1436 J. Opt. Soc. Am. A/Vol. 7, No. 8/August 1990

Analysis of multiplexed-reflection holographic gratings

Xiaohui Ning

American Optical Corporation, 14 Mechanic Street, Southbridge, Massachusetts 01550

Received September 19, 1989; accepted April 19, 1990

Multiplexing is an attractive option for obtaining multiline holographic notch reflectors. We discuss two methodsfor analyzing multiplexed holograms, i.e., coupled-wave theory and thin-film-matrix formulation. A coupled-waveapproach for modeling multiplexed holograms is developed. It is found that the first-order theory of Kogelnik is avalid approach if the multiplexed gratings are nondegenerate. However, if the gratings are near degenerate, theexact thin-film matrix method must be used. Specifically, it is shown that, in the case of degenerate gratings, thediffraction efficiency depends critically on the initial phase of each grating. Numerical results using both methodsare presented and compared.

INTRODUCTION

Multiplexing offers an elegant solution for obtaining multi-line notch reflectors. With this approach the complex pro-cessing required for stacking multiple reflectors is avoided.Thus many potential problems such as optical clarity, dis-tortion, durability, and manufacturability inherent with ap-plying multiple layers onto curved substrates are eliminated.

The subject of multiplexed holograms has been investigat-ed by many researchers in the past.1-3 However, those stud-ies are mostly concerned with transmission gratings or slant-ed reflection gratings. The intended applications are main-ly in the area of data storage and optical interconnect. Theemphasis of this research is on multiline notch reflectors forlaser-attenuation applications. Because holographic reflec-tors that are most suitable for this application are unslantedgratings, the following discussion will be restricted to theunslanted case. In the following sections, we will first devel-op an extension of the coupled-wave theory of Kogelnik4 inorder to model multiplexed holograms. Then we will dis-cuss the applicability of the thin-film-matrix theory formodeling multiplexed holograms. The numerical resultsusing both the coupled-wave theory and thin-film methodare presented and compared.

METHODS OF MODELING

Many theoretical approaches for modeling diffraction grat-ings have been published in the past. Some examples ofthose methods are given in Refs. 5 and 6. In this paper onlytwo better-known methods, i.e., the coupled-wave theory ofKogelnik and the thin-film-matrix methods, will be consid-ered because of their simplicity and practical usefulness.

Coupled-Wave TheoryThe following derivation is essentially an extension of thecoupled-wave theory of the Kogelnik. Let us assume thatthe refractive index and the absorption constant of a multi-plexed hologram are superpositions of multiple-sinusoidalterms. Each term represents a simple sinusoidal gratinghaving a period that is different from that of other gratings.Figure 1 illustrates the grating geometry used for this analy-

sis. The refractive index n and absorption constant a asfunctions of position r in the medium are written as

N

n(r) = no + E Re[Ah exp(iKh r)],h=1

N

a(r) = a0 + E Re[Bh exp(iKh r)],h=1

(1)

(2)

where no and ao are the average refractive index and theaverage absorption constant, respectively, and N is the num-ber of the sinusoidal gratings. The hth grating is character-ized by its grating vector Kh, the index amplitude Ah (com-plex), and the absorption amplitude Bh (complex). Thegrating number is related to the construction wavelength Xh

and the construction angle Oh by the following equation:

27rn0Kh = 2roCOS(Oh)-

Xh

For thick holograms, one can assume that only the refer-ence wave SO and the first-order diffraction waves Sh(h = 1,2, .. ., N) are significantly presented in the medium. For s-polarized light, the total electric field ET and the wave equa-tion are

N

ET= E S exp[-i(a -KI) * r],1=0

V2ET + k

2E, = 0,

(4)

(5)

where a is the wave vector of the reference wave (Ko 0 0) andk2 is given by the following equation:

N

k2 = 2 - 2iaoj# + 2# E Re[Ch exp(iKh r)].h=1

(6)

The new parameters in Eq. (6) are related to the knownparameters by # = 2rno/X and Ch = (27r/X Ah - ibh).

