## Abstract

Integrated optics is a far-reaching attempt to apply thin-film technology to optical circuits and devices, and, by using methods of integrated circuitry, to achieve a better and more economical optical system. The specific topics discussed here are physics of light waves in thin films, materials and losses involved, methods of couplings light beam into and out of a thin film, and nonlinear interactions in waveguide structures. The purpose of this paper is to review in some detail the important development of this new and fascinating field, and to caution the reader that the technology involved is difficult because of the smallness and perfection demanded by thin-film optical devices.

© 1971 Optical Society of America

## I. Introduction

Since the invention of the solid and gas lasers a decade ago, there have been immense advances in optoelectronics. For example, laser transitions now cover the spectral region from ultraviolet light to millimeter waves. Optical modulation, frequency mixing, and parametric oscillations have been extensively studied. Almost all the experiments in the past were performed invariably in bulk materials and with a light beam of nearly gaussian intensity distribution. Recently, however, the introduction of the concept of integrated optics[1],[2] and the development of the prism–film[3] and grating[4],[5] couplers have raised important questions concerning the future needs of optical systems: Can we conform to the idea of the integrated optics and develop optical modulators, frequency converters, and parametric oscillators in a planar thin-film form which are more efficient than their bulk counterparts? Is there any advantage of using a waveguide structure for nonlinear, electrooptical, or light-wave scattering experiments? While many are devoted to the development of ever better bulk crystals for optical devices, could it be that what we really need in the long run are thin crystal films and epitaxial layers?

One cannot answer the above questions until the complex technology involved in the fabrication of thin-film structures is solved. On the other hand, as far as optical systems are considered, certain features of the thin-film devices appear to be definitely advantageous. First, all the elements of a thin-film device are exposed on the surface and are easily accessible for probing, measurement, or modification. Second, compared to microwaves, the optical wavelength is a factor of 10^{4} smaller. The thin-film optical devices can be made very small and they can be placed one next to the other on a single substrate, forming an optical system which is naturally more compact, less vulnerable to the environmental changes, and more economical. Third, since the film has a thickness comparable to the optical wavelength and since most of the light energy is confined within the film, the light intensity inside the film can be very large even at a moderate laser power level. For example, 1 W of laser power can easily result in a power density of 22 MW/cm^{2} in a ZnS film 0.46 *μ*m thick. This large power density is important in nonlinear interactions. Finally, the phase velocity of a light wave in a thin-film waveguide depends on the thickness of the film and the mode of propagation. This provides new possibilities in the design of experiments and devices.

Although this field of integrated optics or thin-film optoelectronics is still in a very elementary stage, in the past two years there have been a number of very important advances which generated substantial excitement. In what follows we will briefly enumerate some of these advances.

The first advance is the great improvement in the method of coupling a light wave propagating in free space into a well-defined mode of the thin-film guide. This was brought about through the invention of the prism–film coupler as described by Tien *et al*.[3] Other theories and experiments of the prism–film coupler have been given by Tien and Ulrich,[6] Ulrich,[7] Midwinter,[8] Harris *et al*.,[9] and Harris and Shubert.[10] Recently, a coupling efficiency of 88% was reported by Ulrich.[11] Another type of coupler, which uses a grating in place of the prism, was reported by Dakss *et al*.[4] and by Kogelnik and Sosnowski.[5] Recently an extremely simple technique of coupling which involves a tapered film edge was described by Tien and Martin.[12] A second advance which has occurred involves the materials of the films used for light-wave propagation. Initial experiments of Tien *et al*.[3] used sputtered ZnO and vacuum-evaporated ZnS films, which have large scattering losses. Then, excellent sputtered glass films were developed by Goell and Standley.[13] Other useful materials include polyurethane and polyester epoxy films developed by Harris *et al*.,[9] Ta_{2}O_{5} films by Hensler *et al*.,[14] and polymerized organosilicon films by Tien *et al*.[15] The organosilicon films made from vinyltrimethylsilane and hexamethyldisiloxane monomers have a loss of the order of 0.04 dB/cm as opposed to a number which was several hundred times larger in some of the earlier experiments. Other advances involve nonlinear and electrooptic effects observed in thin films and laser oscillation in iterated film structures. Tien *et al*.[16] have observed optical second harmonic generation using a ZnS film on a single-crystal ZnO substrate. The nonlinear interaction in this experiment was enhanced by the large concentration of light energy in the vicinity of the film. Kuhn *et al*.[17] succeeded in deflection of an optical guided wave in a thin film by its interaction with a surface acoustic wave. Hall *et al*.[18] have shown that it is possible to control the intensity of the light wave which propagates through a thin film such as a GaAs depletion layer by applying an appropriate bias. The latter two experiments indicate the possibility of efficient modulation of light in thin-film waveguides. Finally, laser oscillation was reported by Kogelnik and Shank[19] in a thin-film grating structure which is doped with a dye of Rhodamine 6G.

The basic problem considered here is simply a dielectric or semiconductor film that is deposited on a substrate. When a light beam in space is fed into a film, the light beam adapts itself so that it is confined within the thickness of the film. However, the transverse dimension of the light beam is not restricted. The film can thus be considered as a slab waveguide. Mathematical solutions of the waveguide show many possible modes of light-wave propagation. Within the plane of the film, the light wave in ay of these waveguide modes is allowed to propagate in any direction and can be reflected or refracted at any given boundary. The problem can be thought of as two-dimensional optics. Moreover, the light wave can be made to interact with the material of the film or of the substrate, or with externally applied electric or microwave fields so that certain functions such as modulation of light, parametric interaction, etc., can be achieved. Integrated optics is therefore an interdisciplinary science; it involves materials, film fabrication, electronics, and physical optics. In this paper, we select a few topics that are essential to the understanding of this new and complex field and discuss them in sufficient detail, hoping that the paper can serve as an introductory review for newcomers as well as a useful reference for those already in the field. The specific topics chosen are waveguide and radiation modes, the light-wave couplers, materials and losses of thin-film waveguides, and nonlinear interactions among optical guided waves. We will not discuss passive optical circuits and their fabrication, which have been reviewed in several excellent papers.[1],[2],[20]

## II. Waveguide and Radiation Modes

The film considered here has a thickness on the order of 1 *μ* or less; it is so thin that it has to be supported by a substrate. We thus consider three media: a film, an air space above, and a substrate below. As shown in Figure. 1, the thickness of the film is in the *X*–*Y* plane. For a thin film to support propagating modes and to act as a dielectric waveguide for the light waves, the refractive index of the film *n*_{1} must be larger than that of the substrate *n*_{0} and naturally that of the air space above *n*_{2}. A typical experiment is shown in Fig. 2. Here the entire surface of a 7.6-cm by 2.5-cm microscope glass slide is coated with a layer of an organic film made from a vinyltrimethylsilane monomer by gas discharge. At a wavelength of 6328 Å of the helium–neon laser, the refractive index of the film is 1.5301, which is larger than that of the glass (1.5125) and also that of the air (1.00). A light beam was fed into the film at the left side of the figure. It propagated through the entire length of the film and then radiated into the free space at the right side of the film. To show that the light wave was truly propagating inside the film, we scratched the film as shown in Fig. 3. The light beam then stopped at the scratched point, which radiated brightly as an antenna. Mathematically, the problem involves a solution of the Maxwell equations that matches the boundary conditions at the film–substrate and film–air interfaces. The solutions indicate three possible modes of propagation. The light wave can be bound and guided by the film as the *waveguide modes*. It can radiate from the film into both of the air and substrate spaces as the *air modes*, or it can radiate into the substrate only as the *substrate modes*. The air and substrate modes are the radiation modes discussed by Marcuse.[21] The modes described above can be explained simply and elegantly by the Snell law of refraction and the related total internal reflection phenomenon in optics.

Let (Fig. 4a) *n*_{0}, *n*_{1}, and *n*_{2} be the refractive indices and *θ*_{0}, *θ*_{1}, *θ*_{2} be the angles measured between the light paths and the normals of the interfaces in the substrate, film, and air, respectively. Here *n*_{1} > *n*_{0} > *n*_{2}. We have then from the Snell law

and

Let us increase *θ*_{1} gradually from 0. When *θ*_{1} is small, a light wave, for example, starts from the air space above the film, can be refracted into the film, and is then refracted again into the substrate (Fig. 4a). In this case, the waves propagate freely in all the three media—air, film, and substrate—and they are the radiation fields that fill all the three spaces (air modes). Next, as *θ*_{1} is increased to a value larger than the critical angle sin^{−1}(*n*_{2}/*n*_{1}) of the film–air interface as shown Fig. 4(b), the impossible condition incurred in Eq. (1), sin*θ*_{2} > 1, indicates that the light wave is totally reflected at the film–air boundary. Now the wave can no longer propagate freely in the air space. We thus describe a solution that the light energy in the film radiates into the substrate only (substrate modes). Finally, when *θ*_{1} is larger than the critical angle sin^{−1}(*n*_{0}/*n*_{1}) of the film–substrate interface, the light wave as shown in Fig. 4(c) is totally reflected at both the upper and lower surfaces of the film. The energy flow is then confined within the film; that is to be expected in the waveguide modes.

