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Optimization and analysis of multi-layer diffractive optical elements in visible waveband
G Model IJLEO-54371; No. of Pages 3
ARTICLE IN PRESS Optik xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Optimization and analysis of multi-layer diffractive optical elements in visible waveband Jinsong Li ∗ , Ke Feng College of Optical and Electronic Technology, China Jiliang University, Hangzhou 310018, PR China
a r t i c l e
i n f o
Article history: Received 14 July 2013 Accepted 6 January 2014 Available online xxx Keywords: Diffraction efficiency MLDOE Relief heights Optimization method
a b s t r a c t An optimization for diffraction efficiency of the multi-layer diffractive optical elements (MLDOEs) is presented, then the construction process of optimization program is introduced in detail. A new optimization initial point is proposed, which contributes to analyze the relationship between the optimal relief heights and the base materials. Through the optimization examples, diffraction efficiency higher than 99.7% from F line (486.1327 nm) to C line (656.2725 nm) of visible waveband can be achieved, and the polychromatic integral diffraction efficiency (PIDE) over the entire waveband is 99.94%. Moreover, this paper compares and analyzes optimization results of different glass pairs, and the relationship between the optimal relief heights pairs and base materials pairs is obtained. © 2014 Published by Elsevier GmbH.
1. Introduction In imaging systems, combining diffractive optical elements (DOEs) with refractive optical elements can bring a new degree of freedom for systems design, the DOEs can improve system performance, simplify system structure, and reduce system weight. Compared with common glass materials, the DOEs have opposite chromatism characteristics. The systems can achieve an achromatic result by properly choosing glass materials and focal power distribution. Single-layer DOEs only have one design wavelength, and only in the design wavelength is diffraction efficiency 100%. When deviating from the design wavelength, the diffraction efficiency drops rapidly, consequently, the other order’s diffraction beam can make the contrast ratio fall in the imaging plane, so the singlelayer DOEs are mainly used in the optical systems in which imaging quality is not high or works in a narrow waveband [1]. In contrast, multi-layer DOEs (MLDOEs) adopt different dispersive materials and different surface relief heights. It can be a good solution to achieve high diffraction efficiency in wide wavebands [2]. The Domestic Changchun Light Technology Institute has done a lot of research [3–6] on design of the MLDOEs. Due to the great difficulty of chromatic aberration correction, the MLDOEs are applied in the infrared imaging systems widely. Besides the infrared imaging systems, telephoto lenses working in the visible waveband often use the MLDOEs to correct chromatic aberration and secondary
∗ Corresponding author. Tel.: +86 571 86914586. E-mail address: [email protected] (J. Li).
spectrum. The structure of a MLDOE is illustrated in Fig. 1. It essentially consists of two harmonic diffractive elements (HDEs) [3] with the same pitch periods and different relief heights. Two HDEs with microns air space join together concentrically. 2. Design principle of the MLDOE In Fig. 1, when a parallel beam vertically enters a MLDOE, maximum optical path difference generated through the MLDOE in a period is shown as Eq. (1) [7] OPD() = (n1 () − 1)h1 − (n2 () − 1)h2
(1)
where n1 () and n2 () are the refractive index of two HDEs in the wavelength of respectively, h1 and h2 are their relief heights. Then the m order diffraction efficiency can be written as Eq. (2) [8]:
m () = sin m −
OPD()
(2)
m = +1, generally +1 order diffraction beam is taken as the imaging beam in the imaging systems. When we design refractive–diffractive imaging systems working in visible waveband, their imagery sensors are CCD or CMOS, usually F and C lines are selected as achromatic lines, waveband from F line (486.1327 nm) to C line (656.2725 nm) is the main imaging waveband of systems, in which the dominant wavelength is d line (587.5618 nm). From Eqs. (1) and (2) we can see, a MLDOE has two design wavelengths, and its diffraction efficiency distribution is bimodally shaped.
http://dx.doi.org/10.1016/j.ijleo.2014.01.088 0030-4026/© 2014 Published by Elsevier GmbH.
