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Analysis of materials selected for multilayer diffractive optical elements
G Model IJLEO-54274; No. of Pages 4
ARTICLE IN PRESS Optik xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Analysis of materials selected for multilayer diffractive optical elements Changxi Xue a,b,∗ , Suet To b a
Department of Optical Engineering, Changchun University of Science and Technology, Changchun 130022, China Partner State Key Laboratory of Ultra-precision Machining Technology, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong b
a r t i c l e
i n f o
Article history: Received 30 June 2013 Accepted 17 December 2013 Available online xxx Keywords: Diffractive optics Micro-optical devices Optical design and fabrication
a b s t r a c t An analysis model to optimize the materials selected for multilayer diffractive elements (MLDOEs) is presented with approximate Cauchy dispersion formula of refractive index and the maximum polychromatic integral diffraction efficiency (PIDE). The analysis model presents that the maximum PIDE of MLDOEs consisting of two materials with large Abbe number difference and small partial dispersion difference can be generated. The scope of application and the relationship between diffraction efficiencies of MLDOEs with different material pairs and different design wavelength pairs are presented and simulated with the analysis model of MLDOEs. Crown Copyright © 2014 Published by Elsevier GmbH. All rights reserved.
1. Introduction Diffractive optical elements (DOEs) have been widely used in hybrid diffractive-refractive optical system for the visible spectrum and IR wavebands to correct the achromatic aberration and various monochromatic aberrations, reduce the secondary and produce an arbitrary phase distribution since DOEs have a unique negative chromatic dispersion characteristic and the optical characteristic of an aspheric surface. However, the diffraction efficiency of conventional single-layer surface relief DOEs decreases quickly when the wavelength deviates from the design wavelength, which hinders the application in broadband optical systems. There are many papers to discuss the design principle of multilayer diffractive optical elements (MLDOEs) for improving the diffraction efficiency in a wideband [1–8]. In Ref. [3], the materials selection principle of MLDOEs is presented that the absolute surface relief structure heights of two harmonic diffractive elements (HDEs) are small when the Abbe number difference of the two materials of MLDOEs is large. Eq. (19) in Ref. [6], the simplified design model for diffraction efficiency of MLDOEs is presented with no consideration of material dispersion. In Ref. [7,8], the analyses of materials selected for MLDOEs are discussed with most design examples. But there has been no an analysis model to optimize materials selected for MLDOEs and no discussion of the relationship between the
∗ Corresponding author. E-mail address: [email protected] (C. Xue).
diffraction efficiencies and the selected material pairs of MLDOEs. In this paper, the principle of optimize materials selected for MLDOEs is presented with consideration of polychromatic integral diffraction efficiency (PIDE) [9]. The maximum PIDEs of MLDOEs composed with different material pairs are compared with consideration of the Abbe number and partial dispersion of the materials selected using the simplify design model with approximate Cauchy dispersion formula of refractive index. At last, the field of application of simplify design model is presented with comparisons of diffraction efficiencies of MLDOEs for different material pairs and different design wavelength pairs which are determined with consideration of maximum PIDE. The result can be used to optimize the materials selected of MLDOEs.
2. Design principle of MLDOEs The MLDOE which consists of two harmonic diffractive elements (HDEs) with different dispersive optical materials is aligned as shown in Fig. 1.The two HDEs have concentric diffraction gratings. According to Ref. [3], the phase retardation () as a function of wavelength at a given zone period as shown in Fig. 1 is () =
2 ((n1 () − 1)H1 + (n2 () − 1)H2 ) = m2,
(1)
where m is diffraction order at the two design wavelengths, H1 and H2 are the surface relief heights of the HDEs, and n1 () and n2 () are the refractive index of the first-layer material and the second-layer
0030-4026/$ – see front matter. Crown Copyright © 2014 Published by Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.12.049
Please cite this article in press as: C. Xue, S. To, Analysis of materials selected for multilayer diffractive optical elements, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2013.12.049
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Fig. 1. Structure of MLDOEs, H1 and H2 are the surface relief heights of the HDEs, D is thick of the air gap between them.
