June 1, 2001 / Vol. 26, No. 11 / OPTICS LETTERS 759

Switchable optical element with Bragg mode diffraction

Mykola Kulishov

Adtek Photomask, Inc., 4950 Fisher Street, Montreal, Quebec H4T 1J6, Canada

Sergey Sarkisov

Alabama A&M University, Normal, Alabama 35762

Yuri Boiko

Department of Electrical Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093

Pavel Cheben

Institute for National Measurement Standards, National Research Council of Canada, Montreal Road,Ottawa, Ontario, K1A 0R6, Canada

Received November 13, 2000

A theoretical model of a new electronically switchable grating design that uses a multilayer structure of anelectro-optic (EO) material with an interdigitated-electrode type of array is proposed as an original techniquefor calculating the induced refractive index. It is shown that asymmetrical distribution of the electric fieldinduces a slanted Bragg grating, which allows the slant angle to be switched electronically among more thantwo switching states. Parameters of the suggested design are calculated for a number of EO materials. Aspecial case of frequency-based switching is anticipated for some polymer-dispersed liquid-crystal materials.© 2001 Optical Society of America

OCIS codes: 120.5710, 060.5060, 060.2760, 160.2100, 130.3120.

Combinations of diffractive structures with electro-optical (EO) materials have recently produced a newclass of switching optical elements (SOEs) with thepotential for a variety of applications. DiffractiveSOEs proposed in the past employed mostly theRaman–Nath, or thin grating, regime of diffraction,with the energy of the light dispersed among multiplediffraction orders.1,2 This dispersion resulted in poorselectivity between ON and OFF states of SOEs. Onlya few thick-diffraction-grating designs of SOEs arecurrently being investigated.3,4 They work in a Braggdiffraction mode with theoretically only one diffrac-tion order and much better switching selectivity. Itappears natural to obtain a thick Bragg grating SOEby vertical stacking of multiple layers of thin SOEs.

Here we theoretically analyze the design and per-formance of such a multilayer SOE with interdigitatedelectrodes that is a thick-grating modification ofthe previously reported thin-grating design.1,2 Itsprinciple of operation is based on switching amongvarious asymmetric distributions of the electric f ieldinduced by a set of regularly positioned transparentelectrodes. One design is based on multiplication ofthe double-sided electrode wafer. As an illustrationof the design concept we show the three-layeredstructure �N � 3� in Fig. 1. However, the number Nof deposited layers has to be sufficient for achieving athick-grating diffraction mode.

Here a rigorous method is presented for calculationof the refractive-index profile inside the EO material,in which the latter is assumed to be crystalline, withthe principal dielectric axes aligned along the x andz directions such that the dielectric properties are de-fined by two diagonal relative permittivities, e11 ande33. Without loss of generality, our analysis is per-formed for a particular case of the three-layered struc-

0146-9592/01/110759-03$15.00/0 ©

ture shown in Fig. 1, where h is the layer thickness,l is the electrode’s spatial period and a is the elec-trode’s width. The electrodes are assumed to be infi-nitely thin and perfectly conducting. Keeping in mindthat the electric potential distribution must be a solu-tion of the Laplace equation, we use an expansion ofthe potential function in terms of harmonic and hyper-bolic eigenfunctions:

w�1��x, z� � 2 V0

X̀n�0

En exp��n 1 1�2�k�3h�2 2 z��

3 sinh��n 1 1�2�dkh�cos��n 1 1�2�kx�,

1 `# z # 3h�2 , (1)264

w�2��x,z�w�3��x, z�w�4��x, z�

375 � V0

3Xn�0

En

264

sinh��n 1 1�2�dk�h�2 2 z��sinh��n 1 1�2�dk�h�2 2 z��sinh��n 1 1�2�dk�3h�2 1 z��

375

3 cos∑µ

n 112

∂kx

∏1

264

sinh��n 1 1�2�dk�3h�2 2 z��sinh��n 1 1�2�dk�h�2 1 z��sinh��n 1 1�2�dk�h�2 1 z��

375

3 cos∑µ

n 112

∂k

µx 2

l2

∂∏, 1 3h�2 # z # h�2,

1 h�2 # z # 2h�2,

2 h�2 # z # 23h�2 . (2)

Equation (2) satisfies the following boundary con-ditions w�1��x, z � 13h�2� � w�2��x, z � 13h�2�,

2001 Optical Society of America

760 OPTICS LETTERS / Vol. 26, No. 11 / June 1, 2001

Fig. 1. Cross section of the SOE multilayer design.

w�2��x, z � 1h�2� � w�3��x, z � 1h�2�, w�3��x, z � 1

h�2� � w�4��x, z � 2h�2�, and w�2��x, z � 13h�2� �w�4��x 2 l�2, z � 23h�2�, where k � 2p�l, d ��e11�e33�

