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Design of diffractiveoptical elements for multiple wavelengths

Yoel Arieli, Salman Noach, Shmuel Ozeri, and Naftali Eisenberg

A method for producing diffractive optical elements ~DOE’s! for multiple wavelengths without chromaticaberration is described. These DOE’s can be designed for any distinct wavelength. The DOE’s areproduced from two different optical materials, taking advantage of their different refractive indices anddispersions. © 1998 Optical Society of America

OCIS codes: 050.1970, 090.1760, 090.2890.

a 4,5

1. Introduction

Conventional computer-generated holograms @dif-ractive optical elements ~DOE’s!# operate at the spe-

cific wavelength for which they were designed.Operating at another wavelength causes chromaticaberration. Since many potential DOE applicationsrequire the simultaneous use of more than one wave-length, correction of the chromatic aberration is es-sential.

In previous papers1–3 a method for eliminating thechromatic aberration of a DOE for two differentwavelengths was presented. In that method theDOE acts in a different manner for each of the twowavelengths. The thickness of the DOE was calcu-lated so that the optical path length at each point ofthe DOE was suitable for each of the two wave-lengths. This was accomplished by the addition ofan integral-multiple phase retardation of 2p to one

avelength until the suitable phase retardation ofhe second wavelength was achieved. The overallhysical thickness of the final DOE is derived fromhe accuracy requirement.

This paper presents another method for designingDOE without chromatic aberration for two or moreifferent wavelengths. The chromatic-aberration-orrected DOE is designed by combination of twoligned DOE’s made of different materials, similar to

Y. Arieli, S. Ozeri, and N. Eisenberg are with the Department ofElectro-optics, Jerusalem College of Technology, Jerusalem 91160,Israel. S. Noach is with the Division of Applied Optics, HebrewUniversity of Jerusalem, Jerusalem 91904, Israel.

Received 3 November 1997; revised manuscript received 20 Feb-ruary 1998.

0003-6935y98y266174-04$15.00y0© 1998 Optical Society of America

6174 APPLIED OPTICS y Vol. 37, No. 26 y 10 September 1998

polarization-selective DOE. Taking advantageof the change in refractive indices of dispersive ma-terials for different wavelengths, one can control theoptical path length of each wavelength. A methodthat uses the dispersion phenomenon of two materi-als for correcting chromatic aberration was reportedearlier.6 That method corrects the chromatic aber-ration in some bandwidths and is limited to specificmaterials. The aim of this paper is to design a DOEthat acts as a different DOE for each individual wave-length without generating chromatic aberration. Inaddition, there are no restrictions to specific materi-als, although the overall DOE thickness can be di-minished by the materials selected.

2. Chromatic-Aberration Correction

As in refractive optics, the chromatic aberration indiffractive optics can be corrected by the combinationof two aligned DOE’s made of differently dispersivematerials. By application of different configura-tions, the chromatic aberration for more than twowavelengths can also be corrected.

A. Chromatic-Aberration Correction for Two Wavelengths

One pixel of the two aligned DOE’s is depicted in Fig.1. The phase retardation f of light of wavelength lpropagating through that pixel is

@n1~l! 2 ng~l!#d1 1 @n2~l! 2 ng~l!#d2 51

2plf, (1)

where n1~l!, ng~l!, and n2~l! are the refractive indicesof the materials along the light path and d1 and d2are the etch depths.

When light of two different wavelengths l1 and l2is propagated through the pixel, the phase retarda-

il

Tcd

ti

@

tions f1 and f2 of the two wavelengths are given bythe matrix equation:

~n 2 ng!d 51

2plf, (2)

where

n 5 Fn1~l1! n2~l1!n1~l2! n2~l2!

G ,

ng 5 Fng~l1! ng~l1!ng~l2! ng~l2!

G ,

d 5 Fd1

d2G ,

lf 5 Fl1~f1 1 m12p!l2~f2 1 m22p!G

The elements of the n matrix are the refractivendices of the dispersive materials for the two wave-engths. The elements of the ng matrix are the re-

fractive indices of the intermediate material for thetwo wavelengths. The elements of the d matrix arethe etched thicknesses of the two DOE’s. The ele-ments of the lf matrix are the required phases forthe two wavelengths multiplied by the wavelengths.The terms m1 and m2 are some arbitrary integers.

he phase values can be the same or different, andonsequently the DOE behavior will be the same orifferent for both wavelengths.Solving Eq. ~2! for the required etched depths d in

he two DOE’s, we obtain

d 51

2p~n 2 ng!