By combining Eqs. (4)-(6) and neglecting the second-order derivatives of Sh with respect to z, i.e., depth of thehologram, one obtains the following coupled-wave equa-tions:

0740-3232/90/081436-05$02.00 © 1990 Optical Society of America

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Vol. 7, No. 8/August 1990/J. Opt, Soc. Am. A 1437

reference wave R

signal wave S

X

Fig. 1. Geometry for analyzing multiplexed holographic gratings.

cZ1 d S + L1 itz~

. N N

+ 2 E [S(K. + Kh - K,)ChSmh=1 m=O

+ S(Km - Kh - KI)Ch*Sm] = 0, (7)

where

(a- )cZ =

0, = 2iao# - a - K112

and

(X) = {0 otherwise}-

These equations can also be written in the following ma-trix form:

d S = MS, (8)

where

S= Si.

The elements of matrix M are

mij =-- (j = 0, . .. , N)Czj

diagonal elements andN

M 2l 2 Z [6(KI + Kh - Kj)Ch

+ 6(KI-KhKj)Ch*]

off-diagonal elements.Equation (8) represents a set of linearly coupled differen-

tial equations. Its general solution can be written as

a l l a12 ... alN exp(Eoz)

S= a21 a22 ... a2N exp(Elz)

aN, aN2 ... aNNJ exp(ENZ)j

(9)

where Ej(j = 0, 1, . . ., N) are the eigenvalues of the matrixM. Numerical approaches are generally required to obtainthe eigenvalues. The elements of matrix A (aij, i = 1, ..,

Z N and i = 1, 2, . . ., n) are not linearly independent. Theyare related to one another by the coupled-wave equations.The elements of A can be uniquely determined once theboundary conditions are specified. Once S is obtained, thediffraction efficiencies can be computed by using the follow-ing equations:

'Oh = Sh(O)Sh*(), h = 1, 2, ... ,N.czo(10)

A quantity often used in laser-attenuation systems is theoptical density (OD). The OD is related to the diffractionefficiency by OD = -log(1 - DE), where DE is diffractionefficiency.

Thin-Film Characteristic Matrix TheoryFor unslanted reflection holograms the refractive index andthe absorption constant depend on only the depth z. There-fore an unslanted hologram is equivalent to a large stack ofthin homogeneous slabs. The characteristic matrix methodfor calculating the properties of thin-film stacks can be ap-plied to modeling the hologram. The detailed mathemati-cal formulation of the thin-film method can be found in anumber of references. 7 8

Because there is no restriction on the shapes of the refrac-tive index and the absorption-constant profiles, this ap-proach is general and powerful. This approach has beenused to model the asymmetries in the spectra observed indichromated gelatin holograms.9 The drawback of this ap-proach is that it applies to only unslanted reflection holo-grams, and, in most cases, lengthy computations are re-quired to obtain convergent solutions. By comparison, thecoupled-wave approach applies to any hologram (reflectionor transmission, slanted or unslanted), and it requires muchless computing power for holograms with only a few sinusoi-dal gratings multiplexed.

NUMERICAL RESULTS AND DISCUSSION

Nondegenerate CaseA multiplexed hologram is nondegenerate if the grating vec-tors Kh are so far apart that there is little interation betweentheir diffraction waves. For unslanted reflection hologramsnondegeneracy implies that the periods of the sinusoidalgratings are significantly different.

As an example let us consider an unslanted, lossless reflec-tion hologram having the following index profile:

n(z) = no + Z nh cos(Khz + Ph),

h=1

(11)

where nh is the real amplitude of the hth grating and Ph

represents its initial phase. For numerical computations

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1438 J. Opt. Soc. Am. A/Vol. 7, No. 8/August 1990

5.0

4.0

-2u.a

cm

C)

3.0

2.0

0.0 -TjT|T450 500 550 600 650 700 750 800 850

Wavelength (nm)Fig. 2. Optical density versus wavelength at normal incidence for amultiplexed hologram.

