Imamura for co-development of the white grating lens. References 1. M. H. Freeman, “Multifocal contact lenses utilizing diffraction and refraction,” U.S. patent 4,637,697 (20 January, 1987).
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2. S. Klein, “Understanding the diffractive bifocal contact lens,” Optom. Vis. Sci. 70, 439–460 (1993). 3. B. Roberto, “Multifocal intraocular lenses,” Curr. Opin. Ophthalmol. 16, 33–37 (2005). 4. T. Nakai and H. Ogawa, “Development of 3-layer diffractive optical elements employed for wide incident angle,” in Technical Digest of ICO’04 International Conference Optics and Photonics in Technology Frontier, Tokyo (2004). 5. W. C. Sweatt, “Describing holographic optical elements as lenses,” J. Opt. Soc. Am. 67, 803–808 (1977). 6. G. H. Spencer and M. V. R. K. Murty, “General ray-tracing procedure,” J. Opt. Soc. Am. 52, 672–676 (1962). 7. M. Muranaka, S. Kawai, T. Akiyama, I. Oki, T. Takamasa, Y. Hayashi, K. Akazu, K. Sakai, N. Miyatake, T. Kawamoto, H. Qi, M. Tanizu, K. Nishizawa, M. Fukuda, T. Fukuda, S. Yamaguchi, and N. Ohya, “Shin Kougaku Lens Gijutsu (Japanese),” Sci. Tech., 71–87 (2013). 8. D. A. Pommet, M. G. Moharam, and E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1834 (1994). 9. T. Ando, T. Korenaga, and M. Suzuki, “Explication of diffraction lights on an optical imaging system from a Fraunhofer diffraction perspective,” Proc. SPIE 8429, 842916 (2012). 10. H. Ichikawa, “Electromagnetic analysis of diffraction gratings by the finite-difference time-domain method,” J. Opt. Soc. Am. A 15, 152–157 (1998). 11. K. Oka, N. Ebizuka, and K. Kodate, “Optimal design of the grating with reflective plate of comb type for astronomical observation using RCWA,” Proc. SPIE 5290, 168 (2004). 12. T. Ando, T. Korenaga, and M. Suzuki, “Phase function design of a diffraction grating lens for an optical imaging system from a Fraunhofer diffraction perspective,” Appl. Opt. 52, 6543–6548 (2013). 13. Y. Okada, A. Murata, T. Ando, T. Suenaga, T. Korenaga, and M. Suzuki, “Development of a diffraction lens for color imaging using a nanocomposite material with high refractive index and high Abbe number (Japanese),” Kobunshi Ronbunshu 67, 390–396 (2010). 14. T. Korenaga, T. Ando, S. Moriguchi, F. Takami, Y. Takasu, S. Nishiwaki, M. Suzuki, and M. Nagashima, “Diffraction grating lens array,” Proc. SPIE 6501, 65010T (2007). 15. G. D. Boreman, Modulation Transfer Function in Optical and Electro-Optical Systems (SPIE, 2001).