It is interesting to note that in the waveguide modes, the light wave in the film follows a zigzag path (Fig. 4c). The light energy is trapped in the film as the wave is totally reflected back and forth between the two film surfaces. We can represent this zigzag wave motion by two wave vectors *A*_{1} and *B*_{1} in Fig. 5(a). We then divide the wave vectors into the vertical and horizontal components in Fig. 5(b). The horizontal components of wave vectors *A*_{1} and *B*_{1} are equal, indicating that the waves propagate with a constant speed in a direction parallel to the film. The vertical component of the wave vector *A*_{1} represents an upward-traveling wave; that of the wave vector *B*_{1}, a downward-traveling wave. When the upward- and downward-traveling waves are superposed, they form a standing wave field pattern across the thickness of the film. By changing *θ*_{1}, we change the direction of the wave vectors *A*_{1} and *B*_{1} and thus their horizontal and vertical components. Consequently, we change the wave velocity parallel to the film as well as the standing wave field pattern across the film.

Since we discuss here a planar geometry, the waves described above are plane waves. They are TE waves if they contain the field components *E** _{y}*,

*H*

*, and*

_{z}*H*

*; they are TM waves if they contain the field components*

_{x}*H*

*,*

_{y}*E*

*and*

_{z}*E*

*. Here*

_{x}*x*is the direction of the wave propagation parallel to the film. The wave vectors

*A*

_{1}and

*B*

_{1}discussed above have thus a magnitude

*kn*

_{1}, where

*k*=

*ω*/

*c*and

*ω*and

*c*are, respectively, the angular frequency of the light wave and the speed of light in vacuum. In the picture of the wave optics, the vectors

*A*

_{1}and

*B*

_{1}are the normals of the wavefronts, when an infinitely wide sheet of plane wave folds back and forth in a zigzag manner between the two film surfaces (Fig. 6a). Now consider an observer who moves with the wave in the direction parallel to the film. He does not see the horizontal components of the wave vectors. What he observes is a plane wave that folds upward and downward, one directly on top of the other as shown in Fig. 6(b). The condition, then, for all those multiple reflected waves to add in phase, as seen by this observer, is that the total phase change experienced by the plane wave for it to travel one round trip, up and down across the film, should be equal to 2

*mπ*, where

*m*is an integer. Otherwise, if after the first reflections from the upper and lower film surfaces, the phase of the reflected wave differs from the original wave by a small phase

*δ*, the phase differences after the second, third, …, reflections would be 2

*δ*,3

*δ*, …, and then the waves of progressively larger phase differences would add finally to zero. As shown in Fig. 5(b), the vertical components of the wave vectors

*A*

_{1}and

*B*

_{1}have a magnitude

*kn*

_{1}cos

*θ*

_{1}. The phase change for the plane wave to cross the thickness

*W*of the film twice (up and down) is then 2

*kn*

_{1}

*W*cos

*θ*

_{1}. In addition, the wave suffers a phase change of −2Φ

_{12}due to the total reflection at the upper film boundary and, similarly, a phase change of −2Φ

_{10}at the lower film boundary. Here, the phases −2Φ

_{12}and −2Φ

_{10}represent, in fact, the Goos-Haenchen shifts.[22] Consequently, in order for the waves in the film to interfere constructively, we have

which is the condition for the waveguide modes. Here *m* = 0, 1, 2, 3, …, is the order of the mode. According to Born and Wolf[23] on the theory of total reflection,

for the TE waves, and

for the TM waves.

It is clear that in spite of the zigzag wave motion described above, the wave in a waveguide mode appears to propagate in the horizontal direction only; the vertical part of the wave motion simply forms a standing wave between the two film surfaces. To avoid confusion, it is desirable to use *β* and *v* exclusively for the phase constant and the wave velocity parallel to the film. Thus,

Another quantity which will also be used frequently is the ratio *β*/*k*. As shown in Eqs. (6), it is the ratio of the speed of light in vacuum to the speed of wave propagation in the waveguide.

After substituting Eqs. (4) or (5) into Eq. (3), we find that both Eqs. (3) and (6) are transcendental equations. Fortunately, the transcendental functions involve *θ*_{1} only. For a given *n*_{0}, *n*_{1}, *n*_{2}, and *m* we may easily compute both *β*/*k* and *W* for a common *θ*_{1}, and then tabulate *β*/*k* and *W* by assigning different values for *θ*_{1}. The curves showing *W* vs *β*/*k* using *m* as the parameter are the mode characteristics of the waveguide. They will be shown later, for example, in Fig. 23.

To summarize, any radius of the quarter-circle shown in Fig. 7 represents a possible direction for the wave vector *B*_{1} described above, and *θ*_{1} is the incident angle measured between the wave vector and the vertical axis. The waveguide modes occur in the range sin^{−1} (*n*_{0}/*n*_{1}) < *θ*_{1} < *π*/2. Within this range of *θ*_{1}, there is a discrete set of the directions which satisfies the equation of the modes (3). Each direction corresponds to one waveguide mode of the film. The horizontal component of the wave vector, *kn*_{1} sin*θ*_{1}, determines the wave motion parallel to the film, while its vertical component, *kn*_{1} cos*θ*_{1}, determines the standing wave field pattern across the film. As shown in the left side of Fig. 7, when *m* = 0, the standing wave pattern has a form similar to a half-sine wave. When *m* = 1, it has a form similar to a full sine wave, and so on. The air and substrate modes occur in the range 0 < *θ*_{1} < sin^{−1}(*n*_{0}/*n*_{1}); they occupy the black region of the quarter-circle. As we vary *θ*_{1} continuously from 0 to sin^{−1}(*n*_{2}/*n*_{1}) for the air modes and sin^{−1}(*n*_{2}/*n*_{1}) to sin^{−1}(*n*_{0}/*n*_{1}) for the substrate modes, the corresponding *θ*_{0} and *θ*_{2} sweep through the entire space of the substrate and the air space. It is thus possible to express any radiation field by superposing waves of the air and substrate modes. What we have discussed here is therefore simply an expansion of the solution of the Maxwell equation into plane waves of all possible directions.

## III. Wave Equation and the Field Distribution

Having described the modes of light-wave propagation purely on an intuitive basis, we may now derive them mathematically. For simplification, assume the light wave in the film to be infinitely wide in the *Y* direction so that ∂/∂*y* = 0 (Fig. 1). Let *X* be the direction of the wave propagation parallel to the film. The Maxwell equations in *E** _{y}* for TE waves (or

*H*

*for TM waves) can be reduced to the wave equation below.*

_{y}where *n** _{j}* is the refractive index of the medium

*j*. The subscripts

*j*= 0,1, and 2 denote the substrate, the film, and the air space, respectively. A time dependence exp(−

*iωt*) is used in Eq. (6), where

*i*= √−1. The solution of the wave equation is in the form of exp(

*ik*

_{xj}*x*) exp(±

*ik*

_{zj}*z*), which may be substituted into Eq. (6) to obtain

The boundary conditions at the film–air and film–substrate interfaces demand a same wave motion parallel to the film in all the three media considered; we may thus put

All the fields thus vary in time and *x* according to the factor exp(−*iωt* + *iβx*). This common factor will be omitted in all the later expressions for simplification. Combining Eqs. (7) and (8), we obtain an important relation,

