Certain transparent materials change their optical properties when subjected to an electric field. This is a result of forces that distort the positions, orientations, or shapes of the molecules constituting the material. The electro-optic effect is a change in the refractive index that results from the application of a steady or low-frequency electric field (Fig. 20.0-1). An clectric field applied to an anisotropic optical material modifies its refractive indexes and thereby the effect that it has on polarized light passing through it, Electric field Figure 20.0-1 A steady electric field applied e to an electro-optic material changes its refractive index. This in tur changes the effect of the Electro-optic material on light traveling through it. The electric material field therefore controls the light. ‘The dependence of the refractive index on the applied electric field usually assumes one of the two following forms: = The refractive index changes in proportion to the applied electric field, an effect known as the linear electro-optic effect or Pockels effect. = The refractive index changes in proportion to the square of the applicd electric field, an effect known as the quadratic electro-optic effect or Kerr effect. The change in the refractive index is typically small. Nevertheless, the phase of an optical wave propagating through an electro-optic medium can be modified sig- nificantly if the distance of travel substantially exceeds the wavelength of light. As an example, if the refractive index is increased by 107° by virtue of the presence of the electric field, an optical wave propagating a distance of 10° wavelengths will experience an additional phase shift of 27. Materials whose refractive index can be modified by means of an applied electric field are useful for producing electrically controllable optical devices, as indicated by the following examples: = A lens comprising a material whose refractive index can be varied is a lens of controllable focal length. = A prism whose beam-bending capability is controllable can be used as an optical scanning device. = Light transmitted through a transparent plate of controllable refractive index un- dergoes a controllable phase shift so that the plate can be used as an optical phase modulator, = An anisotropic crystal whose refractive indexes can be changed serves as a wave retarder of controllable retardation; it may be used to change the polarization properties of light. = A wave retarder placed between two crossed polarizers gives rise to transmitted light whose intensity is dependent on the phase retardation (see Sec. 6.6B). The uansinittance of such a device is therefore electrically controllable so that it can be used as an optical intensity modulator or an optical switch. Controllable components such as these find substantial use in optical communications and in optical signal-processing applications. An electric field can instead modify the optical properties of a material via ab- sorption. A semiconductor material is normally optically transparent to light whose 835,836 © CHAPTER 20 ELECTRO-OPTICS wavelength is longer than the bandgap wavelength (see Sec. 16.2B). However, an applied electric field can reduce the bandgap of the material, thereby facilitating ab- sorption and converting the material from transparent to opaque. This effect, known as electroabsorption, is useful for making optical modulators and switches. This Chapter ‘We begin with a description of the electro-optic effect and the principles of electro- optic modulation and scanning. The initial presentation in Sec. 20.1 is simplified by deferring the detailed consideration of anisotropic effects to Sec. 20.2. Section 20.3 is devoted to the electro-optic properties of liquid crystals. An electric field applied to the molecules of a liquid crystal causes them to alter their orientations. This leads to changes in the optical properties of the medium, i.c., it exhibits an electro- optic effect. The molecules of a twisted nematic liquid crystal are organized in a helical pattern so that they normally act as polarization rotators. An applied electric field can be used to remove the helical pattern, thereby deactivating the polarization rotatory power of the material. Turning the electric field off results in the material regaining its original helical structure and therefore its rotatory power, Thus, the device acts as a dynamic polarization rotator. The use of additional fixed polarizers permits such a polarization rotator to serve as an intensity modulator or a switch. This behavior is the basis of most liquid-crystal display devices. The electro-optic properties of photorefractive media are considered in Sec. 20.4. These are materials in which the absorption of light creates an internal electric field, which, in turn, initiates an electro-optic effect that alters the optical properties of the medium. Thus, the optical properties of the medium are indirectly controlled by the light incident on it. Photorefractive devices therefore permit light to control light. Finally, a brief introduction to electroabsorption is provided in Sec. 20.5. 