21lf. (3)

The etch depths of the two DOE’s as calculated by Eq.~2! cause the suitable phase retardations f1 and f2for the two wavelengths simultaneously.

For investigation of the overall behavior of DOEthickness, Eq. ~3! is expressed explicitly:

efmmtl

The absolute minimum thickness is achieved whenthe denominators of Eqs. ~4! are set to a maximum.The denominators consist of two expressions, andeach expression is made up of two multiples.

For normally dispersive materials in which the re-fractive index is higher for shorter wavelengths, n~l!is a decreasing function. If the intermediate mate-rial is air, ng equals 1. If we assume that l2 . l1,each of the two expressions @Eqs. ~4!# consists of mul-tiples of a high value and a low value. The maxi-mum value of the denominator occurs when one

10

material has high dispersion and the other materialhas low dispersion.

B. Chromatic-Aberration Correction for More than TwoWavelengths

This method can be expanded for more than twowavelengths. More degrees of freedom, and conse-quently more flexibility when using more than twowavelengths, can be obtained when other configura-tions of pixel shapes with multilayer etching areused. Figure ~2! shows one pixel consisting of twoaligned, etched substrates with an etched layer oneach. In this configuration the difference in thephase retardation of light of wavelength l propagat-ng through that pixel relative to the adjacent one is

n1~l! 2 n2~l!#d1 1 @ng~l! 2 n2~l!#d2 1 @ng~l! 2 n3~l!#d3

1 @n4~l! 2 n3~l!#d4 51

2plf, (5)

Four equations similar to Eq. ~5! can be written forfour different wavelengths. Solving these equationsfor the values of d will give the thickness required for

ach layer of the two substrates. The phase valuesor each wavelength can be the same or different, asentioned above. Adding more layers of differentaterials adds more degrees of freedom and enables

he possibility of designing a DOE for more wave-engths.

3. Computational Results

As an example, the thicknesses of a binary DOE fortwo wavelengths was calculated. A binary DOE hasonly two levels, which causes two phase retardations:

Fig. 1. One pixel of the combined DOE.

d1 5

@n2~l2! 2 ng~l2!#l1(f1 1 m12p)

2p2 @n2~l1! 2 ng~l1!#

l2(f2 1 m22p)2p

@n1~l1! 2 ng~l1!#@n2~l2! 2 ng~l2!# 2 @n1~l2! 2 ng~l2!#@n2~l1! 2 ng~l1!#,

d2 5 2

@n1~l2! 2 ng~l2!#l1(f1 1 m12p)

2p2 @n1~l1! 2 ng~l1!#

l2(f2 1 m22p)2p

@n1~l1! 2 ng~l1!#@n2~l2! 2 ng~l2!# 2 @n1~l2! 2 ng~l2!#@n2~l1! 2 ng~l1!#. (4)

September 1998 y Vol. 37, No. 26 y APPLIED OPTICS 6175

f

m

Eti

hase

6

0 or p. The two wavelengths are l1 5 543.5 nm andl2 5 632.8 nm of a He–Ne laser. The materials usedor the two DOE’s are BK7 glass and As2S. The

intermediate material is air. The corresponding re-fractive indices of the materials for the wavelengthsare shown in Table 1.7,8

Equation ~1! was used to calculate the DOE thick-ness for all combinations of phases for the two wave-lengths. The optimized integral numbers m1 and

2 that give the minimal overall thickness of thecombined DOE were chosen for these calculations.Positive and negative thickness values mean addingto or etching the substrate, respectively. If we con-sider that the maximum positive value is the zeropoint and all other values are negative in reference toit, a microlithography etching process can be applied.

Table 2 contains the results of the calculations ofthe thicknesses d1 and d2 in micrometers for the twomaterials for all phase-value combinations. Table 2also shows the integral numbers m1 and m2 and theminimal thicknesses d19 and d29 after optimization.It can be seen that the optimization process dimin-ishes the overall thickness by one order of magnitude.