In the film, *k*_{x}_{1} and *k*_{z}_{1} are the horizontal and vertical components of the wave vector *A*_{1} or *B*_{1} discussed before. They are, respectively, *k*_{x}_{1} = *β* = *kn*_{1} sin*θ*_{1} and *k*_{z}_{1} = *kn*_{1} cos*θ*_{1}. In the waveguide modes, we find from Eq. (9) and from the condition sin^{−1}(*n*_{0}/*n*_{1}) < *θ*_{1} < *π*/2 that *kn*_{0} < *β* < *kn*_{1}, *k*_{z}_{1} is real, and *k*_{z}_{0} and *k*_{z}_{2} are imaginary. The field distribution in Fig. 8(a) is thus a standing wave in the film and exponential in the substrate and in the air space. Next, for the substrate modes, we find from Eq. (9) and from the condition sin^{−1}(*n*_{2}/*n*_{1}) < *θ*_{1} < sin^{−1}(*n*_{0}/*n*_{1}) that *k*_{z}_{1} and *k*_{z}_{0} are real, but *k*_{z}_{2} is imaginary. The fields in this case are standing waves in the film and in the substrate, but exponential in the air space (Fig. 8b). Finally, for the air modes, we find that 0 < *θ*_{1} < sin^{−1}(*n*_{2}/*n*_{1}), and *k*_{z}_{0}, *k*_{z}_{1}, and *k*_{z}_{2} are all real. The fields in all the three media are now standing waves (Fig. 8c). It is convenient to denote *k** _{zj}* by

*b*

*when it is real and by*

_{j}*ip*

*when it is imaginary. For*

_{j}*n*

_{0}≠

*n*

_{2}, such as the case that is considered throughout this paper, the waveguide is asymmetric. We choose

*z*=

*W*

_{12}and

*z*= −

*W*

_{10}as the upper and lower film surfaces. The thickness of the film is then

*W*=

*W*

_{10}+

*W*

_{12}.

The field distributions are derived by choosing *z* = 0 at the position where *E** _{y}* is maximum for any waveguide, substrate, or even air mode, and

*E*

*= 0 for any odd air mode. It is important to note that these positions of*

_{y}*z*= 0 are different for different modes in an asymmetric waveguide. These choices are necessary in order to simplify mathematics so that we can visualize the field distributions of various modes easily. To avoid confusion, we consider below

*E*

*of a TE wave only.*

_{y}For the waveguide modes, as mentioned earlier, the wave suffers a phase change of −2Φ_{12} at the upper film surface, and a phase change of −2Φ_{10} at the lower film surface because of the internal total reflections. The fields at the two film surfaces must therefore be ±*A* cosΦ_{12} and ±*A* cosΦ_{10}, respectively, where *A* is a constant. Let the field at *z* = 0 be a maximum value, *A*. Then, we choose *k*_{z}_{1}*W*_{12} (or *b*_{1}*W*_{12}) = Φ_{12} so that the field at the upper film surface, *z* = *W*_{12}, can be *A* cosΦ_{12}. Similarly we choose *b*_{1}*W*_{10} = Φ_{10} + *mπ* so that the field at the lower film surface, *z* = −*W*_{10}, can be *A* cosΦ_{10} if *m* = even and −*A* cosΦ_{10} if *m* = odd as shown in Fig. 8(a). These choices give *b*_{1}*W* = *b*_{1}*W*_{10} + *b*_{1}*W*_{12} = Φ_{12} + Φ_{10} + *mπ*, which satisfies Eq. (3). The boundary conditions require *E** _{y}* and ∂

*E*

*/∂*

_{y}*to be continuous at the two interfaces. We have, therefore,*

_{z}in the air space and

in the substrate.

For the substrate modes, we again assume a maximum field *A* at *z* = 0 and choose *b*_{1}*W*_{12} = Φ_{12} (Fig. 8b). The field at *z* = *W*_{12} is still *A* cosΦ_{12} and that in the air space is still *A* cosΦ_{12} exp[−*p*_{2}(*z* − *W*_{12})]. The field at the lower film surface is then *A* cos(*b*_{1}*W*_{10}) and that in the substrate is

For the air modes, the even and odd modes must be treated separately. For an asymmetric waveguide, we can choose the *z* = 0 plane anywhere between *z* = *W*_{12} and *z* = −*W*_{10}. However, once it is chosen, the same *z* = 0 plane should be used for all the air modes. For the even modes, the field is a maximum at *z* = 0 and the fields at the two film surfaces are *A* cos*b*_{1}*W*_{12} and *A* cos*b*_{1}*W*_{10}, respectively (Fig. 8c). The boundary conditions require the fields in the substrate and in the air space in the form

where *j* = 0 and 2. For the odd modes the field is zero at *z* = 0 and is *A* sin(*b*_{1}*W*_{12}) and −A sin(*b*_{1}*W*_{10}) at the film surfaces. The fields in the substrate and air space are then

where the plus sign is for *j* = 2 and the minus sign is for *j* = 0. The results discussed above are summarized in Table I.

Mathematically, the field distributions described above are identical to those of the problem of a square potential well in quantum mechanics. Here the air space and the substrate are the potential barriers. We divide the wave energy here into the horizontal and vertical components, keeping the total energy constant. It is the vertical component of the wave energy that negotiates the potential barriers mentioned above. The wave vector represents the momentum and its square, the wave energy. Within the interval *β* = *kn*_{1} and *β* = *kn*_{0}, because of the large horizontal component of the wave vector *β*, the vertical component of the energy is small enough so that the wave, or the particle, is trapped in the potential well. The mode spectrum or the energy level is thus discrete (waveguide modes). As the horizontal component of the momentum is reduced to a value *β* < *kn*_{0}, the vertical component of the wave energy is large enough to overcome the lower potential barrier. The wave function spills over the entire substrate space and we enter into the region of the substrate modes. The mode spectrum or the energy level is now continuous. As we increase further the vertical component of the wave energy by reducing *β* below *kn*_{2}, the wave can spill over the upper and the lower barriers. The mode spectrum remains continuous a ad it belongs to the air modes.

## IV. Light-Wave Couplers

The development of the light-wave couplers in the past two years is an important step forward in thin-film optoelectronics. We can now couple a laser beam efficiently into and out of any thin-film structure and can excite there any single mode of light-wave propagation. In both the prism–film and grating couplers, we feed a light beam into a film through a broad surface of the film and thus avoid the difficult problem of focusing a light beam through a rough film edge. Since the film and the prism (or the grating) are coupled over a length of many optical wavelengths, we can imagine energy transfer taking place continuously between them as waves propagate over the coupled region. It is possible to discuss this type of distributed couplers by a unified theory. We can further show that these couplers have the same optimum coupling efficiency of about 81%, provided that both the coupling strength and the intensity of the incoming laser beam are uniformly distributed over the entire coupling length. An even better efficiency can be achieved by varying the coupling strength along the coupling length in a prescribed manner. By simplifying an earlier theory[6] and by using illustrative figures, we will describe below the principles of the couplers and derive their coupling efficiency in very simple terms.

Figure 9 shows a prism–film coupler. In order to excite all possible waveguide modes in the film, the refractive index of the prism should be larger than *n*_{3} that of the film *n*_{1}. An incoming laser beam enters the prism and is totally reflected at the base of the prism. Because of the total reflection, the field in the prism is a standing wave that continues into an exponentially decreasing function below the base of the prism. The part of the field that extends below the prism base is called the evanescent field, since it decreases rapidly away from the prism and does not represent a free radiation. If we represent the incoming wave in the prism by a wave vector *A*_{3} (Fig. 10), it has a magnitude *kn*_{3} and can be decomposed into a horizontal component *kn*_{3} sin*θ*_{3} and a vertical component *kn*_{3} cos*θ*_{3}. The boundary conditions of the electromagnetic fields at the prism base require that the fields below and above the prism base have the same horizontal wave motion. The evanescent field varies therefore as exp(*ikn*_{3}*x* sin*θ*_{3}) in *x*. Now we place the prism on top of a thin film, maintaining a small but uniform air gap between the base of the prism and the top surface of the film. For effective coupling, the spacing of the air gap is on the order of one-eighth to one-fourth of the vacuum optical wavelength. The evanescent field below the prism then penetrates into the film and excites a light wave into the film. We call this coupling process the optical tunneling. As discussed in Sec. I, the film has many waveguide modes. If the horizontal component of the wave vector *A*_{1} or *B*_{1} of one of the waveguide modes happens to be equal to that of the incoming light wave in the prism *kn*_{3} sin*θ*_{3}, the light wave in the prism is coupled exclusively to this waveguide mode and the laser beam is said to be in a synchronous direction. It is therefore possible to couple the light wave to any waveguide mode by simply choosing a proper direction *θ*_{3} for the incoming laser beam.

When the laser beam is in a synchronous direction, the waves in the prism and in the film have the same horizontal wave motion. The fields at the two opposite sides of the air gap are in phase at every point along *x*. As shown in Fig. 9, the field in the waveguide mode has an exponential tail extending upward above the film. The evanescent field of the prism is an exponential extending downward below the prism. These two exponential tails overlap in the air gap. The parts of the fields that overlap are common to the prism and the film and constitute the coupling between them.