material. After selecting two different materials and 1 and 2 , the two design wavelengths in the design waveband, the surface relief heights of the two HDEs, according to Refs. [3,5], are H1 = H2 =
m1 (n2 (2 ) − 1) − m2 (n2 (1 ) − 1) , (n1 (1 ) − 1)(n2 (2 ) − 1) − (n1 (2 ) − 1)(n2 (1 ) − 1) m2 (n1 (1 ) − 1) − m1 (n1 (2 ) − 1) . (n1 (1 ) − 1)(n2 (2 ) − 1) − (n1 (2 ) − 1)(n2 (1 ) − 1)
(2)
So, when the Abbe number difference of the two materials of MLDOEs is large, the absolute denominators of Eq. (2) are relatively big. The absolute surface relief structure heights of two HDEs, thus, are small [3]. Using Eqs. (1) and (2), the mth order diffraction efficiency of the MLDOEs can be written as
m = sin c 2 m −
() 2
,
(3)
where sinc(x) = sinx/x, and the corresponding PIDE of the MLDOEs for a given wideband is [9]
max −
m
1 int (1 , 2 ) = max − min
sin c 2 m −
() 2
d,
(4)
min
where min and max are the minimum and the maximum wavelengths of the design waveband. The simplify design model of MLDOEs is presented with the approximate Cauchy dispersion formula in Ref. [10], and the formula and the phase retardation expression Eq. (1) of MLDOEs with these two materials are ni () = ai +
bi , 2
(5) 2
() =
2
2 2 1 (1 + 2 ) + 2 (2 − 1 ) m . 2 (1 + 2 )
(6)
where, ni () is the refractive index of the material, ai and bi are dispersion coefficients. The refractive index of the two different materials are expressed with Eq. (5), so the Abbe number vd of the two materials are different and the partial dispersion of the two materials are both Pgf = 0.54096, and the absolute value of the ratio of partial dispersion difference to Abbe number difference is zero. Eq. (6) shows that the phase retardation of the MLDOE is independent of all materials’ dispersion coefficients.
Fig. 2. Refractive index versus wavelength.
difference of diffraction efficiency at wavelength 465 nm is 29.42%, the biggest among three MLDOEs when the design wavelength pair is 400 and 700 nm. The difference of diffraction efficiency, with Fand C-lines as design wavelength pair, is smaller than 1.82% when the wavelength is longer than 471 nm, but the difference is large in short wavelength region. Based on this analysis, the difference of diffraction efficiency is too big to give out the principle of materials selected for MLDOEs. Based on the method for designing MLDOEs with accurate formula of refractive index in Ref. [5], we, however, used approximate Cauchy dispersion formula of refractive index. The PIDEs for different design wavelength pairs are shown in Fig. 4a, and the maximum PIDEs for the first design wavelength, changed from 400 to 700 nm, are shown in Fig. 4b. From Fig. 4, the maximum PIDE is 99.620%, and the corresponding design wavelength pair is 440.3 and 606.1 nm. Furthermore, the relationship between maximum PIDEs and material pairs selected is found. Firstly, the Abbe number and partial dispersion of the six materials and the absolute values of the ratio of partial dispersion difference to Abbe number difference for any two materials used to design MLDOEs are calculated. Secondly, the maximum PIDEs and corresponding design wavelength pairs are determined according to Ref. [5]. The relationship between maximum PIDEs and selected material pairs is shown in Fig. 5.The maximum PIDE of the MLDOEs composed of N-SSK8 and PMMA is 99.59%, the largest among these MLDOEs examples, design wavelengths being 438.3 and 602.5 nm. At the same time, the absolute value of the ratio of partial dispersion difference to Abbe number difference is the smallest among these material pairs, which is 0.062 × 10−3 . The maximum PIDE and this absolute ratio are near those that composed of two materials expressed with approximate Cauchy dispersion formula. We also found that the maximum PIDE of the MLDOE composed of PC and PS is the lowest, 97.37%, and the absolute ratio of partial dispersion difference to Abbe number difference is the largest among these material pairs, and the corresponding design wavelengths are 424.8 and 573.8 nm. According to this analysis and Fig. 5, the conclusion is drawn that maximum PIDE
3. Analysis of materials selected for multilayer diffractive optical elements Three MLDOEs, as examples for visible waveband, are designed with m = 1.The three MLDOEs are composed of three pairs of different materials, respectively, which are polycarbonate (PC) and polystyrene (PS), N-SSK8 and poly-methyl methacrylate (PMMA), N-FK51A and P-SF67 [11,12], and the refractive index of the materials are shown in Fig. 2. With Eqs. (1), (3) and (6), the difference of diffraction efficiencies of MLDOEs with two different design wavelength pairs, 400 and 700 nm, F- and C-lines, are shown in Fig. 3. From Fig. 3, the
Fig. 3. Diffraction efficiency difference of MLDOEs with three different material pairs at two different design wavelength pairs, 400 and 700 nm, 486.1 and 656.3 nm.