1/2, and En are the Fourier coeff icients tobe found. The boundary conditions for the electricpotential and the normal component of the dielectricdisplacement vector at z � 13h�2 lead to the followingdual series equations:

X̀n�0

En�Hn cos��n 1 1�2�kx� � 1 , 0 # x # a�2 ,

X̀n�0

�n 1 1�2�En��cos��n 1 1�2�kx�2

Gncos��n 1 1�2�k�x 2 l�2��� � 0 , a�2 # x # l�2 , (3)

where Gn � �cosh��n 1 1�2�dkh� 1 �e��de33��sinh��n 11�2�dkh��21, En

� � En�Gn, an d Hn � sinh��n 1

l�2�dkh�Gn, where e is the dielectric constant of thesurrounding medium. One can see that the solutionis reduced to the dual series of Eqs. (3), which weresolved in the recent paper by Kulishov et al..2 Havingfound the expansion coeff icients En, we can calculatethe electric potential and the electric field distributioninside the EO material. Once the EO response of theparticular material is known, the EO-induced refrac-tive index can be derived from the field distribution.

Though our technique can be used for a varietyof materials, including EO polymers and polymer-dispersed liquid crystals (PDLCs),4 some materialsmay require quite cumbersome calculation that isbeyond the scope of this Letter. For this study, simplecases of EO crystals and EO polycrystalline ceramicsare used. In Fig. 2 we present a contour plot of therefractive-index distribution of the structure shownin Fig. 1 when a 30% molar ratio Disperse Red 1chromophore side chain poly(methyl methacrylate)polymer is used as the EO material. We consider thepropagation of a y-polarized incident wave. Refractiveindex ny for this wave is described by the well-knownrelationship ny�x, z� � no�1 2 r13no

2Ez�x, z��2�,1where no is the intrinsic ordinary refractive in-dex of the material, r13 is the EO coefficient, andEz�x, z� �2≠w�x, z��≠z is the z component of the elec-tric field strength vector. A similar distribution will

be achieved for any material that belongs to one of 4,6, and `mm crystallographic point-group symmetries.

The distributions in Fig. 2 demonstrate that fringescompose a slanted grating. One can reverse the slantangle simply by inverting the potentials on the elec-trodes of Fig. 1 in the planes z �1h�2 and z �23h�2between 1V0 and 2V0. Inversion of the potentials onthe electrodes in the planes z �2h�2 and z �23h�2between 1V0 and 2V0 results in a periodic refractive-index distribution in which the angle of the slant isrepeated every second layer. The suggested schemefor application of the electric potential creates onlya variable component of the electric field without aconstant component. Thus the applied voltage affectsonly the induced change in distribution of the refrac-tive index within each layer, leaving intact the aver-age value of the refractive index of the EO material.Therefore the average value of the refractive index in-side the EO material does not depend on the appliedelectric f ield, and this refractive index needs to be cho-sen to be as close as possible to the refractive indexof indium tin oxide (ITO) on the operating wavelength.The results depicted in Fig. 2 are determined by ourselection of the EO material. For our design it wouldbe advantageous to use a material with an EO indexdifference Dn�E� � 0.01 0.05, which exceeds the in-dex mismatch between the electrode material (ITO)and the EO material. The promising candidates forsuch a design are the PDLC materials4 that have theadvantage that no electric f ield poling is required (incontrast to EO polymers and ceramics) for EO activity.Another advantage of PDLCs as EO media is an addi-tional possibility for refractive-index control based onPDLC frequency-dependent EO response. Dependingon the frequency of the applied ac voltage, such mate-rials exhibit a positive or a negative EO response (thedielectric anisotropy, De, is positive when liquid-crys-tal molecules rotates in a direction perpendicular to anapplied electric f ield). The frequency of the appliedvoltage at which De changes sign is called the crossoverfrequency.

For this design it is important to have a clearpicture of the electric f ield distribution in the lay-ers. The electric field vector plots inside a singlelayer for three types of electric potential appli-cation are presented in Fig. 3. The lengths of

Fig. 2. Contour plot of the refractive-index distributionfor the following parameter set: a�l � 0.5, h � 1�2, e � l,e11 � 8.5, e33 � 2.9, n0 � 1.6, V0 � 6 V, h � 4 mm, and r13 �30 pm�V. The induced refractive index varies: 1.5992 #n # 1.6008.