Let *a*_{3} and *b*_{3} be the field amplitudes of the incoming and reflected waves in the prism and let *a*_{1} and *b*_{1} be the field amplitudes of the zigzag waves in the waveguide mode of the film (Fig. 10). The *a*_{1} and *b*_{1} waves are represented by the wave vectors *A*_{1} and *B*_{1} in the earlier discussion, and *a*_{1} may be considered as the reflection of the *b*_{1} wave, or vice versa, so that |*a*_{1}| = |*b*_{1}| at any *x*. Let all the wave amplitudes be normalized such that *a*_{j}*a** _{j}** or

*b*

_{j}*b*

** is the Poynting vector in the direction normal to the film, where*

_{j}*j*= 1 or 3. Because of the coupling described above, the energy is continuously transferred from the prism to the film along the coupling length which starts from

*x*= 0. Since the Maxwell equations are linear in field amplitudes, we expect that

*a*

_{1}(or

*b*

_{1}) increases in

*x*according to

*a*

_{3}, or that

*da*

_{1}/

*dx*should be linearly proportional to

*a*

_{3}. On the other hand, as soon as the wave energy in the film builds up, it continuously leaks into the prism, since the energy transfer is possible in both ways between the prism and the film. We ought to expect, then, that

*da*

_{1}/

*dx*is also proportional (−

*a*

_{1}). We have thus,

where *T* and *S* are the coupling constants that depend on the geometrical configuration and the refractive indices of the media. Near *x* = 0, *a*_{1} is small and so is the term *Sa*_{1} in Eq. (10); *a*_{1} increases linearly from *x* = 0 according to *Ta*_{3}*x*. At a large *x*, *a*_{1} grows to an amplitude so that *Sa*_{1} approaches a value that nearly cancels the term *Ta*_{3} in Eq. (10); *da*_{1}/*dx* = 0 and *a*_{1} reaches a saturation. The wave amplitude in the film cannot therefore increase indefinitely by simply increasing the coupling length.

In Fig. 11(a), we assume that *a*_{3} is uniformly distributed between *x* = 0 and *x* = *l*, which are the left and right edges of the laser beam incident on the prism base. The amplitude *a*_{1} increases in *x* until the point *x* = *l*. Beyond that point, *a*_{3} = 0 in Eq. (10) and the equation indicates that *a*_{1} should decrease exponentially to zero according to exp[−*Sx*]. All the energy fed into the film between *x* = 0 and *l* is returned to the prism at *x* > *l* and therefore the net energy transfer from the prism to the film is zero. If the film is not perfect and scatters the light, a more complex phenomenon occurs. Since the incident laser beam is in the synchronous direction, the light energy is coupled into one of the waveguide modes of the film. However, the energy in the original waveguide mode can be rapidly scattered into other waveguide modes before it is coupled back to the prism. The returned light wave in the prism therefore consists of many waveguide modes; each of them appears in its own synchronous direction. We thus observe a series of bright lines at the right side of the prism. They are called the *m* lines.[3] For good film, the *m* lines are thin and weak.

In Fig. 11(b), we use a rectangular prism. Here the rectangular corner of the prism is placed at *x* = *l*. Contrary to the earlier case, here the coupling between the prism and the film no longer exists beyond *x* = *l*. The wave energy which is fed into the film between *x* = 0 and *l* is retained in the film as the wave continues to propagate beyond *x* = *l*. Therefore, for coupling light energy into or out of a film, we always use a rectangular prism and place the right edge of the laser beam as close as possible to the rectangular corner of the prism.

We notice that because of their directions, the *a*_{3} wave in the prism is coupled only with the *a*_{1} wave in the film and, similarly, the *b*_{1} wave is coupled only to the *b*_{3} wave. Of course, *a*_{1} and *b*_{1} must increase and decrease together in *x*, since each of them is the reflection of the other. To calculate the coupling efficiency, we consider first the prism-film coupler in Fig. 12, which is used as an output coupler, to couple light energy out of the film. There is a light wave propagating in the film, which is represented by the *a*_{1} and *b*_{1} waves in the film. There is no input laser beam and thus *a*_{3} = 0 everywhere. Beyond *x* = 0, the film is coupled to the prism; consequently, the amplitude of the *b*_{1} (or *a*_{1}) wave decreases as light energy in the film leaks into the prism. By replacing *a*_{1} by *b*_{1} and putting *a*_{3} = 0 in Eq. (10), we have

It is easy to see that the total power flow in the film, which is proportional to *b*_{1}(*x*)*b*_{1} *(*x*), must decrease as exp[−2*Sx*]. The power lost in the film in a small distance *dx* is then proportional to ∂[*b*_{1}(*x*)*b*_{1}*(*x*)]/∂*x* and it must reappear in the prism as the *b*_{3} wave. We have thus immediately

Both *b*_{1}(*x*) and *b*_{3}(*x*) are plotted in Fig. 12. We see from Eq. (11) that eventually all the power in the film will be transferred into the prism. An output coupler is always at perfect output coupler, provided that the coupling length is sufficiently long. Now the reciprocity of the linear optics indicates that we can reverse the process. Consequently, if we apply a laser beam in the prism in the direction opposite to that of the *b*_{3} wave discussed above, and if it has an amplitude distribution exactly as *b*_{3}(*x*) in Fig. 12, all the applied laser energy should then enter into the film as to be expected in a perfect input coupler. On the other hand, if we apply a laser beam which is uniform over the cross section as shown in *B*_{3}(*x*) in Fig. 12, we expect that a part of *B*_{3}(*x*) which matches *b*_{3}(*x*) is accepted into the film and the rest is reflected at the prism base. We can therefore define an overlap integral[11]

which specifies the correlation between the amplitude distribution of the input laser beam and that required for a perfect coupler. In fact, *η* in Eq. (13) is the coupling efficiency. Since here *B*_{3}(*x*) is constant between *x* = 0 and *l*, and *b*_{3}(*x*) is exponential beyond *x* = 0, the integral in Eq. (13) can easily be performed. We have

which is identical to the formula given in the earlier papers.[6],[7] By maximizing the expression (14) with respect to *Sl*, we find the optimum coupling length *Sl* = 1.25, and the optimum coupling efficiency *η* = 81%. We notice that all the properties of the coupler depend on the parameter *S* which can be computed from the geometrical configuration and the refractive indices of the coupler, or it can be determined experimentally. A detailed calculation[6] shows

where
${p}_{2}={[{\beta}^{2}-{(k{n}_{2})}^{2}]}^{{\scriptstyle \frac{1}{2}}}$; *d* is the spacing of the air gap between the prism and the film, *W*_{eff} is the effective thickness of the film, and Φ_{32} may be obtained from Eq. (4) or (5) by replacing the subscript 1 by 3. The effective thickness *W*_{eff} will be discussed later in Sec. VII. The use of *W*_{eff} instead of the actual thickness of the film *W* is due to the Goos-Haenchen shifts.[24] The coupling efficiency does not differ significantly if a gaussian input field distribution is used.

From the above discussion, we realize that when the prism–film coupler is used as an output coupler, it can easily be made 100% efficient. The light which is not coupled out, say up to the point *x* = *x** _{a}*, remains in the film. It thus can always be coupled out at

*x*>

*x*

*, provided that the coupling length is sufficiently long. In contrast, an input coupler is 100% efficient only when the input light is properly distributed along the coupling gap, since the uncoupled light is immediately lost upon being reflected at the prism base.*

_{a}The field distribution of a perfect coupler, *b*_{3}(*x*), in Fig. 12 tends to be more uniform and thus matches better the uniform input field distribution, if we can decrease the coupling strength at *x* = 0 and increase it at *x* = *l* (refer to Fig. 12) by properly varying the gap spacing between film and prism. We accomplish this in the following way.

Figure 13 shows a prism–film coupler which is mounted on a turntable so that a laser beam can enter the prism at any angle. The film is coated on a glass slide and pressed against the base of the prism by a knife edge, while the dust particles between the prism and the film act as the spacers. The pressure point is about 1 mm or less away from the rectangular corner of the prism. The pressure applied to the back side of the glass slide actually bends the slide slightly so that the gap between the prism base and the film is smaller at the pressure point and larger at the corner of the prism, as shown in Fig. 14. This provides a stronger coupling at *x* = *l* and weaker coupling at *x* = 0 (Fig. 12) and between these two points the input laser beam is located. We thus approach the ideal condition mentioned earlier. In practice, it is easy to obtain a coupling efficiency of about 60%. Beyond that percentage, one needs a prism having a sharp rectangular corner, since any imperfection in the corner would radiate light energy and thus limit the attainable coupling efficiency. Figure 15 shows a laser beam that enters an organic film through the prism–film coupler at the right. The light beam propagated through the film and then was taken out of the film by another prism–film coupler at the left.