Please cite this article in press as: C. Xue, S. To, Analysis of materials selected for multilayer diffractive optical elements, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2013.12.049
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Fig. 6. Diffraction efficiency difference of MLDOEs with different material pairs and different design wavelength pairs which are determined with consideration of maximum PIDEs. The solid lines represent the difference of diffraction efficiencies of MLDOEs at design wavelength pairs which are determined with consideration of maximum PIDE using accurate formula of refractive index for selected materials, and the dashed lines represent the difference of diffraction efficiencies of MLDOEs at the design wavelengths, 440.3 and 606.1 nm.
Fig. 4. (a) PIDE of MLDOEs with different design wavelength pairs and the refractive index is expressed with approximate Cauchy dispersion formula. (b) Maximum PIDEs of MLDOEs versus to the first design wavelength when the second design wavelength changed from 400 nm to 700 nm.
of MLDOEs is dependant on the Abbe number and partial dispersion of the materials pairs selected. More detail discussion shows the relationship between diffraction efficiency and material pair selected. We use three MLDOEs examples which are working in the visible waveband, from 400 to
700 nm, and the corresponding design wavelength pairs are determined with consideration of maximum PIDE with accurate formula of refractive index. The diffraction efficiencies of three MLDOEs examples, whose refractive indices are expressed with accurate formula of refractive index, are used to subtract those with approximate Cauchy dispersion formula for different design wavelengths pairs, respectively, and the difference are shown in Fig. 6. In Fig. 6, the solid lines represent the difference of diffraction efficiencies of MLDOEs at design wavelength pairs which are determined with consideration of maximum PIDE using accurate formula of refractive index for selected materials, and the dashed lines represent the difference of diffraction efficiencies of MLDOEs at the design wavelengths, 440.3 and 606.1 nm. From Fig. 6, the difference of diffraction efficiency of the MLDOEs, composed of N-SSK8 and PMMA, is smaller than 1% in the wideband; and the difference to the MLDOEs, composed of N-FK51A and P-SF67 is smaller than 0.47% in the waveband from 422.3 to 674.8 nm; however, the difference to the MLDOEs, composed of PS and PC, is quite obvious in the visible waveband. Base on this analysis, when the partial dispersion difference of the two materials is small, correspondently the difference of diffraction efficiency of MLDOEs designed with consideration of maximum PIDEs for a given waveband is quite small. We can conclude that the diffraction efficiency is independent of material pairs selected, the simplify design model is valid only when the difference of partial dispersion is small and the design wavelengths pair is determined with consideration of the maximum PIDE of MLDOEs.
4. Conclusion
Fig. 5. (a) Maximum PIDE of MLDOEs versus the absolute ratio of partial dispersion difference to Abbe number difference for any two materials. (b) Partial enlarged detail of (a).
An analysis model to optimize the materials selected for multilayer diffractive elements (MLDOEs) is presented to show the scope of application and the relationship between diffraction efficiency and material pairs selected which affect the surface relief heights of the HDEs. The maximum PIDE affected by the Abbe number and partial dispersion of material pairs selected have been found. The MLDOEs designed with two materials are suggested which have large Abbe number difference and small partial dispersion difference, so that the maximum PIDE and the small surface relief heights of the HDEs can be reached. Meanwhile the diffraction efficiency is independent of selected materials when the design wavelength pair is determined in consideration of the maximum PIDE. This method can be used to optimize materials selected for MLDOEs.
Please cite this article in press as: C. Xue, S. To, Analysis of materials selected for multilayer diffractive optical elements, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2013.12.049
G Model IJLEO-54274; No. of Pages 4
ARTICLE IN PRESS C. Xue, S. To / Optik xxx (2014) xxx–xxx
4
Acknowledgment This work is partially supported by the Research Committee of the The Hong Kong Polytechnic University (Project code: G-YJ34) and the National Natural Science Foundation of China (NSFC) (grant no. 61108044).