20.1 PRINCIPLES OF ELECTRO-OPTICS A. Pockels and Kerr Effects The refractive index of an electro-optic medium is a function n(£) of an applied steady (or slowly varying) electric field 2. The function n(£) varies only slightly with E so that it can be expanded in a Taylor series about n(B) =n+a8 + hak? +---, (20.1-1) where the coefficients of expansion aren = n(0), a1 = (dn/dB)|p~0, and a2 = (@n/dE*)|r-0. For reasons that will become apparent below, it is conventional to write (20.1-1) in terms of two new coefficients, t = —2a,/n3 and 5 = —a2/n3, known as the electro-optic coefficients, so that n(B) =n — hrn'B — pen B+. (20.1-2) The second- and higher-order terms of this series are typically many orders of magni- tude smaller than n. Terms higher than the third can safely be neglected. For future use it is convenient to derive an expression for the electric impermeability, €,/€ = 1/n®, of the electro-optic medium as a function of E. The parameter 17 is useful in describing the optical propertics of anisotropic media (sce See. 6.3A). The incremental change An = (dn/dn)An = (—2/n®)(—}en9E — }sn3E*) =20.1. PRINCIPLES OF ELECTRO-OPTICS 837 vE | 5 E%,so that n(E) Sn +tE+sE%, (20.1-3) where 1 = (0). The electro-optic coefficients and s are therefore simply the coef- ficients of proportionality of the (wo terms of An with B and £?, respectively, This explains the seemingly odd definitions of t and s in (20. 1-2). The values of the coefficients t and s depend on the direction of the applied electric field and the polarization of the light, as will be discussed in Sec. 20.2. Pockels Effect In many materials the third term of (20.1 2) is negligible in comparison with the second, whereupon n(B) © n — pen, (20.1-4) Pockels Effect as illustrated in Fig. 20.1-1(a). The medium is then known as a Pockels medium (or a Pockels cell). The coefficient t is called the Pockels coefficient or the linear electro- optic coefficient. Typical values of r lie in the range 10-"? to 107! m/V (1 to 100 pm/V). For FE = 10° V/m (10 kV applied across a cell of thickness 1 cm), for example, the term Sen B in (20.1-4) is on the order of 10~° to 1074. Changes in the refractive index induced by electric fields are indeed very small. Common crystals used as Pockels cells include NHqH2POs (ADP), KH2POq (KDP), LiNbOs, LiTaOs, and CdTe. n(E) n(B) > n (a) 0 E (b) oO E Figure 20.1-1 Dependence of the refractive index on the electric field: (a) Pockels: medium; (b) Kerr medium. Kerr Effect If the material is centrogymmetric, as is the case for gases, liquids, and certain crystals, n(E) must be an even symmetric function [see Fig. 20.1-1(b)] since it must be invariant to the reversal of #. Its first derivative then vanishes, so that the coefficient ¢ must be zeto, whereupon n(E) x n—-dsr3B. (20.1-5) Kerr Effect ‘The material is then known as a Kerr medium (ora Kerr cell). The parameter s is called the Kerr coefficient or the quadratic electro-optic coefficient. Typical values of s are 10-8 to 10-4 m?/V? in crystals and 10? to 10" m?/V? in liquids. For 2 = 10°838 CHAPTER20 ELECTRO-OPTICS ‘Vim the term $523? in (20.1-5) is on the order of 107° to 107? in crystals and 107! to 1077 in liquids. B. Electro-Optic Modulators and Switches Phase Modulators A beam of light traversing a Pockels cell of length L to which an elecwic field E is applied undergoes a phase shift y = n(E)kol = 2rn(#)L/Ao, Where Ap is the free- space wavelength. Using (20.1-4), we have 3 exe ne, (20.1-6) where yo = 2nnF/Ao. If the electric field is obtained by applying a voltage V across two faces of the cell separated by distance d, then E = V/d, and (20.1-6) gives (20.1-7) Phase Modulation (201-8) Half-Wave Voltage ‘The parameter V;, known as the half-wave voltage, is the applied voltage at which the phase shift changes by 7. Equation (20. 