The prism–film coupler serves many useful functions in an experiment. By correlating the measured values of the synchronous directions with a theoretical calculation on the waveguide modes, one can independently determine the refractive index and the thickness of the film.[3] The prism–film coupler has been extensively used to determine the refractive index of organosilicon films.[15] The accuracy obtained is within 1 part in 1000 for the refractive index and 1% for the thickness. We often use a rutile prism for semiconductor films and a glass prism for organic and glass films.

Figure 16 shows a grating coupler.[4],[5] A phase grating made of photoresist or dichromated gelatin is fabricated by holographic technique on a thin film. A laser beam incident on the phase grating at an angle *θ* has a phase variation in the *x* direction according to exp[*i*(2*π*/λ_{0}) (sin*θ*)*x*], where λ_{0} is the vacuum laser wavelength. As the beam passes through the grating, it, obtains an additional spatial phase modulation ΔΦ sin(2*πx*/*d*), where ΔΦ is the amplitude of the spatial phase modulation caused by the grating and is sometimes called the phase depth of the grating; *d* is the periodicity of the grating. The light wave reaching to the top surface of the film contains many fourier components, exp{*i*(2*π*/λ_{0}) [sin*θ* + *m*(2*π*/*d*)]*x*}, where *m* is an integer. If one of these components matches the wave motion of one of the waveguide modes of the film, the light beam is exclusively coupled to this mode and the light energy is fed into the film. We can analyze the grating coupler by the same method used previously for the prism–film coupler. The only difference is the calculation of the parameter *S* which, in practice, is determined experimentally anyway.

The tapered-film coupler[12] is operated on an entirely different principle. It utilizes the cutoff property of an asymmetric waveguide. We remember in the discussion of the waveguide modes that the thickness of the film *W* can be divided into two parts *W*_{12} and *W*_{10} and that they are proportional to Φ_{12} and (Φ_{10} + *mπ*), respectively. As *β* varies from *kn*_{1} to *kn*_{0}, both Φ_{12} and Φ_{10} decrease. At the cutoff point of the waveguide modes, *β* = *kn*_{0}, Φ_{10} = 0, and *W* is minimum. Therefore, there is a minimum thickness of the film, less than which a waveguide mode of the order *m* cannot propagate. Now consider, in Fig. 17(a), a film that is deposited on a substrate and is tapered to nothing in a distance between *x* = *x** _{a}* and

*x*=

*x*

*, typically in the order of 10 to 100 vacuum wavelengths. The film is in the*

_{b}*X*–

*Y*plane and the tapered edge is parallel to the

*Y*axis. Consider a light wave in a waveguide mode of the film propagating toward the tapered edge in the direction normal to it. Since the thickness of the film decreases continuously in the tapered region, the waveguide mode is cut off at

*x*=

*x*

*. A detailed calculation shows that within a distance of about eight vacuum wavelengths in the vicinity of*

_{c}*x*

*, the waveguide mode is gradually converted into the substrate modes and the light wave reappears in the substrate as the radiation field. The far-field pattern of the radiation field shows that more than 80% of its energy is concentrated within an angle of 15° below the film–substrate interface. We can understand the problem better by considering again the ray optics. Figure 17(b) shows a light wave that propagates in a zigzag path from the left side of the film toward the tapered film edge. As it enters into the tapered region, the angle between the light path and the*

_{c}*Z*axis becomes smaller and smaller, and eventually, near

*x*=

*x*

*, the angle becomes smaller than the critical angle of the film–substrate interface. The light beam is then refracted into the substrate. In the experiments,[12] it is very easy to couple all the light energy out of a film through the tapered film edge. By reversing the process, we can also feed a light beam into the film by focusing it on the tapered edge through the substrate. The tapered-film coupler is simply the film itself and is particularly useful to the study of the semiconductor epitax layers. Here the refractive index of the film such as the GaAs layer is so large that it becomes difficult to find a prism of a higher refractive index that is also transparent to the radiation used in the experiment.*

_{c}## V. Materials and Losses

Sputtered ZnO films were first used in the light-guiding experiments.[3] They were deposited on a heated glass substrate in an argon–oxygen atmosphere. The method of deposition was developed previously by Foster *et al*.[25],[26] for ultrasonic transducers. The films discussed below were grown at a substrate temperature of 400°C. They have very high resistivities. After heat treatment in vacuum, nitrogen, or hydrogen, their resistivities can be reduced from 10^{6} to 10^{−2} Ω-cm. The largest mobility measured is 40 cm^{2}/V-sec as compared with that of bulk 200 cm^{2}/V-sec. The films have the hexagonal or Wurzite crystal structure with the *c* axis normal to the surface of the film and the (0002) planes parallel to it. As determined from x-ray diffraction by using a Debye-Scherrer camera, the *c* axis of the crystallites is oriented within 5° from the normal of the film. The refractive index of the film is 1.973 ± 0.001 at 6328 Å of the helium–neon laser wavelength as compared with the value 1.988 of bulk ZnO.

It was indeed a surprise when we found that those seemingly perfect films had a loss of more than 60 dB/cm in the light propagation experiment.[3] The excessive loss in the film was not understood until we took electron micrographs of the films. Figure 18(a) shows the surface profile of a ZnO film 1.5 *μ* thick. This electron micrograph was taken from a platinum-shadowed carbon replica. It can be seen that the average grain size is on the order of 0.5 *μ*. This grain size is comparable to the optical wavelength used in the experiment and thus causes excessive scattering to the propagating light wave in the film. The film surface (Fig. 18a) has an irregular profile; the peak-to-peak surface roughness is estimated on the order of 1000 Å. It is interesting to note that it was possible to polish the surface of the film by lapping it with chromium oxide dispersed in water. Figure 18(b) shows the surface of the ZnO film after being lapped. We estimated that the residual roughness was reduced to less than 100 Å. After being polished, the films still have a loss of more than 20 dB/cm.

To avoid scattering caused by large crystals, we started to evaporate ZnS films on glass substrates which were held at room temperature. They are polycrystal films with very small crystal sites. Although we have significantly reduced the scattering loss, unfortunately additional loss was found due to the absorption arising from the long tail of the fundamental band gap. The refractive indices of the film are 2.404, 2.342, and 2.289 at the wavelengths 5322.5 Å, 6328 Å, and 1.0645 Å, respectively. The loss measured at 6328-Å. wavelength is on the order of 5 dB/cm.

Now, trying to avoid both the scattering and absorption losses, we chose Ta_{2}O_{5}, which has a larger energy gap, 4.6 eV. The Ta_{2}O_{5} films[14] were prepared by first sputtering high purity tantalum in an argon atmosphere. The *β*-tantalum thus deposited was then heated in pure oxygen at 500°C until completely converted to Ta_{2}O_{5}. Debye-Scherrer x-ray diffraction patterns show the films to be amorphous. In spite of many visible defects, the same conclusion can be drawn from the electron micrograph in Fig. 19. The refractive indices of the film are 2.2136, 2.2423, and 2.2767 at the wavelengths 6328 Å, 5145 Å, and 4880 Å, respectively. At the above wavelengths, the losses measured are 0.9 dB/cm for the red, 2.5 dB/cm for the green, and 4.1 dB/cm for the blue.

As shown in Fig. 20, all the low-loss films presently used in the light-guide experiments are amorphous. This includes sputtered glass,[13] polyurethane and polyester epoxy,[9] and organic polymer[15] films. All of these have a loss less than 1 dB/cm. To the author’s knowledge, large transparent single-crystal semiconductor films other than epitax layers are not available at the present, although they are essential to the development of thin-film devices.