Please cite this article in press as: J. Li, K. Feng, Optimization and analysis of multi-layer diffractive optical elements in visible waveband, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.01.088
G Model
ARTICLE IN PRESS
IJLEO-54371; No. of Pages 3
J. Li, K. Feng / Optik xxx (2014) xxx–xxx
2
Incident beam
h2
h1 Fig. 1. Structure of a MLDOE.
3. The method of optimization
Fig. 2. K9 – ZF4 diffraction efficiency distribution.
Here a new optimization is put forward. RMS (Root Mean Square) deviation between actual distribution and ideal distribution is adopted as the optimization objective, where RMS deviation means calculating the root mean square of deviation values of many sampling points. RMS deviation can reflect comprehensive diffraction efficiency of the entire waveband. For construction of the optimization program, the first step is to calculate the functional relationship between the refractive index of base materials and the wavelength. The second step is to structure the expression of diffraction efficiency deviation. The final step is to establish the optimization algorithm of RMS. In MATLAB the least square optimization is equivalent to the optimization of RMS. So the optimization function is adopted as the optimization algorithm, which is suitable for the nonlinear least square optimization with constraints. It is worth noting that most optimization functions can only search and optimize in a local scope, therefore, optimization results will have more or less difference when adopting different initial points of optimization. In the previous research, researchers generally take the edge wavelengths as initial design wavelengths, and the optimization process can be regarded as searching the optimal design wavelengths [4], which correspond to the optimal surface relief heights and the optimal diffraction efficiency distribution. According to [4], the relationship between relief heights and design wavelengths is given, once base materials and design wavelengths are selected, corresponding relief heights are also determined. Both design wavelengths and relief heights can be selected as optimization variables. Because the expression of diffraction efficiency has direct relationship with the relief heights, in this paper the surface relief heights are taken as the optimization variables replacing the design wavelengths. Next a new optimization initial point will be introduced. If the diffraction efficiency distribution conforms with ideal distribution, diffraction efficiency over the entire waveband is 100%, by putting the three wavelengths of F, d and C lines into Eq. (2), Eqs. (3) and (4) can be obtained (n1d − 1)hp1 − (n2d − 1)hp2 = d
(3)
(n1F − n1C )hp1 − (n2F − n2C )hp2 = −␦
(4)
where hp1 and hp2 are the ideal surface relief heights, ␦ is the bandwidth from F line to C line. Next, Eq. (5) can be obtained by putting the Abbe number formula into Eq. (4): n1d − 1 n −1 hp1 − 2d hp2 = −␦ 1 2
(5)
By solving Eqs. (3) and (5), we can obtain Eqs. (6) and (7): hp1 =
d + ␦ · 2 (n1d − 1)(1 − (2 /1 ))
(6)
hp2 =
d + ␦ · 1 (n2d − 1)((1 /2 ) − 1)
(7)
According to Eqs. (6) and (7), when base materials of a MLDOE are selected, hp1 and hp2 can be calculated. In this paper hp1 and hp2 are firstly selected as initial points of optimization. In addition, values of relief heights are limited between 5 and 30 m, which determine the machining difficulty and cost. This optimization program will output the optimal relief heights hm1 and hm2 . The quality of the optimization result can be evaluated by the minimum diffraction efficiency (MNDE) in the entire waveband and the polychromatic integral diffraction efficiency (PIDE) [4], PIDE can be obtained by Eq. (8) [6]: m (h1 , h2 ) =
1 max − min
max
sin c 2 m − min
OPD()
(8)
d