1-7) expresses a linear relation between the optical phase shift and the voltage. One can therefore modulate the phase of an optical wave by varying the voltage V that is applied across a material through which the light passcs. The parameter V; is an important characteristic of the modulator. It depends on the material properties (n and t), on the wavelength ,, and on the aspect ratio d/L. The electric field may be applied in a direction perpendicular to the direction of light propagation (Iransverse modulators) or parallel thereto (longitudinal modulators), in which case d = L (Fig, 20.1-2). The value of the electro-optic coefficient + depends on the directions of propagation and the applied field since the crystal is, in general, anisotropic (as explained in Sec. 20.2). Typical values of the half-wave voltage are in the vicinity of 1 to a few kilovolts for longitudinal modulators. and hundreds of volts for transverse modulators, The speed at which an clectro-optic modulator operates is limited by electrical capacitive effects and by the transit time of the light through the material. If the electric field F(t) varies significantly within the light transit time T, the traveling optical wave will be subjected to different electric fields as it traverses the crystal. The modulated20.1. PRINCIPLES OF ELECTRO-OPTICS 839 @ o o Figure 20.1-2 (a) Longitudinal modulator. The electrodes may take the shape of washers or bands, or may be transparent conductors, (b) Transverse modulator, (c) Traveling-wave transverse modulator. phase at a given time ¢ will then be proportional to the average electric field E(t) at times from ¢ — T to 6. As a result, the tansit-time-limited modulation bandwidth is = 1/T. One method of reducing this time is to apply the voltage V at one end of the crystal while the electrodes serve as a transmission line, as illustrated in Fig. 20.1- 2c). If the velocity of the traveling electrical wave matches that of the optical wave, uansit (ime effects can, in principle, be eliminated. Commercial modulators in the forms shown in Fig. 20.1-2 generally operate at several hundred MHz, but modulation speeds of several GHz are possible. Electro-optic modulators can also be constructed as integrated-optical devices. These devices operate at higher speeds and lower voltages than do bulk devices. An optical waveguide is fabricated in an electro-optic substrate (often LiNbOs) by indiffusing a material such as titanium to increase the refractive index. The electric field is applied to the waveguide using electrodes, as shown in Fig. 20.1-3. Because the configuration is transverse and the width of the waveguide is much smaller than its length (o' < L), the half-wave voltage can be as smalll as a few volts. These modulators have been operated at speeds in excess of 100 GHz. Light can be conveniently coupled into, and out of, the modulator by the use of optical fibers. Input Waveguide Figure 20.1-3 An integrated-optival phase modulator using the electro-optic effect, Dynamic Wave Retarders An anisotropic medium has two linearly polarized normal modes that propagate with different velocities, say c./n1 and co/7i2 (see Sec. 6.3B). If the medium exhibits the840 CHAPTER20 ELEGTHO-OPTICS Pockels effect, then in the presence of a steady electrical field E the two refractive indexes are modified in accordance with (20.1-4), ic., m(B) & m — FuunjB (20.1-9) n2(E) = nz — dren3B, (20.1-10) where ty and rp are the appropriate Pockels coefficients (anisotropic effects are exam- ined in detail in Sec. 20.2). After propagation a distance L, the two modes undergo a relative phase retardation given by T= kolra(E) — n(B)|L = ko(m, — na) L — Bko(tynt — tend) EL. 20.1-11) If E is obtained by applying a voltage be V between two surfaces of the medium that are separated by a distance d, (20.1-11) can be written in compact form as pon, (20.1-12) Ve Phase Retardation and 5 (20.1-13) tiny — tang Retardation Halt-Wave Voltage is the applied voltage necessary to obtain a phase retardation 7. Equation (20.1-12) in- dicates that the phase retardation is linearly related to the applied voltage. The medium serves as an electrically controllable dynamic wave retarder. Intensity Modulators: Use of a Phase Modulator in an Interferometer Phase delay (or retardation) alone does not affect the intensity of a light beam. How- ever, a phase modulator placed in one branch of an interferometer can function as an intensity modulator. Consider, for example, the Mach-Zehnder interferometer illus- trated in Fig. 20.1-4. If the beamsplitters divide the optical power equally, the intensity transmitted through one output port of the interferometer [, is related to the incident intensity I; by I, = 4h + fli cosy = [cos (y/2), (20.1-14) where y = v1 — is the difference between the phase shifts encountered by light as it travels through the two branches (see Sec. 2.5A). The transmittance of the interfer ometer is T — [p/I; — cos?