Figure 3 shows the simulated response of the DOEfor different wavelengths, where l1 and l2 are thewavelengths for which it was designed. The phaseretardations required in this case are p for the twowavelengths. It can be seen that the phase retarda-tions for these wavelengths are p, as required, butthe phases change for other wavelengths. Simula-tion shows that, for a light source with a bandwidth

Fig. 2. One pixel of the comb

Table 1. Refractive Indices of the DOE Materials for Two Wavelengths

Wavelength

Glass l1 5 543.5 nm l2 5 632.8 nmBK7 n1~l1! 5 1.51885 n1~l2! 5 1.51509As2S n2~l1! 5 2.71445 n2~l2! 5 2.60615

Table 2. Required Etch Depths of the Combined DOE for Different P

f1 ~rad! F2 ~rad! m1 m2 d1 ~1026 m!

0 0 0 0 00 p 0 0 10.9064p 0 0 0 28.7755p p 0 0 2.1308

176 APPLIED OPTICS y Vol. 37, No. 26 y 10 September 1998

of 2 nm, the phase-retardation changes are less than1%. All other wavelengths will suffer chromatic ab-erration and degradation in efficiency. Other phasecombinations show similar behavior.

The response of the phase values to errors in fab-rication is calculated by

Df 52p

l$@n1~l! 2 ng~l!#Dd1 1 @n2~l! 2 ng~l!#Dd2% , (6)

quation ~6! shows that the error response is similaro or less than that in other types of DOE because its a function of the two independent variables d1 and

d2.

4. Conclusions

In conclusion, we have shown that it is possible todesign a DOE without chromatic aberration for twoor more different wavelengths by combining twoDOE’s in various pixel configurations. This com-bined DOE can act differently for the different wave-lengths. This method is not restricted to specificmaterials, although the minimum thickness is

DOE etched from two sides.

Fig. 3. Wavelength response of the combined DOE.

Combinations of the Two Wavelengths before and after Optimization

026 m! m1 m2 d19 ~1026 m! d29 ~1026 m!

0 0 0 0.3007 23 23 21.8786 20.3825.8143 23 22 0.2523 20.8689.4864 0 0 2.1308 20.4864

ined

d2 ~1

023

220

2

3

4. F. Xu, J. E. Ford, and Y. Fainman, “Polarization selective

achieved by use of coupled high- and low-dispersionmaterials. Because of the sharp response to a par-ticular wavelength, one can use it as a wavelengthselector when used with a wideband light source.

References1. Y. Arieli, Y. Z. Lauber, and N. P. Eisenberg, “Kinoforms for

wideband use,” in 9th Meeting on Optical Engineering in Israel,I. Shaldov, ed., Proc. SPIE 2426, 443–445 ~1994!.

. S. Noach, A. Lewis, Y. Arieli, and N. Eisenberg, “Integrateddiffractive and refractive elements for spectrum shaping,” Appl.Opt. 35, 3635–3539 ~1996!.

. J. E. Ford, F. Xu, and Y. Fainman, “Wavelength-selective pla-nar holograms,” Opt. Lett. 21, 80–82 ~1996!.

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computer-generated holograms: design, fabrication, and ap-plications,” Appl. Opt. 34, 256–266 ~1995!.

5. N. Nieuborg, A. Kirk, B. Morlion, H. Thienpont, and I. Vereten-nicoff, “Polarization-selective diffractive optical element with anindex-matching gap material,” Appl. Opt. 36, 4681–4685~1997!.

6. S. M. Ebstein, “Achromatic diffractive optical elements,” in Dif-fractive and Holographic Optics Technology II, I. Cindrich andS. H. Lee, eds., Proc. SPIE 2404, 211–216 ~1995!.

7. Schott optical glass ~Schott Optical Glass, P.O. Box 2480,D-Mainz, Germany, 1980!.

8. W. S. Rodney, H. Haltison, and T. A. King, “Refractive index ofarsenic trisulfide,” Opt. Soc. Am. 48, 633–636 ~1958!.

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