References [1] T. Nakai, H. Ogawa, Research on multilayer diffractive optical elements and their application to camera lenses, in Diffractive Optics and Micro-Optics, in: R. Magnusson (Ed.), Trends in Optics and Photonics Series, 75, Optical Society of America, 2002, Paper DMA2. [2] S. Noach, Y. Arieli, N. Eisenberg, Wave-front control and aberration correction with a diffractive optical element, Opt. Lett. 24 (1999) 333–335.
[3] Y. Arieli, S. Noach, S. Ozeri, N. Eisenberg, Design of diffractive optical elements for multiple wavelengths, Appl. Opt. 37 (1998) 6174–6177. [4] Y. Arieli, S. Noach, S. Ozeri, N. Eisenberg, Design of a diffractive optical elements for wide spectral bandwidth, Opt. Lett. 23 (1998) 823–824. [5] C. Xue, Q. Cui, Design of multilayer diffractive optical elements with polychromatic integral diffraction efficiency, Opt. Lett. 35 (2010) 986–988. [6] B.H. Kleemann, M. Seesselberg, J. Ruoff, Design concepts for broadband highefficiency DOEs, J. Europ. Opt. Soc. Rap. Public. 3 (2008) 08015. [7] H. Ukuda, Yokohama, Optical material and optical system using it, U.S. Patent No. 6759471B1 (2004). [8] H. Iwasa, Kawasaki-shi, Laminated diffractive optical elements and resin composition therefore, U.S. Patent No.2009/0190224 A1 (2009). [9] D.A. Buralli, G.M. Morris, Effects of diffraction efficiency on the modulation transfer function of diffractive lenses, Appl. Opt. 31 (1992) 4388–4396. [10] W.J. Smith, Modern Optical Engineering, McGraw-Hill, Inc., 2008. [11] M. Bass, Optical Society of America, Handbook of Optics, II, McGraw-Hill, New York, 1995. [12] Schott optical glass catalog (SCHOTT AG Division SCHOTT Lithotec Otto-SchottStrasse 1307745 Jena Germany, 2007).
Please cite this article in press as: C. Xue, S. To, Analysis of materials selected for multilayer diffractive optical elements, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2013.12.049
ARTICLE IN PRESS Optik xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Analysis of materials selected for multilayer diffractive optical elements Changxi Xue a,b,∗ , Suet To b a
Department of Optical Engineering, Changchun University of Science and Technology, Changchun 130022, China Partner State Key Laboratory of Ultra-precision Machining Technology, Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong b
a r t i c l e
i n f o
Article history: Received 30 June 2013 Accepted 17 December 2013 Available online xxx Keywords: Diffractive optics Micro-optical devices Optical design and fabrication
a b s t r a c t An analysis model to optimize the materials selected for multilayer diffractive elements (MLDOEs) is presented with approximate Cauchy dispersion formula of refractive index and the maximum polychromatic integral diffraction efficiency (PIDE). The analysis model presents that the maximum PIDE of MLDOEs consisting of two materials with large Abbe number difference and small partial dispersion difference can be generated. The scope of application and the relationship between diffraction efficiencies of MLDOEs with different material pairs and different design wavelength pairs are presented and simulated with the analysis model of MLDOEs. Crown Copyright © 2014 Published by Elsevier GmbH. All rights reserved.
1. Introduction Diffractive optical elements (DOEs) have been widely used in hybrid diffractive-refractive optical system for the visible spectrum and IR wavebands to correct the achromatic aberration and various monochromatic aberrations, reduce the secondary and produce an arbitrary phase distribution since DOEs have a unique negative chromatic dispersion characteristic and the optical characteristic of an aspheric surface. However, the diffraction efficiency of conventional single-layer surface relief DOEs decreases quickly when the wavelength deviates from the design wavelength, which hinders the application in broadband optical systems. There are many papers to discuss the design principle of multilayer diffractive optical elements (MLDOEs) for improving the diffraction efficiency in a wideband [1–8]. In Ref. [3], the materials selection principle of MLDOEs is presented that the absolute surface relief structure heights of two harmonic diffractive elements (HDEs) are small when the Abbe number difference of the two materials of MLDOEs is large. Eq. (19) in Ref. [6], the simplified design model for diffraction efficiency of MLDOEs is presented with no consideration of material dispersion. In Ref. [7,8], the analyses of materials selected for MLDOEs are discussed with most design examples. But there has been no an analysis model to optimize materials selected for MLDOEs and no discussion of the relationship between the
∗ Corresponding author. E-mail address: [email protected] (C. Xue).