(y/2). Because of the presence of the phase modulator in branch 1, according to (20.1- 7) we have v1 = Yi, — TV/V;. so that ¢ is controlled by the applied voltage V in20.1 PRINCIPLES OF ELECTRO-OPTICS 841. Branch 2 Figure 20.1-4 A phase modulator placed in one branch of a Mach-Zehnder interferometer can serve as an intensity modulator. The transmittance of the interferometer T(V) — Iy/J; varies periodically with the applied voltage V. By operating in a limited region near point B, the device acts as a linear intensity modulator. If V is switched between points A and C;, the device serves as an optical switch. accordance with the linear relation yo = 1 — ¢2 = Yo ~ TV /Vzy where the constant (Po = 1, — 2 depends on the optical path difference. The transmittance of the device is therefore a function of the applied voltage V, go TV 100) = 008 (2-55). (20.1-15) 22% Transmittance This function is plotted in Fig. 20.1-4 for an arbitrary value of «zo. The device may be operated as a linear intensity modulator by adjusting the optical path difference so that Yo = 7/2 and operating in the nearly linear region around J = 0.5. Alternatively, the optical path difference may be adjusted so that yo is a multiple of 27. In this case (0) = 1 and 3(V,,) = 0, so that the modulator switches the light on and off as V is switched between 0 and A Mach-Zehnder intensity modulator may also be constructed in the form of an integrated-optical device. Waveguides are placed on a substrate in the geometry shown, in Fig. 20.1-5. The beamsplitters are implemented by the use of waveguide Y's. The optical input and output may be carried out by optical fibers. Commercially available integrated-optical modulators generally operate at speeds of a few GHz but modulation speeds exceeding 25 GHz have been achieved. ee wi . 8 Figure 20.1-5 An integrated-optical in- tensity modulator (or optical switch). A Mach-Zehnder interferometer and an electro- optic phase modulator are implemented using optical waveguides fabricated from a material such as LINDOs. © taut ian Intensity Modulators: Use of a Retarder Between Crossed Polarizers As described in Sec. 6.6B, a wave retarder (retardation I’) sandwiched between two crossed polarizers, placed at 45° with respect to the retarder’s axes (see Fig. 6.6- 4), has an intensity transmittance T = sin’(I’'/2). If the retarder is a Pockels cell,842 CHAPTER 20 ELECTRO-OPTICS then T’ is linearly dependent on the applied voltage V as provided in (20.1-12). The transmittance of the device is then a periodic function of V, (20.1-16) Transmittance as shown in Fig. 20.1-6. By changing V, the transmittance can be varied between 0 (shutter closed) and 1 (shutter open). The device can also be used as a linear modulator if the system is operated in the region near T(V) = 0.5. By selecting I) = /2 and V<«V,, Vv v0 (20.1-17) F(V) = sin? G z) = 3(0) + 2 dv so that T(V/) is a linear function with slope */2Vq representing the sensitivity of the modulator, The phase retardation 'g can be adjusted either optically (by assisting the modulator with an additional phase retarder, a compensator) or electrically by adding a constant bias voltage to V. @ ® t Figure 20.1-6 (a) An optical intensity modulator using a Pockels cell placed between two crossed polarizers. ()) Optical transmittance versus applied voltage for an arbitrary value of Uy; for linear operation the cell is biased near the point B. In practice, the maximum transmittance of the modulator is smaller than unity because of losses caused by reflection, absorption, and scattering. Furthermore, the minimum transmittance is greater than 0 because of misalignments of the direction of propagation and the directions of polarizations relative to the crystal axes and the polarizers. The ratio between the maximum and minimum transmittances is called the extinction ratio, Ratios higher than 30 dB (1000: 1) are possible. C. Scanners An optical beam can be deflected dynamically by using a prism with an electrically controlled refractive index. The angle of deflection introduced by a prism of smail apex angle a and refractive index n is @ © (n — 1a [see (1.2-7)]. An incremental change of the refractive index An caused by an applied electric field F corresponds to an incremental change of the deflection angle, AO = aAn = —jarn*E = —jarn*V/d, (20.1-18)20.1 PRINCIPLES OF ELECTRO-OPTICS 843 where V is the applied voltage and d is the prism width [Fig. 20.1-7(a)]. By varying the applied voltage V, the angle AQ varies proportionally, so that the incident light is scanned. @ Lt Figure 20.1-7 (a) An clectro-optic prism. The deflection angle @ is controlled by the applied voltage. (2) An electro-optic double prism, Itis often more convenient to place triangularly shaped electrodes defining a prism on the rectangular crystal. Two, or several, prisms can be cascaded by alternating the direction of the clectric field, as illustrated in Fig. 20.1-7(6). An important parameter that characterizes a scanneris its resolution, i.e., the number of independent spots it can scan. An optical beam of width D and wavelength , has an angular divergence 56 = q/D [see (4.3-7)]. To minimize that angle, the beam should be as wide as possible, ideally covering the entire width of the prism itself. For a given maximum voltage V corresponding to a scanned angle A@, the number of independent spots is given by (20.1-19) Neo (20.1-20) from which V ~ 2VV,. This is a discouraging result. To scan NV independent spots, a voltage 21V times greater than the half-wave voltage is necessary. Since V, is usually large, making a useful scanner with N ‘>> 1 requires unacceptably high voltages. More commonly used scanners therefore include mechanical and acousto-optic scanners (see Secs. 19.2B and 23.3B). The process of double refraction in anisotropic crystals (see Sec. 6.3E) introduces a lateral shift of an incident beam parallel to itself for one polarization and no shift for the other polarization. This effect can be used for switching a beam between two parallel positions by switching the polarization. A linearly polarized optical beam is transinitted first dhrough an electro-optic wave retarder acting as a polarization rotator and then through the crystal. The rotator controls the polarization electrically, which determines whether the beam is shifted laterally, as illustrated in Fig. 20.1-8. D. Directional Couplers An important application of the electro-optic effect is in controlling the coupling be- tween two parallel waveguides in integrated-optical device. An electric field can be used to transfer the light from one waveguide to the other, so that the device serves as an electrically controlled directional coupler.844 CHAPTER20 ELECTRO-OPTICS “ges Hlectooptic 4 Birefringent polarization rotator crystal Figure 20.1-8 A position switch based on electro-optic phase retar- dation and double refraction. The coupling of light between two parallel single-mode planar waveguides [Fig. 20.1-9(a)] was examined in Sec. 8.5B. It was shown that the optical powers carried by the two waveguides, P,(z) and P2(z), are exchanged periodically along the direction of propagation z. Two parameters govern the strength of this coupling process: the coupling coefficient © (which depends on the dimensions, wavelength, and refractive indexes), and the mismatch of the propagation constants AB = f, — , = 27 An/d., where An is the difference between the refractive indexes of the waveguides. If the waveguides are identical, with Af = 0 and P2(0) = 0, then at a distance z = Lo = 7/2C, called the transfer distance or coupling length, the power is transferred completely from waveguide 1 into waveguide 2, ie., P:(To) = 0 and Po(Io) = P, (0), as illustrated in Fig. 20.1-9(a).. 1 \ a) 0 Br Asly @ % to o Figure 20.1-9 (0) Exchange of power between two parallel weakly coupled waveguides that are identical, with the same propagation constant 3. At z — 0 all of the power is in waveguide 1. At Lp all of the power is transferred into waveguide 2. (b) Dependence of the power-transfer ratio P2(Lo)/P,(0) on the phase mismatch parameter AG Lo. Fora waveguide of length Ly and AG ¥ 0, the power-transfer ratio J = P2(Lo)/P1(0) is a function of the phase mismatch [see (8.5-12a)], (20.1-21) where sine(r) = sin(r)/(m2). Figure 20.1-9(}) illustrates this dependence. The ratio has its maximum value of unity at A Lo = 0, decreases with increasing Af Lo, and vanishes when AG Lo = 377, at which point the optical power is not transferred to waveguide 2.20.1 PRINCIPLES OF ELECTRO-OPTICS 845 ‘A dependence of the coupled power on the phase mismatch is the key to making