## VI. Simple Theory of the Surface Scattering

The losses of the films quoted above are the losses of the *m* = 0 waveguide mode. In fact, the loss increases rapidly in the higher order modes. The circles in Fig. 21 are the measurement made in a Ta_{2}O_{5} film by Hensler *et al*.[14] for a TE wave at 6328-Å light wavelength. The data show that the loss expressed in decibels per cm (or intensity attenuation in cm^{−1}) for the *m* = 3 mode is as much as fourteen times that of the *m* = 0 mode. In this figure, the losses of different modes are plotted vs *β*/*k*, where *β* is the horizontal component of the wave vector *A*_{1} or *B*_{1} as discussed at the end of Sec. II. It was pointed out earlier that the waveguide modes occur between *β*/*k* = *n*_{0} and *n*_{1} and that the value of *β*/*k* associated with each mode depends on the thickness of the film and the refractive indices of the film and substrate. It is evident in Fig. 21 that the loss for the *m* = 0 mode depends on the value of *β*/*k;* it represents neither volume loss of the material nor scattering loss caused by surface roughness. Therefore, it seems that there must be a better way to define the loss in a film other than that of a particular waveguide mode.

The losses in Fig. 21 were measured by a system of lenses, filter, slit, and detector which collected the light scattered from a small area of the film into a detector through an adjustable slit. By moving the detector system away from the input coupler but keeping it along a path parallel to the light beam in the film, we measured a decrease of scattered light along the light path, which should represent the intensity attenuation of the light wave propagating inside the film. The losses measured therefore included the volume absorption and scattering as well as the surface scattering. The same method of measurement was used by Goell and Standley[13] for their glass films.

Another method which was used extensively to measure losses in organosilicon films is the transmission measurement.[15] It involved two prism–film couplers in an arrangement similar to that shown in Fig. 15. One prism–film coupler excited a light streak in the film and a second coupler, several centimeters from the first, coupled the light wave out of the film. The efficiency of the input coupler was not measured, but it remained intact during the entire experiment. We accomplished this by monitoring the light scattered from a small section of the streak near the input end so that any change of the input conditions could be detected and corrected. The efficiency of the output coupler was always adjusted to be 100%. We mentioned earlier in Sec. IV that the output coupler can easily be made 100% efficient. In our experiment, the output coupler was applied at different points along the light streak. At each point the coupling was adjusted until the streak disappeared completely beyond the coupling point. The light emerging from the output prism was then detected. The measurements thus obtained at different points along the streak were used to evaluate the loss of the film.

It is possible to estimate the loss of a film based on the sensitivity of the eye. The sensitivity of the eye covers a range of about 27 dB. Thus if the length of the light streak as observed by the naked eye is *x* cm, the loss should be 27/*x* dB/cm.

The surface scattering of a symmetric slab waveguide has been calculated by Marcuse[21] based on the radiation modes. His results after certain approximations and evaluated to the limit of long correlation length agree with the simple theory that we will develop below. For an asymmetric waveguide (*n*_{2} ≠ *n*_{0}) which is considered in this paper, the two surfaces of the film scatter differently. Since it is practically impossible to measure the scattering losses of the two surfaces separately, we hope to develop a crude theory in which we can lump all the surface properties of the film into a single parameter. We wish further to use this parameter to calculate the losses of different waveguide modes and compare them with the measurement. Of course, the theory that we will discuss is very crude. However, it establishes a guideline by which we can gain insight into this complex problem.

Returning to Fig. 21, first, we must separate the volume loss from surface scattering. As *β*/*k* approaches n, the fields at two film surfaces vanish. We should then expect surface scattering to vanish and the residue loss at *β*/*k* = *n*_{1} to be volume loss only. At the other values of *β*/*k*, the volume loss should be proportional to the length of the zigzag path. We showed earlier that the light wave, in the waveguide mode, is a plane wave which propagates along a zigzag path and a zigzag path is considerably longer than the actual length of the film. We have ignored here, for simplification, the fields extending outside the film. According to the above argument, the volume loss is proportional to (sin*θ*_{1})^{−1} or (*β*/*kn*_{1})^{−1}. It is plotted as the dashed line in Fig. 21. We find that the volume loss of the *m* = 3 mode is merely 30% larger than that of the *m* = 0 mode as compared with a factor of 14 in the total losses. Consequently, almost all the losses in the higher order modes and the large variation of the losses among different waveguide modes are due to surface scattering only.

To develop a crude theory for surface scattering, we resort to the hundred-year-old Rayleigh criterion.[27] Figure 22 shows a plane wave incident on the upper surface of the film. To cover a unit length of the film in the *x* direction, the plane wave has a width cos*θ*_{1} in the direction parallel to the wavefront. Considering a TE wave, the power carried by the incident beam is (*c*/8*π*)*n*_{1}*E*_{y}^{2} cos*θ*_{1}, in gaussian units, where *E** _{y}* is the field amplitude. Again, we are taking ∂/∂

*y*= 0 and considering a space of unit length in the

*y*direction. According to the Rayleigh criterion, the specularly reflected beam from the upper film surface has a power

We use the double subscripts 12 and 10 to denote the film–air and film–substance interfaces, respectively. Note that λ_{1} is the wavelength in the film. The surface scattering is usually characterized by two statistical quantities: the statistical variation of the surface about the mean and the correlation length of the surface variation. Here *σ*_{12} in Eq. (16) is the variance of the surface roughness. A recent calculation by Mlarcuse[21] shows that the Rayleigh criterion applies only to the case of long correlation length. Since scattering observed in our loss measurements was always dominated by forward scattering, the assumption of the long correlation length may be correct. The limitation of the expression exp[−(4*πσ*_{12} cos*θ*_{1}/λ_{1})^{2}] is also discussed by Beckmann and Spizzichino.[27] In spite of its shortcomings, the expression is widely used because of its simplicity. The power lost by surface scattering at two surfaces of the film is therefore

where we have assumed that loss per unit length of the film is small, and

It can be easily shown from the field distribution discussed in Sec. III that the total power flow in the film for any waveguide mode is

where *W* is the thickness of the film, and *p*_{10} and *p*_{12} have been defined earlier. Dividing Eq. (17) by Eq. (19), we obtain the power attenuation per unit length of the film

We can also express the attenuation in decibels per unit length after multiplying Eq. (20) by 4.343. It is the loss caused by surface scattering only. We have thus expressed the loss as a product of three independent factors. The first factor, *K*, depends solely on the surface properties of the film and is a dimensionless quantity. The second factor involves *θ*_{1} only and thus depends on the waveguide mode considered. The third factor is the reciprocal of the effective film thickness and shows explicitly that the loss is inversely proportional to the thickness, as is to be expected. From this theory, it appears that the single parameter *K* defines all the surface properties of the film. It is a dimensionless parameter and quantitatively it compares surface roughness with the optical wavelength.

The crosses in Fig. 21 are the results calculated from Eq. (20). The parameter *K* = 1.27 × 10^{−2} is evaluated so that the loss of the *m* = 3 mode computed by the theory and that obtained by the measurement coincide. For this Ta_{2}O_{5} film, the agreement between the theory and measurement is excellent. For some other films, the agreement is only moderate. The details will be discussed elsewhere.

The large scattering loss observed in the experiment, particularly that of the higher order waveguide modes, is to be expected. Again, we return to the picture of the zigzag waves. For the Ta_{2}O_{5} film considered in Fig. 21, the light wave is reflected back and forth between the two film surfaces about 2000 times in a length of 1 cm of the film for the *m* = 0 mode and about 10,000 times for the *m* = 3 mode. Thus, for this film to have a scattering loss of 1 dB/cm in the *m* = 0 mode, the loss per reflection should be 1.5 × 10^{−4}, which should be compared with the loss per reflection as large as 1 × 10^{−3} for the better dielectric mirrors used in lasers.

## VII. Field Concentration and Mode Characteristics

Before discussing nonlinear interactions in thin films, it may be necessary to learn more about mode characteristics. Also included in this section are the discussion on the cutoff of the waveguide mode, field concentration in a film, and finally the concept of the effective film thickness. Throughout this section, we will use a ZnS film on a glass substrate as the example. The light wavelength is 1.06 *μ*m of the YAG:Nd laser. At this wavelength, *n*_{1} = 2.2899 and *n*_{0} = 1.5040.

We have shown at the end of Sec. II how to calculate *W* and *β*/*k* for a given *n*_{0}, *n*_{1}*n*_{2}, and *m*. The result of the calculation for both TE and TM waves is shown in Fig. 23. These *W* vs *β*/*k* curves are the mode characteristics of the waveguide. The ratio *β*/*k* is called the effective refractive index, since it measures the ratio of the speed of light in vacuum to that in the waveguide in the same way as the ordinary refractive index measures the ratio of the speed of light in vacuum to that in a dense medium. First, we notice *β*/*k* ranging from *n*_{0} to *n*_{1} for the waveguide modes. Let us concentrate on the *m* = 0 TE mode in Fig. 23. When *W* is large, the effective index (*β*/*k*) approaches the refractive index of the film *n*_{1}. To this limit, the film acts as a bulk medium and all the light energy is contained within the film. The fields therefore vanish at the two film surfaces. When *β*/*k* varies from *n*_{1} to *n*_{0}, *W* decreases continuously as the fields extend more and more outside the film. At *β*/*k* → *n*_{0}, the mode becomes cut off and *W* is the minimum thickness that can support this waveguide mode. At the cutoff, the waveguide mode is turned into a substrate mode as the fields extend infinitely into the substrate.