The optimal design wavelengths can be obtained by calculating the maximum position of the diffraction efficiency distribution. 4. Examples of optimization Through analyzing the structure of MLDOEs, it is known that the convex HDE is equivalent to the convex lens, and corresponds to crown glass [8]. Adversely, concave HDE corresponds to flint glass. Glass materials are chosen according to the rules above and all glass materials are from CHINA glass base. The first time K9 (nd = 1.519, = 64.1) and ZF4 (nd = 1.728, = 28.3) are selected as base materials, after running the optimization program the optimal diffraction efficiency distribution can be obtained. As shown in Fig. 2, the dotted and solid curves represent the diffraction efficiency distribution before and after optimization, respectively. After optimization the diffraction efficiency of the entire waveband is more than 99.7% and the PIDE is 99.94%. The corresponding optimal relief heights pair and optimal design wavelengths pair are 18.755 and 12.482, 0.512 and 0.619 m. In order to compare the optimization results of different initial points, the edge wavelengths are taken as the design wavelengths pair, through optimization, its MNDE and PIDE is 99.69% and 99.94%, respectively. Therefore, the new initial points of optimization has not brought descent of optimization result. Next, optimization results of different base materials pairs will be studied. K9 is fixed, flint glass is replaced with F7, ZF1, ZF2, ZF3, ZF6, ZF7 in turn. After optimization, their diffraction efficiency distributions are almost the same as ZF4 and the fluctuation of PIDE is ±0.005%. But the optimal relief heights with different flint glass differ greatly. Table 1 shows the optimal relief heights with Table 1 The optimal relief depths with different flint glass. Glass
F7
ZF1
ZF2
ZF3
ZF4
ZF6
ZF7
nd hm1 hp1 hm2 hp2
1.636 35.36 28.481 28.476 22.176 22.183
1.647 33.88 26.063 26.057 19.860 19.867
1.673 32.22 23.604 23.599 17.235 17.242
1.717 29.49 20.082 20.077 13.625 13.632
1.728 28.34 18.755 18.750 12.482 12.489
1.755 27.52 17.864 17.859 11.426 11.433
1.806 25.37 15.701 15.696 9.318 9.324
Please cite this article in press as: J. Li, K. Feng, Optimization and analysis of multi-layer diffractive optical elements in visible waveband, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.01.088
G Model IJLEO-54371; No. of Pages 3
ARTICLE IN PRESS J. Li, K. Feng / Optik xxx (2014) xxx–xxx
3
Fig. 4 shows the relationship between hm2 and n2d , 2 , in which colors from red to blue correspond relief heights from large to small. Usually the refractive index of flint glass is between 1.6 and 1.8 and the Abbe number is between 24 and 36. In Fig. 4, the larger 2 or the smaller n2d , the larger hm2 , or vice versa. The optimization program for different material pairs can obtain equally good optimization results, but the size of relief heights directly affects the machining difficulty and cost. Therefore, in order to guarantee the processing feasibility of the design, the material pair should be chosen appropriately according to the current processing level. Fig. 3. The relationship between hm1 and 2 .
Fig. 4. The relationship between hm2 and n2d , 2 .
different flint glass, in which the unit of the relief heights is m. 5. Analysis of the optimal relief heights According to Table 1, firstly by comparing hm1 , hm2 of different flint glass, the smaller 2 , the smaller hm1 and hm2 too. However, when 2 decreases and n2d would increase, which cannot make sure whether hm1 and hm2 are correlated to n2d Then compare hm1 , hp1 and hm2 , hp2 , a difference of only 0.007 m can be found, they are very close in value. Therefore, the new initial points of optimization help analyze the relationship between the optimal relief heights and base materials greatly. Based on Eq. (6), when the crown glass K9 is selected, in other words, n1d and 1 is fixed, hp1 is only correlated to 2 . Likewise, hp2 is correlated to n2d and 2 from Eq. (7). In summary, the relationship between hp1 , hp2 and n2d , 2 can represent the relationship between hm1 , hm2 and n2d , 2 . According to Eqs. (6) and (7), their relationship diagrams can be drawn as shown Figs. 3 and 4, respectively. In Fig. 3, hm1 increases nonlinearly with 2 .