June 1, 2001 / Vol. 26, No. 11 / OPTICS LETTERS 761

Fig. 3. Electric-f ield vector plot inside a single layer for(a), (b) two different layer thicknesses and for (c) differentschemes for the potential application: e11 � 8.5 and e33 �5.9; (a) a�l � 0.5, (b) h � l�2, and (c) h � l.

the bars are proportional to the f ield strength,E�x, z� �

pEx�x, z�2 1 Ez�x, z�2, and the bars are

oriented along the field c � arctan�Ez�x, z��Ex�x, z��.The bars do not show the field direction; for example,the fields within 2l # x # 0 and 0 # x # 1l have thesame orientation but opposite direction. However, itis the f ield’s orientation rather than its direction thatdefines the PDLC’s EO properties. The symbols ©and ™ denote the voltage polarity, 6V0, that allows usto decrease the periodicity of the PDLC-based gratingL � 2h�l 1 4�h�l�2�21�2 to two times less then that ofthe Pockels-effect based grating. The electrodes inFigs. 3(a) and 3(b) are electrically connected in sucha way that the electric f ields induce gratings withopposite tilts (roughly 145± and 230±, respectively).The field distributions in Figs. 3(a) and 3(b) showthat a thinner layer provides a greater EO orientationanisotropy. However, it is preferable to use thickerlayers to reduce the number of layers required forachieving the Bragg diffraction regime. The questionis how many layers we need to achieve a thick-gratingdiffraction mode. The most useful rule of thumb isthe Nath parameter r � �l�L�2��n0n1�, where L is thegrating period, l is the free-space wavelength of light;n0 is the average refractive index of the EO material,and n1 is the index modulation that compose thegrating. It was shown5 that the maximum intensitythat diffracted beams of orders 21 and 12 can attainis 2r22 and 1��4r2�, respectively. For example, thecontributions to intensity from these higher diffractionorders will not exceed 4% and 0.5%, respectively, forr � 7, providing a diffraction eff iciency of as much96.5% for the ON state. We took this value of r,together with operating wavelength l � 1 mm andaverage refractive index n0 � 1.5, and made a simpleestimation of the number of layers. Substitutinggrating period L into the formula for the r factor,we obtained that the condition n1h2 # 0.12 has to

be satisfied for h�l � 1, where the layer thicknessis expressed in micrometers. The light is switchedinto the f irst diffracted order when the dimensionlessgrating strength n � pn1Nh��l cos�Q�� reaches p�2,where Nh is the grating thickness and Q is the angleof incidence. For normal incidence, we arrive atn1h2 � 1��4N2n1�. Therefore N2n1 $ 1��4 3 0.12�,or n1 $ 2.08�N2. For example, for a five-layergrating, N � 5, n1 $ 0.083, h # 1.2 mm, andl # 1.2 mm. All these parameters are achievable withmicrofabrication.

An important issue in this design is how the periodicelectrodes will affect the SOE’s performance. Evenwith a high degree of ITO transparency, the transmit-ted light experiences phase modulation owing to theelectrode’s finite thickness. It has been shown that,1

provided that the electrode has constant thickness inall layers with a duty ratio a�l � 0.5, for normally in-cident light the phase delay from electrode f ingers inone plane is compensated for by the phase delay fromthe electrode finger in the next plane as long as thenumber of layers is kept uneven.

It is evident that, for gratings with the refractive-index mismatch between ITO electrodes and EOmaterial (in the OFF state) much lower than themagnitude of the EO-induced refractive-index grat-ing, the contribution from ITO electrode diffractionwill be negligible, allowing for an arbitrary angle ofincidence. For oblique light incidence we can alsoemploy an unslanted distribution of the inducedrefractive index. This distribution can be inducedby use of the potential application scheme shown inFig. 3(c), where the electric field distribution is alsoshown. The electric f ield distribution in Fig. 3 allowsus to anticipate that gradual tuning of the slant canbe achieved here through dielectric anisotropy De

control by adjustment of ac voltage about the crossoverfrequency (a discussion of which is beyond the scopeof the present Letter).

In conclusion, we have used the effective calculationmethod to analyze a novel design for an electronicallyswitchable thick grating based on a multiple-layer EOstructure with a periodic array of transparent elec-trodes. The design has the advantage of asymmetricdistribution of the electric f ield and the induced re-fractive index, which inf luences the slant of the grat-ing. Reversal of the slant is an envisaged result ofthe suggested design. Combined with high selectivityof the Bragg diffraction, it provides superior switchingcontrast.

References

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J. Opt. Soc. Am. B 18, 457 (2001).3. M. L. Jepsen and H. J. Gerritsen, Opt. Lett. 21, 1081

(1996).4. R. T. Pogue, R. L. Sutherland, M. G. Schmitt, L. V.

Natarajan, S. A. Siwecki, V. P. Tondiglia, and T. J. Bun-ning, Appl. Spectros. 54, 1 (2000).

5. K.-K. Lee and R. P. Kenan, Appl. Opt. 28, 74 (1989).