diffraction efficiencies and the selected material pairs of MLDOEs. In this paper, the principle of optimize materials selected for MLDOEs is presented with consideration of polychromatic integral diffraction efficiency (PIDE) [9]. The maximum PIDEs of MLDOEs composed with different material pairs are compared with consideration of the Abbe number and partial dispersion of the materials selected using the simplify design model with approximate Cauchy dispersion formula of refractive index. At last, the field of application of simplify design model is presented with comparisons of diffraction efficiencies of MLDOEs for different material pairs and different design wavelength pairs which are determined with consideration of maximum PIDE. The result can be used to optimize the materials selected of MLDOEs.
2. Design principle of MLDOEs The MLDOE which consists of two harmonic diffractive elements (HDEs) with different dispersive optical materials is aligned as shown in Fig. 1.The two HDEs have concentric diffraction gratings. According to Ref. [3], the phase retardation () as a function of wavelength at a given zone period as shown in Fig. 1 is () =
2 ((n1 () − 1)H1 + (n2 () − 1)H2 ) = m2,
(1)
where m is diffraction order at the two design wavelengths, H1 and H2 are the surface relief heights of the HDEs, and n1 () and n2 () are the refractive index of the first-layer material and the second-layer
0030-4026/$ – see front matter. Crown Copyright © 2014 Published by Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.12.049
Please cite this article in press as: C. Xue, S. To, Analysis of materials selected for multilayer diffractive optical elements, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2013.12.049
G Model
ARTICLE IN PRESS
IJLEO-54274; No. of Pages 4
C. Xue, S. To / Optik xxx (2014) xxx–xxx
2
Fig. 1. Structure of MLDOEs, H1 and H2 are the surface relief heights of the HDEs, D is thick of the air gap between them.
material. After selecting two different materials and 1 and 2 , the two design wavelengths in the design waveband, the surface relief heights of the two HDEs, according to Refs. [3,5], are H1 = H2 =
m1 (n2 (2 ) − 1) − m2 (n2 (1 ) − 1) , (n1 (1 ) − 1)(n2 (2 ) − 1) − (n1 (2 ) − 1)(n2 (1 ) − 1) m2 (n1 (1 ) − 1) − m1 (n1 (2 ) − 1) . (n1 (1 ) − 1)(n2 (2 ) − 1) − (n1 (2 ) − 1)(n2 (1 ) − 1)
(2)
So, when the Abbe number difference of the two materials of MLDOEs is large, the absolute denominators of Eq. (2) are relatively big. The absolute surface relief structure heights of two HDEs, thus, are small [3]. Using Eqs. (1) and (2), the mth order diffraction efficiency of the MLDOEs can be written as
m = sin c 2 m −
() 2
,
(3)
where sinc(x) = sinx/x, and the corresponding PIDE of the MLDOEs for a given wideband is [9]
max −
m
1 int (1 , 2 ) = max − min
sin c 2 m −
() 2
d,
(4)
min
where min and max are the minimum and the maximum wavelengths of the design waveband. The simplify design model of MLDOEs is presented with the approximate Cauchy dispersion formula in Ref. [10], and the formula and the phase retardation expression Eq. (1) of MLDOEs with these two materials are ni () = ai +
bi , 2
(5) 2
() =
2
2 2 1 (1 + 2 ) + 2 (2 − 1 ) m . 2 (1 + 2 )
(6)
where, ni () is the refractive index of the material, ai and bi are dispersion coefficients. The refractive index of the two different materials are expressed with Eq. (5), so the Abbe number vd of the two materials are different and the partial dispersion of the two materials are both Pgf = 0.54096, and the absolute value of the ratio of partial dispersion difference to Abbe number difference is zero. Eq. (6) shows that the phase retardation of the MLDOE is independent of all materials’ dispersion coefficients.