Next, we notice that for the same *β*/*k*, *W’s* of different TE or TM modes are equally spaced. That is, the difference between *W*(TE, *m* = 1) and *W*(TE, *m* = 0) is equal to the difference between *W*(TE, *m* = 2) and *W*(TE, *m* = 1), and so on. We also notice that for a given *m* and *β*/*k*, *W*(TM) is always larger than *W*(TE) simply because the TM wave has larger Φ’s [see Eqs. (4) and (5)]. The Φ’s of the TM wave increase with the ratio *n*_{1}/*n*_{0}.

In designing an experiment particularly for nonlinear interactions, we often want to calculate the wave velocity accurately. This is difficult in practice. The film may not be homogeneous, and the materials may change with the environment. We do not really know the rafractive indices exactly. Moreover, the film may not be uniform and there are difficulties in measuring the film thickness exactly. It is then important to know the wave velocity, or *β*/*k*, varies with small increments in *W*, *n*_{0}, *n*_{1} and *n*_{2}. For this purpose, have calculated [*dW*/*d*(*β*/*k*)], [*d*(*β*/*k*)/*dn*_{1}], [*d*(*β*/*k*)/*dn*_{0}], and [*d*(*β*/*k*)/*dn*_{2}]. They are given below for the TE wave only.

where *j* = 0 or 2. We have calculated the above expressions for the TE *m* = 0 waveguide mode. The results are shown in Fig. 24. In Fig. 24(a), we notice that *dW*/*d*(*β*/*k*) is large for *β*/*k* near *n*_{1} and also *n*_{0}. A large *dW*/*d*(*β*/*k*) means that the wave velocity is not sensitive to small variations in *W*. In Fig. 24(b), we find that *n*_{1} does not affect *β*/*k* very much for *β*/*k* → *n*_{0}, and similarly *n*_{0} does not affect *β*/*k* very much for *β*/*k* → *n*_{1}, as is to be expected. The influence of *n*_{2} on *β*/*k* is generally 10 times smaller than that exercised by *n*_{1} or *n*_{0}. In spite of its smallness, it is relatively easy to vary *n*_{2} in order to obtain a fine adjustment in the wave velocity, for example, by using a liquid of index of refraction to replace the air space on top of the film.

One can easily calculate the power carried by a TE waveguide mode based on field distribution discussed in Sec. III. We find

where *E** _{y}* is the field amplitude of the

*A*

_{1}or

*B*

_{1}wave in the film and

*d*is the width of the light wave in the

*y*direction. The equation is written in gaussian units. Because the field extends outside the film, [

*W*+ (1/

*p*

_{0}) + (1/

*p*

_{2})] is the effective thickness of the film,

*W*

_{eff}. For a given

*P*, the light intensity

*E*

_{y}^{2}is inversely proportional to

*W*

_{eff}instead of

*W*. Both

*W*and

*W*

_{eff}are plotted in Fig. 25 for the

*m*= 0 and m = 1 modes. For large concentration of light intensity inside the film, we prefer the

*m*= 0 mode. We see that even for the

*m*= 0 mode, we cannot increase light intensity indefinitely by simply reducing thickness of the film. When the film becomes too thin, the fields penetrate deep into the substrate and

*W*

_{eff}no longer decreases with

*W*. It is interesting to note that for the

*m*= 0 mode,

*W*

_{eff}approaches λ

_{1}/2 but is never smaller than it, where λ

_{1}is the optical wavelength in the film medium. The minimum

*W*

_{eff}of 0.458

*μ*m occurs at

*β*/

*k*→ 1.82, where the average power density and the maximum field amplitude in the film are, respectively, 21.8 MW/cm

^{2}and 6.78 × 10

^{4}V/cm. Here we assume that

*d*= 10

*μ*m and that 1 W of the laser power is being fed into the

*m*= 0 waveguide mode of the film.

## VIII. Phase-Match and Nonlinear Interactions Between Guided Waves

The advantages of performing nonlinear optics in a thin film are many. A thin film can concentrate laser energy for a long distance, whereas a focused gaussian beam diffracts rapidly away from the focused point. The phase velocity of a light wave in a waveguide mode depends on the thickness of the film and the mode of propagation. Thus, for example, by using different waveguide modes for the signal, idler, and pump waves in a parametric oscillator, we can obtain a phase match condition without relying upon the birefringence of the crystal. The crystals such as GaAs, GaP, ZnS, ZnTe, etc., which have large nonlinear coefficients but little birefringences can then be used for nonlinear experiments. The film and the substrate can be immersed in a liquid and the phase-match condition-can be varied by varying the refractive index of the liquid. Finally, the nonlinear interaction can take place in the film, in the substrate, or both. All these advantages provide many alternatives to the design of the experiment. Of course, the primary purpose of developing nonlinear devices in thin-film form is that they can be used in integrated optical circuitry. In spite of these advantages, the development of thin-film parametric devices is handicapped by the lack of single-crystal films and by the difficulties in obtaining a long coherence length. We shall illustrate this fully by considering the problem of optical second harmonic generation (SHG).

In a parametric oscillator, the frequency of the oscillation adjusts itself so that the phase velocities of the signal, idler, and pump are matched. In the SHG, however, we do not have this kind of flexibility, and the film must have exactly the thickness required for the fundamental and the harmonic waves to propagate at the same wave velocity. Any nonhomogeneity in refractive indices and nonuniformity in thickness would reduce the efficiency of the nonlinear interaction. The phase-match condition in nonlinear optics involves only the wave velocities parallel to the film. The phase-match condition of SHG is (*β*/*k*)^{(1)} = (*β*/*k*)^{(2)}, where the superscripts (1) and (2) denote the fundamental and the harmonic, respectively. Basically, if we plot *W* vs (*β*/*k*) for the fundamental and for the harmonic, the crossing point of the curves is the phase-match condition. The problem becomes simpler if we follow the following simple rules:

- 1.
*dW*/*d*(*β*/*k*) for the fundamental and the harmonic are always positive and the*W*vs (*β*/*k*) curves of the fundamental can cross that of the harmonic only once or not at all. - 2. In general, when the
*W*vs (*β*/*k*) curves show that*W*^{(1)}>*W*^{(2)}near*β*/*k*=*n*_{0}, a phase-match condition can be obtained only when*n*_{1}^{(2)}<*n*_{1}^{(1)}. Similarly, when*W*^{(1)}<*W*^{(2)}near*β*/*k*=*n*_{0}, the phase-match can be obtained only when*n*_{1}^{(2)}>*n*_{1}^{(1)}.

We shall illustrate these rules in the following examples.

As the first example, we consider nonlinear interaction in the substrate. In this case, we use the electric fields that extend into the substrate. The efficiency of the interaction depends on the amount of the fundamental and the harmonic electric fields in the substrate as compared with their distributions over the film, substrate, and air space. In order to obtain a large efficiency, it is obvious that we must operate at *β*/*k* near *n*_{0}. Since the field distribution in the substrate of any waveguide mode is always exponential, it does not really matter if the fundamental and the harmonic are in the same or different waveguide modes. A good choice is *m* = 0 mode for the fundamental and *m* = 1 mode for the harmonic. In this case, birefringence of the crystal is not required. Both the fundamental and the harmonic can be TE or TM, or one of them TE and the other TM. For example, in Fig. 26, we consider a ZnS film on a single-crystal ZnO substrate. The refractive indices are *n*_{1}^{(1)} = 2.2899; *n*_{1}^{(2)} = 2.4038; *n*_{0}^{(1)} = 1.9562; *n*_{0}^{(2)} = 2.0521 for the wavelengths 1.064 *μ*m and 0.532 *μ*m, respectively. Here the *c* axis of ZnO is oriented in the *y* direction, and the refractive indices quoted are those of the extraordinary ray. To use nonlinear coefficient *d*_{33} of ZnO, both the fundamental and the harmonic must be TE waves. Phase-match condition is obtained at (*β*/*k*) = 2.0877 where the thickness of the film is 0.314 *μ*m.