electrically activated directional couplers. If the mismatch A Lo is switched from 0 Sr, the light remains in waveguide 1, Electrical control of Af is achieved by use of the electro-optic effect. An electric field E applied to one of two, otherwise identical, waveguides alters the refractive index by An = —4n%r F, where t is the Pockels coefficient. This results in a phase shift AB Lo = An(2nLo/'y ) = -(n/Ay)n° LoB. A typical electro-optic directional coupler has the geometry shown in Fig. 20.1-10. ‘The electrodes are laid over two waveguides separated by a distance d. An applied voltage V creates an electric field E ~ V/d in one waveguide and —V//d in the other, where d is an effective distance determined by solving the electrostatics problem, (the electric-field lines go downward at one waveguide and upward at the other). The refractive index is incremented in one guide and decremented in the other. The result, is a net refractive index difference 2An = —nt(V/d), corresponding to a phase mismatch AG Lo = —(27/A,)n*x(Lo/d)V, which is proportional to the applied voltage V. Pu(0). =. a Figure 20.110 An _ integrated electro-optic directional coupler. ‘The voltage Vp necessary to switch the optical power is that for which |A3 Lo| = V3r, ie., qd do _ V3 0rd ST One nw ont Yo= : (20.1-22) where Lo = 7/2C and @ is the coupling coefficient. This is called the switching voltage Since |AZ Lo| = V3 x V/V, (20.1-21) gives (20.1-23) Coupling Efficiency This equation (plotted in Fig. 20.1-11) governs the coupling of power as a function of the applied voltage V. ‘An electro-optic directional coupler is characterized by its coupling length Lo, which is inversely proportional to the coupling coefficient ©, and its switching voltage Vo, which is directly proportional to @. The key parameter is therefore , which is governed by the geometry and the refractive indexes. Integrated-optic directional couplers may be fabricated, for example, by diffusing titanium into high-purity LiNbO. substrates. The switching voltage Vp is typically less846 CHAPTER20 ELECTRO-OPTICS 1 1 Figure 20.1-11 Dependence of the coupling effi- ciency on the applied voltage V. When V = 0, all of the optical power is coupled from waveguide 1 into d waveguide 2; when V = Vg, all of the optical power 0 Vo v remains in waveguide 1. than 10 V, and the operating speeds can exceed 10 GHz. The light beams are focused to spot sizes of a few zm. The ends of the waveguide may be permanently attached to single-mode polarization-maintaining optical fibers (sce Sec. 9.1C). Increased band- Widths can be obtained by making use of a traveling-wave version of this device. a PI EE eee EXERCISE 20.1-1 Coupling-Etticiency Spectral Response. Equation (20.1-22) indicates that the switching voltage Vo is proportional to the wavelength. Assume that the applied voltage V = Vo for a particular value of the wavelength A... so that the coupling efficiency T = 0 at A.. If. instead, the incident wave hhas wavelength 2, plot the coupling efficiency J as a function of 1, — A... Assume that the coupling coefficient @ and the material parameters n and ¢ are approximately independent of wavelength. E. Spatial Light Modulators A spatial light modulator is a device that modulates the intensity of light at different positions by prescribed factors (Fig. 20.1-12). It is a planar optical element of control- lable intensity transmittance T(., y). The transmitted light intensity /,(, y) is related to the incident light intensity J;(z,y) by the product I,(x,y) = I;(x, y)T(«,y). If the incident light is uniform [i.e., 1;(«, y) is constant], the transmitted light intensity is proportional to T(x, y). The “image” T(x, y) is then imparted to the transmitted light, much like “reading” the image stored in a transparency by uniformly illuminating it in a slide projector. In a spatial light modulator, however, T(x, y) is controllable. In an electro-optic modulator the control is electrical. Figure 20.1-12 The spatial light modulator. To construct a spatial light modulator using the electro-optic effect, some mecha- nism must be devised for creating an electric field E(x, y) proportional to the desired transmittance T(:r, y) at each position. This is not easy. One approach is to place an array of transparent electrodes on small plates of electro-optic material placed between crossed polarizers and to apply on each electrode an appropriate voltage (Fig. 20.1- 13). The voltage applied to the electrode centered at the position (2;,9);), t= 1,2,..-