6. Summary In conclusion, an optimization for diffraction efficiency of the MLDOEs has been presented, and then the construction process of the optimization program has been introduced in detail. A new optimization initial point has been put forward, which contributes to analyze the relationship between the optimal relief heights and base materials. Through the optimization examples, quite high diffraction efficiency can be achieved in waveband from F line (486.1327 nm) to C line (656.2725 nm), which is very close to the case using the edge wavelengths pair as the initial point of optimization. By comparing and analyzing optimization results of different material pairs, the relationship between the optimal relief heights and base materials is obtained. The relationship has a guiding significance for the choice of base materials. Acknowledgements This work was supported by National Basic Research Program of China (973 Program) under Grant No. 2010CB327804. References [1] A. Dale, G. Buralli, M. Michael, Effects of diffraction efficiency on the modulation transfer function of diffractive lenses, Appl. Opt. 31 (22) (1992) 4389–4396. [2] Y. Arieli, S. Noach, S. Ozeri, N. Eisenberg, Design of diffractive optical elements for multiple wavelengths, Appl. Opt. 37 (1998) 6174–6177. [3] C. Xue, Q. Cui, Optimal design of a multilayer diffractive optical element for dual wavebands, Opt. Lett. 35 (24) (2010) 4157–4159. [4] C. Xue, Q. Cui, Design of multilayer diffractive optical elements with polychromatic integral diffraction efficiency, Opt. Lett. 35 (7) (2010) 986–988. [5] T. Liu, Q. Cui, Evaluation of narcissus for multilayer diffractive optical elements in IR systems, Appl. Opt. 50 (33) (2011) 6146–6152. [6] X. Pei, Q. Cui, Design and diffraction efficiency of a multi-layer diffractive optical element, Acta Photon. Sin. 38 (5) (2009) 1126–1131. [7] S. Noach, Y. Arieli, N. Eisenberg, Wave-front control and aberration correction with a diffractive optical element, Opt. Lett. 24 (1999) 333. [8] J.B. Cohen, Narcissus of diffractive optical surfaces, Proc. SPIE 2426 (1995) 380–385.
Please cite this article in press as: J. Li, K. Feng, Optimization and analysis of multi-layer diffractive optical elements in visible waveband, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.01.088
ARTICLE IN PRESS Optik xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Optimization and analysis of multi-layer diffractive optical elements in visible waveband Jinsong Li ∗ , Ke Feng College of Optical and Electronic Technology, China Jiliang University, Hangzhou 310018, PR China
a r t i c l e
i n f o
Article history: Received 14 July 2013 Accepted 6 January 2014 Available online xxx Keywords: Diffraction efficiency MLDOE Relief heights Optimization method
a b s t r a c t An optimization for diffraction efficiency of the multi-layer diffractive optical elements (MLDOEs) is presented, then the construction process of optimization program is introduced in detail. A new optimization initial point is proposed, which contributes to analyze the relationship between the optimal relief heights and the base materials. Through the optimization examples, diffraction efficiency higher than 99.7% from F line (486.1327 nm) to C line (656.2725 nm) of visible waveband can be achieved, and the polychromatic integral diffraction efficiency (PIDE) over the entire waveband is 99.94%. Moreover, this paper compares and analyzes optimization results of different glass pairs, and the relationship between the optimal relief heights pairs and base materials pairs is obtained. © 2014 Published by Elsevier GmbH.
1. Introduction In imaging systems, combining diffractive optical elements (DOEs) with refractive optical elements can bring a new degree of freedom for systems design, the DOEs can improve system performance, simplify system structure, and reduce system weight. Compared with common glass materials, the DOEs have opposite chromatism characteristics. The systems can achieve an achromatic result by properly choosing glass materials and focal power distribution. Single-layer DOEs only have one design wavelength, and only in the design wavelength is diffraction efficiency 100%. When deviating from the design wavelength, the diffraction efficiency drops rapidly, consequently, the other order’s diffraction beam can make the contrast ratio fall in the imaging plane, so the singlelayer DOEs are mainly used in the optical systems in which imaging quality is not high or works in a narrow waveband [1]. In contrast, multi-layer DOEs (MLDOEs) adopt different dispersive materials and different surface relief heights. It can be a good solution to achieve high diffraction efficiency in wide wavebands [2]. The Domestic Changchun Light Technology Institute has done a lot of research [3–6] on design of the MLDOEs. Due to the great difficulty of chromatic aberration correction, the MLDOEs are applied in the infrared imaging systems widely. Besides the infrared imaging systems, telephoto lenses working in the visible waveband often use the MLDOEs to correct chromatic aberration and secondary
∗ Corresponding author. Tel.: +86 571 86914586. E-mail address: [email protected] (J. Li).