A detailed calculation shows that at (*β*/*k*) = 2.0877, *d* [(*β*/*k*)^{(2)} − (*β*/*k*)^{(1)}]/*dW* = 0.444. For a coherence length of 1000 *μ*m, *d*[(*β*/*k*)^{(2)} − (*β*/*k*)^{(1)}] should be equal or less than 2.65 × 10^{−3}. We find immediately that the average thickness of the film should be kept within the limits that the average thickness of the film should be kept within the limits

It is not impossible to evaporate a ZnS film within a thickness tolerance of 0.006 *μ*m. Because of the small birefringence, the coherence length in a bulk ZnO is less than 2 *μ*m. Now we have obtained a coherence length of 1000 *μ*m by using the waveguide modes. We have thus improved the coherence length by a factor of 500, which is remarkable. On the other hand, a 1.064-*μ*m light wave can propagate at least a distance of 2.5 cm in ZnS film without suffering appreciable loss. Because of the phase-match problem we have used only a section of 1000 *μ*m of this 2.5-cm-long beam. The advantage of the thin-film waveguide is thus not fully utilized. We have used here the problem of SHG to illustrate the difficulties involved in the integrated optics. We can gain some advantages by using the waveguide principle, but in this and other problems, the full potential of the integrated optics cannot be realized without developing techniques to control the homogeneity and uniformity of the film.

Next, as the second example, we consider nonlinear interaction in film. Although one can use different waveguide modes for the fundamental and the harmonic mismatch between the distribution of nonlinear polarization and that of the fields would reduce the efficiency of interaction sharply. We thus prefer the use of TE and TM waves of the same mode order *m* for the fundamental and harmonic. A study of the mode characteristics shows that *W* of the harmonic is always smaller than that of the fundamental at a same value of *β*/*k* near *n*_{0}. According to our second rule given earlier, we cannot obtain a phase-match condition *unless the refractive index of the film at the harmonic frequency is less than that at the fundamental frequency*. Since most of nonlinear materials have a normal dispersion in the visible spectrum, to satisfy the above condition, we must choose a material of sufficient birefringence for the film. As an example, we consider in Fig. 27 a single-crystal LiNbO_{3} film on a quartz substrate. The *c* axis of LiNbO_{3} is normal to the film. A TE (*m* = 0) wave is used for the fundamental and a TM (*m* = 0) wave for the harmonic. The nonlinear coefficient used is the *d*_{31} of LiNbO_{3}. The refractive indices involved are than *n*_{1}^{(1)} = 2.365; *n*_{0}^{(1)} = 1.4614; *n*_{1}^{(2)} = 2.300; *n*_{0}^{(2)} = 1.4745. Again we consider SHG from 1.064 *μ*m to 0.532 *μ*m. We find from Fig. 27 that a phase-match condition is obtained at (*β*/*k*) = 2.2274 where *W* = 2.47 *μ*m. Unfortunately, the single-crystal LiNbO_{3} film is not available at present. The discussion is therefore academic. We could use the *d*_{31} nonlinear coefficient of oriented ZnO or CdS films that have been developed for ultrasonic transducers, but they are too lossy, as discussed in Sec. V.

From our earlier discussion on nonlinear interaction in the substrate, we realize that the requirement set by the phase-match condition for the thickness and uniformity of the film is very stringent. To circumvent this problem, a novel method has been used by Tien *et al*.[16] by generating second harmonic in the form of Cerenkov radiation. The nonlinear interaction by their method is not as efficient as that under the phase-match condition, but the interaction extends to the full length of the fundamental wave. As in the case discussed before, a polycrystal ZnS film on a single-crystal ZnO substrated was used and nonlinear interaction took place in the substrate. The *c* axis of the ZnO was oriented in the *y* direction, and both the fundamental and the harmonic waves were polarized parallel to the *c* axis.

The experimental arrangement is shown in Fig. 28. A light beam of a YAG: Nd laser at 1.06 *μ*m was fed into the *m* = 0 waveguide mode of the ZnS film as the fundamental wave. It propagated as exp [−*iω*^{(1)}*t* + *iβx*], where *n*_{0}^{(1)} < (*β*/*k*) < *n*_{1}^{(1)}. The fundamental wave excited a wave of second harmonic nonlinear polarization in the substrate via the *d*_{33} nonlinear coefficient of ZnO. Here the nonlinear polarization wave was a forced wave and thus it varied as exp [−*iω*^{(2)}*t* + 2*iβx*]. Because of the normal dispersion of ZnO, *n*_{0}^{(2)} > *n*_{0}^{(1)}. Therefore, by using a proper thickness *W* for the film, it was possible to obtain

Under this condition, the phase velocity of the nonlinear polarization *ω*^{(2)}/2*β* exceeds the phase velocity *c*/*n*_{0}^{(2)} of the free second harmonic radiation in the substrate. Consequently, Cerenkov radiation at the second harmonic frequency is emitted and it is emitted at a Cerenkov angle *α* where.

To review, the fundamental wave propagating in the ZnS film generates a sheet of nonlinear polarization wave in the substrate immediately below the film–substrate interface. The nonlinear polarization wave then generates a second harmonic radiation in the form of Cerenkov radiation that may be considered simply as a plane wave propagating in the substrate at an angle *α* below the interface. If the wave vector *C* represents this plane wave, its horizontal component is equal to the wave vector of the nonlinear polarization wave and is also twice the fundamental wave vector *β*. The process described above is illustrated in Fig. 29, in which any horizontal line drawn from a value of (*β*/*k*) at the middle column gives the corresponding Cerenkov angle a in the left column and the thickness of the film *W* in the right column. We notice that at the upper limit of the inequality (24), *β*/*k* → *n*_{0}^{(2)}, the Cerenkov angle vanishes, and the phase velocity of the nonlinear polarization wave is equal to that of the free wave in the substrate. At the lower limit of (24), *β*/*k* → *n*_{0}^{(1)}, the waveguide mode of the fundamental wave becomes cut off. Figure 30 is a photograph of the experiment. The bright star in the photograph is the Cerenkov radiation. Figure 31 shows the radiation as it emerged from the side surface of the ZnO crystal. The photograph was taken through a microscope by focusing it on the side surface of the crystal.

## IX. Conclusions

Most of the material presented in this paper is drawn from unpublished notes accumulated during the past two years. These notes were prepared for talks and lectures given on various occasions. Much of the time and effort has been spent in developing a method whereby one can visualize easily the waveguide and radiation modes without having to derive the Maxwell equations. A theory that consideres a zigzag plane wave has been developed for that purpose. We have used this theory in Sec. II to derive the mode equation. It was used again in Sec. IV for a unified theory of the prism and grating couplers, and in Sec. VI for a simple theory of surface scattering. This theory turned out to be most useful in the analysis of complex optical devices. For example, the prism–film coupler involves four coupled media: the prism, the air gap, the film, and the substrate. A direct solution of the Maxwell equations for four simultaneously coupled media is not simple. Here, using the theory of the zigzag wave, we only have to consider two sets of the waves, and each set consists of only two coupled waves.

We have not discussed in this paper the Goos-Haenchen shift, but it is included in the derivation of the mode equation by introducing the phase changes −2Φ_{10} and −2Φ_{12}. It is also implied in the expression for the power flow and in the coupling constant of the prism–film coupler by introducing the effective film thickness. A detailed study of the Goos-Haenchen shift should include the discussion of energy flow and is beyond the scope of this paper.

The experiments described in this paper are difficult; one has to learn how to control the small dimensions involved in these experiments. For example, the spacing of the air gap between the prism and the film can be determined by observing Newton’s rings near the pressure point. Usually, one should accurately measure the refractive indices of the prism and the substrate, tabulate the incident angles against the values of *β*/*k*, and compute the mode characteristics for each film–substrate combination.

The technology involved in integrated optics may be more difficult than that we can visualize today. Large single-crystal films are needed for the development of electrooptical and nonlinear thin-film devices. These single-crystal films, with the exception of epitax layers, simply do not exist today. Methods to control the uniformity and the thickness of the film within the accuracy of one or two atomic layers and techniques for the fabrication of structures 10^{4} times smaller than their microwave counterparts are also needed. We are still observing the *m* lines in our best films; this indicates that considerable energy stored in the film and in the substrate is not in the main mode of propagation, and it would eventually distort the signal carried by the integrated optical circuitry. These problems described above will continue to challange us for some time to come. Looking into the future, we expect the field of integrated optics to grow rapidly, simply because there are needs for optical systems in the electronics and communication industries and there are needs for integrated optics in optical systems.

## Figures and Table

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