4.0 -thin film

coupled wave

3.0 - Al \ thin film3 2 0 ! 0 coupled wave

C

1.0

0.0~ 650 700 750 800 850

Wavelength (nm)

Fig. 3. Comparison of the multicoupled-wave theory with the thin-film calculation.

the average index no and the thickness d are assumed to be1.5 and 25 ,gm, respectively. The construction wavelengthsare chosen to be 500, 600, 700, and 800 nm, and the construc-tion angles are all 0. The index modulation amplitude andthe initial phase are 0.04 and 0, respectively, for all gratings.

The OD as a function of incident wavelength can be calcu-lated by using either method discussed above. Figure 2shows the result using the thin-film-matrix method (normalincidence is assumed). The effect of multiplexing is clearlydemonstrated in this figure. For comparison purposes, thespectrum is also computed by using the coupled-wave the-ory. In this calculation only two of the four peaks are mod-eled. The comparison is shown in Fig. 3. From these re-sults, the three observations discussed below are made.

(1) For nondegenerate holograms the coupled-wave re-sults are essentially identical to the combined results ob-tained by applying Kogelnik's theory to each sinusoidalgrating. Because the gratings are widely separated the inci-dent wave can never satisfy (or nearly satisfy) the Braggcondition for more than one grating simultaneously. Thussignificant diffraction is predominately caused by just one

grating for which the Bragg condition is satisfied (or nearlysatisfied). The diffraction waves from other gratings arenegligible. This property permits Kogelnik's theory for asingle sinusoidal grating to be used to estimate the peak OD,peak wavelength, and width for each grating. A more quan-titative definition for nondegeneracy can be derived basedon Fig. 3. A multiplexed grating may be called nondegener-ate if there is no overlapping between the peaks at the fullwidth at half-maximum points.

(2) Because the coupled-wave theory is an approximationwhereas the thin-film results are more accurate (in principle,the thin-film results can approach the exact solutions indefi-nitely), the OD versus wavelength curves obtained by usingthe two methods do not exactly agree. Specifically, thethin-film formulation is sensitive to the initial phases of thegratings whereas the coupled-wave theory is not. To provethis, let us examine the coupled-wave equations for unslant-ed reflection holograms. In this case Eq. (7) can be simpli-fied because all the grating vectors are parallel. The cou-pled-wave equations can be reduced to the following form:

d NCz dSo + 0S + E ChSh 0,h=1

cZA d SI + S + I CISO = 0 for 1 0.

(12a)

(12b)

To prove that the coupled-wave theory is independent ofthe initial phases, let us assume that the hth sinusoidal termin Eq. (1) is shifted by a phase factor of Ph. It can be shownthat the coupled-wave equations for the new index profileare analogous to Eqs. (12) with C1 replaced by Cl exp(ipl).The new coupled-wave equations will become completelyidentical to Eqs. (12) if one makes the transformation So -SO and Si - St exp(ip1). Since the diffraction efficienciesare not affected by this transformation, we have proved thatthe coupled-wave theory is independent of the phase shifts.(This conclusion holds for only unslanted gratings. Forslanted multiplexed gratings, the diffraction efficiency maydepend on the initial phases.10 ) As an example let us com-pare the spectrum of a sine (90-deg phase shift) grating withthat of a cosine grating. The spectra calculated by using thethin-film theory are shown in Fig. 4. The differences in