spectrum. The structure of a MLDOE is illustrated in Fig. 1. It essentially consists of two harmonic diffractive elements (HDEs) [3] with the same pitch periods and different relief heights. Two HDEs with microns air space join together concentrically. 2. Design principle of the MLDOE In Fig. 1, when a parallel beam vertically enters a MLDOE, maximum optical path difference generated through the MLDOE in a period is shown as Eq. (1) [7] OPD() = (n1 () − 1)h1 − (n2 () − 1)h2
(1)
where n1 () and n2 () are the refractive index of two HDEs in the wavelength of respectively, h1 and h2 are their relief heights. Then the m order diffraction efficiency can be written as Eq. (2) [8]:
m () = sin m −
OPD()
(2)
m = +1, generally +1 order diffraction beam is taken as the imaging beam in the imaging systems. When we design refractive–diffractive imaging systems working in visible waveband, their imagery sensors are CCD or CMOS, usually F and C lines are selected as achromatic lines, waveband from F line (486.1327 nm) to C line (656.2725 nm) is the main imaging waveband of systems, in which the dominant wavelength is d line (587.5618 nm). From Eqs. (1) and (2) we can see, a MLDOE has two design wavelengths, and its diffraction efficiency distribution is bimodally shaped.
http://dx.doi.org/10.1016/j.ijleo.2014.01.088 0030-4026/© 2014 Published by Elsevier GmbH.
Please cite this article in press as: J. Li, K. Feng, Optimization and analysis of multi-layer diffractive optical elements in visible waveband, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.01.088
G Model
ARTICLE IN PRESS
IJLEO-54371; No. of Pages 3
J. Li, K. Feng / Optik xxx (2014) xxx–xxx
2
Incident beam
h2
h1 Fig. 1. Structure of a MLDOE.
3. The method of optimization
Fig. 2. K9 – ZF4 diffraction efficiency distribution.
Here a new optimization is put forward. RMS (Root Mean Square) deviation between actual distribution and ideal distribution is adopted as the optimization objective, where RMS deviation means calculating the root mean square of deviation values of many sampling points. RMS deviation can reflect comprehensive diffraction efficiency of the entire waveband. For construction of the optimization program, the first step is to calculate the functional relationship between the refractive index of base materials and the wavelength. The second step is to structure the expression of diffraction efficiency deviation. The final step is to establish the optimization algorithm of RMS. In MATLAB the least square optimization is equivalent to the optimization of RMS. So the optimization function is adopted as the optimization algorithm, which is suitable for the nonlinear least square optimization with constraints. It is worth noting that most optimization functions can only search and optimize in a local scope, therefore, optimization results will have more or less difference when adopting different initial points of optimization. In the previous research, researchers generally take the edge wavelengths as initial design wavelengths, and the optimization process can be regarded as searching the optimal design wavelengths [4], which correspond to the optimal surface relief heights and the optimal diffraction efficiency distribution. According to [4], the relationship between relief heights and design wavelengths is given, once base materials and design wavelengths are selected, corresponding relief heights are also determined. Both design wavelengths and relief heights can be selected as optimization variables. Because the expression of diffraction efficiency has direct relationship with the relief heights, in this paper the surface relief heights are taken as the optimization variables replacing the design wavelengths. Next a new optimization initial point will be introduced. If the diffraction efficiency distribution conforms with ideal distribution, diffraction efficiency over the entire waveband is 100%, by putting the three wavelengths of F, d and C lines into Eq. (2), Eqs. (3) and (4) can be obtained (n1d − 1)hp1 − (n2d − 1)hp2 = d
(3)
(n1F − n1C )hp1 − (n2F − n2C )hp2 = −␦
(4)