Fig. 2. Refractive index versus wavelength.
difference of diffraction efficiency at wavelength 465 nm is 29.42%, the biggest among three MLDOEs when the design wavelength pair is 400 and 700 nm. The difference of diffraction efficiency, with Fand C-lines as design wavelength pair, is smaller than 1.82% when the wavelength is longer than 471 nm, but the difference is large in short wavelength region. Based on this analysis, the difference of diffraction efficiency is too big to give out the principle of materials selected for MLDOEs. Based on the method for designing MLDOEs with accurate formula of refractive index in Ref. [5], we, however, used approximate Cauchy dispersion formula of refractive index. The PIDEs for different design wavelength pairs are shown in Fig. 4a, and the maximum PIDEs for the first design wavelength, changed from 400 to 700 nm, are shown in Fig. 4b. From Fig. 4, the maximum PIDE is 99.620%, and the corresponding design wavelength pair is 440.3 and 606.1 nm. Furthermore, the relationship between maximum PIDEs and material pairs selected is found. Firstly, the Abbe number and partial dispersion of the six materials and the absolute values of the ratio of partial dispersion difference to Abbe number difference for any two materials used to design MLDOEs are calculated. Secondly, the maximum PIDEs and corresponding design wavelength pairs are determined according to Ref. [5]. The relationship between maximum PIDEs and selected material pairs is shown in Fig. 5.The maximum PIDE of the MLDOEs composed of N-SSK8 and PMMA is 99.59%, the largest among these MLDOEs examples, design wavelengths being 438.3 and 602.5 nm. At the same time, the absolute value of the ratio of partial dispersion difference to Abbe number difference is the smallest among these material pairs, which is 0.062 × 10−3 . The maximum PIDE and this absolute ratio are near those that composed of two materials expressed with approximate Cauchy dispersion formula. We also found that the maximum PIDE of the MLDOE composed of PC and PS is the lowest, 97.37%, and the absolute ratio of partial dispersion difference to Abbe number difference is the largest among these material pairs, and the corresponding design wavelengths are 424.8 and 573.8 nm. According to this analysis and Fig. 5, the conclusion is drawn that maximum PIDE
3. Analysis of materials selected for multilayer diffractive optical elements Three MLDOEs, as examples for visible waveband, are designed with m = 1.The three MLDOEs are composed of three pairs of different materials, respectively, which are polycarbonate (PC) and polystyrene (PS), N-SSK8 and poly-methyl methacrylate (PMMA), N-FK51A and P-SF67 [11,12], and the refractive index of the materials are shown in Fig. 2. With Eqs. (1), (3) and (6), the difference of diffraction efficiencies of MLDOEs with two different design wavelength pairs, 400 and 700 nm, F- and C-lines, are shown in Fig. 3. From Fig. 3, the
Fig. 3. Diffraction efficiency difference of MLDOEs with three different material pairs at two different design wavelength pairs, 400 and 700 nm, 486.1 and 656.3 nm.
Please cite this article in press as: C. Xue, S. To, Analysis of materials selected for multilayer diffractive optical elements, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2013.12.049
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Fig. 6. Diffraction efficiency difference of MLDOEs with different material pairs and different design wavelength pairs which are determined with consideration of maximum PIDEs. The solid lines represent the difference of diffraction efficiencies of MLDOEs at design wavelength pairs which are determined with consideration of maximum PIDE using accurate formula of refractive index for selected materials, and the dashed lines represent the difference of diffraction efficiencies of MLDOEs at the design wavelengths, 440.3 and 606.1 nm.
Fig. 4. (a) PIDE of MLDOEs with different design wavelength pairs and the refractive index is expressed with approximate Cauchy dispersion formula. (b) Maximum PIDEs of MLDOEs versus to the first design wavelength when the second design wavelength changed from 400 nm to 700 nm.
of MLDOEs is dependant on the Abbe number and partial dispersion of the materials pairs selected. More detail discussion shows the relationship between diffraction efficiency and material pair selected. We use three MLDOEs examples which are working in the visible waveband, from 400 to
700 nm, and the corresponding design wavelength pairs are determined with consideration of maximum PIDE with accurate formula of refractive index. The diffraction efficiencies of three MLDOEs examples, whose refractive indices are expressed with accurate formula of refractive index, are used to subtract those with approximate Cauchy dispersion formula for different design wavelengths pairs, respectively, and the difference are shown in Fig. 6. In Fig. 6, the solid lines represent the difference of diffraction efficiencies of MLDOEs at design wavelength pairs which are determined with consideration of maximum PIDE using accurate formula of refractive index for selected materials, and the dashed lines represent the difference of diffraction efficiencies of MLDOEs at the design wavelengths, 440.3 and 606.1 nm. From Fig. 6, the difference of diffraction efficiency of the MLDOEs, composed of N-SSK8 and PMMA, is smaller than 1% in the wideband; and the difference to the MLDOEs, composed of N-FK51A and P-SF67 is smaller than 0.47% in the waveband from 422.3 to 674.8 nm; however, the difference to the MLDOEs, composed of PS and PC, is quite obvious in the visible waveband. Base on this analysis, when the partial dispersion difference of the two materials is small, correspondently the difference of diffraction efficiency of MLDOEs designed with consideration of maximum PIDEs for a given waveband is quite small. We can conclude that the diffraction efficiency is independent of material pairs selected, the simplify design model is valid only when the difference of partial dispersion is small and the design wavelengths pair is determined with consideration of the maximum PIDE of MLDOEs.