4.0 -

3.0-cosine~

~~~ 2.0 -~~~~~~~sine0

680 690 700 710

Wavelength (nm)Fig. 4. Spectra for two gratings with different phase factors.

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Vol. 7, No. 8/August 1990/J. Opt. Soc. Am. A 1439

4.0

3.0

.r_a

To2.0

.0 S

0 .0 I ---------

300 350 400 450 500 550 600 650 700 750 800 850Wavelength (nm)

Fig. 5. The cross coupling between two diffraction waves causesthe third peak at 375 nm.

peak OD, peak wavelength, and sidebands are clearly shownin this figure. This is, of course, due to the fact that thecoupled-wave approach retains only the first-order diffrac-tion waves. If higher-order diffraction waves are also in-cluded in Eq. (4),5 the results should be identical to the thin-film-matrix results.

As a second example illustrating the accuracy of the thin-film calculation, we consider an index profile consisting ofjust two sinusoidal gratings. The full spectrum calculatedby using thin-film theory is shown in Fig. 5. As can be seenfrom this figure, in addition to the two primary diffractionpeaks, there is also a third peak at approximately 375 nm.This is the sum-frequency contribution due to the weakcross coupling between the two primary gratings. This can-not be predicted by the coupled-wave theory because thecoupled-wave theory assumes that only the reference waveand the first-order diffraction waves exist in the medium.

(3) All the above discussions are based on the assumptionthat the dynamic range of the refractive index of the mediumis unlimited. However, the dynamic ranges of real holo-graphic materials are always finite and are determined bythe intrinsic properties of the media. Therefore, it is ofinterest to explore the effect of limited dynamic range on thespectrum. The dynamic range may be defined as the maxi-mum permissible index modulation, i.e., I=,nh in Eq. (11).An index profile represented by Eq. (11) is chosen to illus-trate once again the physical significance of limited dynamicrange. The index profile is first generated by using Eq. (11),and the dynamic range is 0.16. The ac amplitude of thisindex profile is then numerically truncated to +0.08. Thespectrum for this truncated index profile is shown in Fig. 6.By comparison with Fig. 2 one can see that even though thegeneral shape remains, the OD's and the spectral widths aresignificantly reduced. It is possible to compensate for thereduction in peak OD's by increasing the thickness of thefilm. However, this will not increase the spectral or angularwidths. Therefore the dynamic range is the most importantparameter for determining the number of reflection linesthat can be potentially achieved in a given medium. Ingeneral, materials with larger dynamic ranges are bettersuited for making multiplexed holograms. At present dich-romated gelatin offers the greatest dynamic range. The

theoretical value is approximately 0.5, i.e., the difference inindex between the gelatin and air. Therefore it is expectedthat dichromated gelatin is the best medium for multiplex-ing a large number of gratings.

Nearly Degenerate CaseA similar analysis can be performed for nearly degenerateholograms. A multiplexed hologram is referred to as nearlydengerate if the grating vectors are nearly equal such thatthere is significant interaction between the diffracted waves.As an example let us calculate the spectrum of a hologramwith an index profile consisting of two sinusoidal gratings.The two construction wavelengths chosen are 700 and 705nm. Figure 7 shows the calculated OD as a function ofwavelength for this hologram for two different combinationsof the initial phases. The spectrum calculated by using thecoupled-wave theory is also shown for comparison purposes.From this figure, we make the two observations discussedbelow.

(1) The peak OD value and the shape of the spectrum aremuch more sensitive to the initial phases of the gratings.

4.0 -

. _

0

-I

.i_

CD

3.0 -

2.0 -

1.0 -

nn450 500

g~~~~~~~~5 I

550 600 650 700 750 800 850Wavelength (nm)

Fig. 6. Effect of limited dynamic range on the optical density andspectral width.

5.0

4-0 __la4 ... _unia /hon\\ < cos - sin

.;,

a)

.Y

0L

3.0

2.0

1.0

0.0650 660 670 680 690 700 710 720 730 740 750

Wavelength (nm)Fig. 7. Optical density versus wavelength at normal incidence fornear-degenerate multiplexing.

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I __

1440 J. Opt. Soc. Am. A/Vol. 7, No. 8/August 1990