where hp1 and hp2 are the ideal surface relief heights, ␦ is the bandwidth from F line to C line. Next, Eq. (5) can be obtained by putting the Abbe number formula into Eq. (4): n1d − 1 n −1 hp1 − 2d hp2 = −␦ 1 2
(5)
By solving Eqs. (3) and (5), we can obtain Eqs. (6) and (7): hp1 =
d + ␦ · 2 (n1d − 1)(1 − (2 /1 ))
(6)
hp2 =
d + ␦ · 1 (n2d − 1)((1 /2 ) − 1)
(7)
According to Eqs. (6) and (7), when base materials of a MLDOE are selected, hp1 and hp2 can be calculated. In this paper hp1 and hp2 are firstly selected as initial points of optimization. In addition, values of relief heights are limited between 5 and 30 m, which determine the machining difficulty and cost. This optimization program will output the optimal relief heights hm1 and hm2 . The quality of the optimization result can be evaluated by the minimum diffraction efficiency (MNDE) in the entire waveband and the polychromatic integral diffraction efficiency (PIDE) [4], PIDE can be obtained by Eq. (8) [6]: m (h1 , h2 ) =
1 max − min
max
sin c 2 m − min
OPD()
(8)
d
The optimal design wavelengths can be obtained by calculating the maximum position of the diffraction efficiency distribution. 4. Examples of optimization Through analyzing the structure of MLDOEs, it is known that the convex HDE is equivalent to the convex lens, and corresponds to crown glass [8]. Adversely, concave HDE corresponds to flint glass. Glass materials are chosen according to the rules above and all glass materials are from CHINA glass base. The first time K9 (nd = 1.519, = 64.1) and ZF4 (nd = 1.728, = 28.3) are selected as base materials, after running the optimization program the optimal diffraction efficiency distribution can be obtained. As shown in Fig. 2, the dotted and solid curves represent the diffraction efficiency distribution before and after optimization, respectively. After optimization the diffraction efficiency of the entire waveband is more than 99.7% and the PIDE is 99.94%. The corresponding optimal relief heights pair and optimal design wavelengths pair are 18.755 and 12.482, 0.512 and 0.619 m. In order to compare the optimization results of different initial points, the edge wavelengths are taken as the design wavelengths pair, through optimization, its MNDE and PIDE is 99.69% and 99.94%, respectively. Therefore, the new initial points of optimization has not brought descent of optimization result. Next, optimization results of different base materials pairs will be studied. K9 is fixed, flint glass is replaced with F7, ZF1, ZF2, ZF3, ZF6, ZF7 in turn. After optimization, their diffraction efficiency distributions are almost the same as ZF4 and the fluctuation of PIDE is ±0.005%. But the optimal relief heights with different flint glass differ greatly. Table 1 shows the optimal relief heights with Table 1 The optimal relief depths with different flint glass. Glass
F7
ZF1
ZF2
ZF3
ZF4
ZF6
ZF7
nd hm1 hp1 hm2 hp2
1.636 35.36 28.481 28.476 22.176 22.183
1.647 33.88 26.063 26.057 19.860 19.867
1.673 32.22 23.604 23.599 17.235 17.242
1.717 29.49 20.082 20.077 13.625 13.632
1.728 28.34 18.755 18.750 12.482 12.489
1.755 27.52 17.864 17.859 11.426 11.433
1.806 25.37 15.701 15.696 9.318 9.324
Please cite this article in press as: J. Li, K. Feng, Optimization and analysis of multi-layer diffractive optical elements in visible waveband, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.01.088
G Model IJLEO-54371; No. of Pages 3
ARTICLE IN PRESS J. Li, K. Feng / Optik xxx (2014) xxx–xxx
3
Fig. 4 shows the relationship between hm2 and n2d , 2 , in which colors from red to blue correspond relief heights from large to small. Usually the refractive index of flint glass is between 1.6 and 1.8 and the Abbe number is between 24 and 36. In Fig. 4, the larger 2 or the smaller n2d , the larger hm2 , or vice versa. The optimization program for different material pairs can obtain equally good optimization results, but the size of relief heights directly affects the machining difficulty and cost. Therefore, in order to guarantee the processing feasibility of the design, the material pair should be chosen appropriately according to the current processing level. Fig. 3. The relationship between hm1 and 2 .