4. Conclusion
Fig. 5. (a) Maximum PIDE of MLDOEs versus the absolute ratio of partial dispersion difference to Abbe number difference for any two materials. (b) Partial enlarged detail of (a).
An analysis model to optimize the materials selected for multilayer diffractive elements (MLDOEs) is presented to show the scope of application and the relationship between diffraction efficiency and material pairs selected which affect the surface relief heights of the HDEs. The maximum PIDE affected by the Abbe number and partial dispersion of material pairs selected have been found. The MLDOEs designed with two materials are suggested which have large Abbe number difference and small partial dispersion difference, so that the maximum PIDE and the small surface relief heights of the HDEs can be reached. Meanwhile the diffraction efficiency is independent of selected materials when the design wavelength pair is determined in consideration of the maximum PIDE. This method can be used to optimize materials selected for MLDOEs.
Please cite this article in press as: C. Xue, S. To, Analysis of materials selected for multilayer diffractive optical elements, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2013.12.049
G Model IJLEO-54274; No. of Pages 4
ARTICLE IN PRESS C. Xue, S. To / Optik xxx (2014) xxx–xxx
4
Acknowledgment This work is partially supported by the Research Committee of the The Hong Kong Polytechnic University (Project code: G-YJ34) and the National Natural Science Foundation of China (NSFC) (grant no. 61108044).
References [1] T. Nakai, H. Ogawa, Research on multilayer diffractive optical elements and their application to camera lenses, in Diffractive Optics and Micro-Optics, in: R. Magnusson (Ed.), Trends in Optics and Photonics Series, 75, Optical Society of America, 2002, Paper DMA2. [2] S. Noach, Y. Arieli, N. Eisenberg, Wave-front control and aberration correction with a diffractive optical element, Opt. Lett. 24 (1999) 333–335.
[3] Y. Arieli, S. Noach, S. Ozeri, N. Eisenberg, Design of diffractive optical elements for multiple wavelengths, Appl. Opt. 37 (1998) 6174–6177. [4] Y. Arieli, S. Noach, S. Ozeri, N. Eisenberg, Design of a diffractive optical elements for wide spectral bandwidth, Opt. Lett. 23 (1998) 823–824. [5] C. Xue, Q. Cui, Design of multilayer diffractive optical elements with polychromatic integral diffraction efficiency, Opt. Lett. 35 (2010) 986–988. [6] B.H. Kleemann, M. Seesselberg, J. Ruoff, Design concepts for broadband highefficiency DOEs, J. Europ. Opt. Soc. Rap. Public. 3 (2008) 08015. [7] H. Ukuda, Yokohama, Optical material and optical system using it, U.S. Patent No. 6759471B1 (2004). [8] H. Iwasa, Kawasaki-shi, Laminated diffractive optical elements and resin composition therefore, U.S. Patent No.2009/0190224 A1 (2009). [9] D.A. Buralli, G.M. Morris, Effects of diffraction efficiency on the modulation transfer function of diffractive lenses, Appl. Opt. 31 (1992) 4388–4396. [10] W.J. Smith, Modern Optical Engineering, McGraw-Hill, Inc., 2008. [11] M. Bass, Optical Society of America, Handbook of Optics, II, McGraw-Hill, New York, 1995. [12] Schott optical glass catalog (SCHOTT AG Division SCHOTT Lithotec Otto-SchottStrasse 1307745 Jena Germany, 2007).
Please cite this article in press as: C. Xue, S. To, Analysis of materials selected for multilayer diffractive optical elements, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2013.12.049