5000 0000 15000Depth Z (nm)

(a)

20000 25

000 15000

Depth Z (ni)

(2) The departure of the coupled-wave theory from theexact thin-film theory is more significant. The sensitivity ofthe OD curve to the initial phases of the gratings is notreflected by the coupled-wave theory. Furthermore, Kogel-nik's theory for a single sinusoidal grating can no longer beused to estimate the spectral properties because of thestrong interaction between the two nearly degenerate grat-ings. Nevertheless, for practical applications [the indexprofiles are more likely to be like the one shown in Fig. 8(b)than that shown in Fig. 8(a)] the coupled-wave theory is stilla useful method for estimating the spectral properties.

x0

(b)Fig. 8. Refractive-index profiles for two combinations of the initialphases: (a) cos + cos; (b) cos - sin.

This perhaps can be better understood by inspecting theshape of the actual index profile within the medium. Theprofile can be considered as the product of two periodicfunctions: a slow varying envelope and a rapid oscillatingfunction. The frequency of the envelope is half the frequen-cy difference between the two sinusoidal gratings, and thefrequency of the fast oscillating functions is half the fre-quency sum of the two gratings. If the periods of the twosinusoidal gratings are nearly equal, the period of the enve-lope can be greater than the thickness of the hologram.Depending on the initial phases of the two sinusoidal grat-ings, the index profile within the medium can be either nearthe valleys (envelope = 0) or near the peaks (envelope = h1)of the envelope function. In this specific case the cos + cosprofile corresponds to a valley case, and the cos - sin corre-sponds to a peak case. The actual profiles are shown in Figs.8(a) and 8(b), respectively. The overall reduction in theindex modulation for the cos + cos profile accounts for theoverall lower OD in Fig. 7.

SUMMARY AND CONCLUSION

Two methods for modeling multiplexed-holographic reflec-tors were discussed. Specifically the coupled-wave theoryof Kogelnik was extended to cover multiplexed hologramshaving index profiles as superpositions of mutiple-sinusoi-dal terms. Numerical results were presented for nondegen-erate and nearly degenerate cases. We concluded that thecoupled-wave approach is a valid method for estimating thepeak OD's, peak wavelengths, and widths if the sinusoidalgratings are nondegenerate. For nearly degenerate holo-grams the numerical results indicate that the shape of thespectrum depends sensitively on the initial phases of thesinusoidal gratings and that the thin-film-matrix method isrequired for accurate results.

REFERENCES

1. S. K. Case, "Coupled-wave theory for multiply exposed thickholographic gratings," J. Opt. Soc. Am. 65, 724-729 (1975).

2. H. J. Caulfield, Handbook of Optical Holography (Academic,New York, 1979).

3. R. K. Kostuk, J. W. Goodman, and L. Hesselink, "Volume re-flection holograms with multiple gratings: an experimentaland theoretical evaluation," Appl. Opt. 25, 4362-4369 (1986).

4. H. Kogelnik, "Coupled wave theory for thick hologram grat-ings," Bell Syst. Tech. J. 48, 2909-2947 (1969).

5. For example, R. Magnusson and T. K. Gaylord, "Analysis ofmultiwave diffraction of thick gratings," J. Opt. Soc. Am. 67,1165-1170 (1977); M. G. Moharam and T. K. Gaylord, "Rigor-ous coupled-wave analysis of planar-grating diffraction," J.Opt. Soc. Am. 71,811-818 (1981); Z. Zylberberg and E. Marom,"Rigorous coupled-wave analysis of pure reflection gratings," J.Opt. Soc. Am. 73, 392-398 (1983).

6. X. Y. Chen, "Using the finite element method to solve coupledwave equations in volume holograms," J. Mod. Opt. 35, 1383-1391 (1988).

7. M. G. Moharam and T. K. Gaylord, "Chain-matrix analysis ofarbitrary-thickness dielectric reflection gratings," J. Opt. Soc.Am. 72, 187-190 (1982).

8. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon,New York, 1975), p. 66.

9. J. D. Masso, "Multilayer thin film simulation of volume holo-grams," in Holographic Optics: Design and Applications, I.Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 883, 68-74(1986).

10. N. Tsukada, R. Tsujinishi, and K. Tomishima, "Effects of therelative phase relationships of gratings on diffraction from thickholograms," J. Opt. Soc. Am. 69, 705-711 (1979).

I..1.60

1.55

150

1.45

X.a

1.40 +0

1.60

1.55

1.50

x

U)U)

1.45

1.40

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