Fig. 4. The relationship between hm2 and n2d , 2 .
different flint glass, in which the unit of the relief heights is m. 5. Analysis of the optimal relief heights According to Table 1, firstly by comparing hm1 , hm2 of different flint glass, the smaller 2 , the smaller hm1 and hm2 too. However, when 2 decreases and n2d would increase, which cannot make sure whether hm1 and hm2 are correlated to n2d Then compare hm1 , hp1 and hm2 , hp2 , a difference of only 0.007 m can be found, they are very close in value. Therefore, the new initial points of optimization help analyze the relationship between the optimal relief heights and base materials greatly. Based on Eq. (6), when the crown glass K9 is selected, in other words, n1d and 1 is fixed, hp1 is only correlated to 2 . Likewise, hp2 is correlated to n2d and 2 from Eq. (7). In summary, the relationship between hp1 , hp2 and n2d , 2 can represent the relationship between hm1 , hm2 and n2d , 2 . According to Eqs. (6) and (7), their relationship diagrams can be drawn as shown Figs. 3 and 4, respectively. In Fig. 3, hm1 increases nonlinearly with 2 .
6. Summary In conclusion, an optimization for diffraction efficiency of the MLDOEs has been presented, and then the construction process of the optimization program has been introduced in detail. A new optimization initial point has been put forward, which contributes to analyze the relationship between the optimal relief heights and base materials. Through the optimization examples, quite high diffraction efficiency can be achieved in waveband from F line (486.1327 nm) to C line (656.2725 nm), which is very close to the case using the edge wavelengths pair as the initial point of optimization. By comparing and analyzing optimization results of different material pairs, the relationship between the optimal relief heights and base materials is obtained. The relationship has a guiding significance for the choice of base materials. Acknowledgements This work was supported by National Basic Research Program of China (973 Program) under Grant No. 2010CB327804. References [1] A. Dale, G. Buralli, M. Michael, Effects of diffraction efficiency on the modulation transfer function of diffractive lenses, Appl. Opt. 31 (22) (1992) 4389–4396. [2] Y. Arieli, S. Noach, S. Ozeri, N. Eisenberg, Design of diffractive optical elements for multiple wavelengths, Appl. Opt. 37 (1998) 6174–6177. [3] C. Xue, Q. Cui, Optimal design of a multilayer diffractive optical element for dual wavebands, Opt. Lett. 35 (24) (2010) 4157–4159. [4] C. Xue, Q. Cui, Design of multilayer diffractive optical elements with polychromatic integral diffraction efficiency, Opt. Lett. 35 (7) (2010) 986–988. [5] T. Liu, Q. Cui, Evaluation of narcissus for multilayer diffractive optical elements in IR systems, Appl. Opt. 50 (33) (2011) 6146–6152. [6] X. Pei, Q. Cui, Design and diffraction efficiency of a multi-layer diffractive optical element, Acta Photon. Sin. 38 (5) (2009) 1126–1131. [7] S. Noach, Y. Arieli, N. Eisenberg, Wave-front control and aberration correction with a diffractive optical element, Opt. Lett. 24 (1999) 333. [8] J.B. Cohen, Narcissus of diffractive optical surfaces, Proc. SPIE 2426 (1995) 380–385.
Please cite this article in press as: J. Li, K. Feng, Optimization and analysis of multi-layer diffractive optical elements in visible waveband, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.01.088