Light-propagation and imaging in planar optical waveguides

This article has been downloaded from IOPscience. Please scroll down to see the full text article.

1975 Nouvelle Revue d'Optique 6 253

(http://iopscience.iop.org/0335-7368/6/5/302)

Download details:

IP Address: 221.6.159.251

The article was downloaded on 18/02/2011 at 13:58

Please note that terms and conditions apply.

View the table of contents for this issue, or go to the journal homepage for more

Home Search Collections Journals About Contact us My IOPscience

Nouv. Rev. Optigue, 1975, t. 6, no 5 , pp. 253-262

MAX-PLANCK-INSTITUT Fm FESTK~RPERFORSCHUNG, 7000 Stuttgart (R.F.A.)

LIGHT-PROPAGATION AND IMAGING IN PLANAR OPTICAL WAVEGUIDES (*)

by

R. ULRICH

KEY WORDS : Optical waveguide Image formation

Mors CLES : Guide d’onde optique Formation d’images

SUMMARY : Planar optical waveguides, permitting light-propaga- Propagation de la lumikre et imagerie tion in only two dimensions, can be used to form images of one- d m l e guide #on& plans optique. dimensional objects. In one arrangement, the paths of light-rays on the guide are determined by the distribution of the effective RESUME : Les guides d’ondes plans permettent a la lumiere de se refractive index over the regions of the guide, resulting in two- propager seulement dans 2 directions. 11s peuvent &re utilisks pour dimensional optical systems. Another possibility, existing in former des images d’objets a une dimension. Aprb avoir passe thick, multimode planar guides is the utilisation of self-focussing en revue les proprietes des guides d’ondes plans, on etudie plus par- or self-imaging properties. All these imaging methods are reviewed, ticulierement la formation des images d’objets dans lesquels I’infor- with particular emphasis on the most recent method of self-imaging. mation est repartie dans une direction normale au plan du guide.

INTRODUCTION

Planar optical guides are conceptually the simplest type of waveguide, and they have been the subject of a large number of publications [l-81. The present paper, opening the session on planar guides, will therefore discuss only very generally the planar waveguide as a two-dimensional propagation medium, comparing it with the unrestricted, three-dimensional propagation in the volume and with the one-dimensio- nal propagation along an optical fibre. Following a review of methods for manipulating light beams in planar guides, the bulk of the paper will deal

with a new form of imaging : the self-imaging effect existing in multimode planar guides. Theory and experiments of this effect are described, and some applications for integrated optics are proposed.

I. - THE ROLE OF PLANAR OPTICAL GUIDES

Comparing light propagation media of different dimensionality (fig. l), we recognise a unique feature of the planar optical guide : It combines the possibi- lities of free light propagation, typical for bulk optics, with the characteristics of guiding. Freely

W

[ f r e e propagation in x y directions 1 [confinement in z d i r e c t i o n ]

transmission 2-dim. l -d im. - of images

FIG. I . - Comparison of optical propagation media of different dimensionality.

(*) Invitedpaper presented at the Colloquium on (( the Optic of Guided Waves )), (Paris. April 8-1 1, 1975).

N o w Rev Opfrque, 1975, t 6, no 5 23

254 R. ULRICH Nouv. Rev. Optique, 1975, t. 6, no 5

propagating waves obey the laws of geometrical optics [9]. They permit the design of planar optical systems, using all the experience existing in ordinary, bulk optics. The absence of the third dimension of propagation restricts these systems, of course, to the processing of only one-dimensional images. The characteristic properties of the guide, on the other hand, add a new dimension to the possibilities of the design : the planar guide may be viewed as an artificial dielectric. whose (effective) index of refraction, bire- fringence, and dispersion can be adjusted freely within wide limits by the choice of the guide para- meters. This freedom permits new solutions to pro- blems of linear [lo] and non-linear optics [l 13.

Compared to ordinary, 3-dimensional propagation, it is another advantage of all optical guides, including the planar one, that they can maintain a high concen- tration of light energy over an extended length of propagation, because they prevent the spreading of light by diffraction. By employing a guide, therefore, the interaction volume can be considerably reduced in such active optical components as e.g. modulators based on electro-optic, acousto-optic, magneto-optic, and electro-absorptive effects [13]. The resulting reduction of the modulator drive power is by a factor of the order of (j./L)1’2 for the planar guide, and by (A/L) for the one-dimensional guide, where L denotes the length of the modulator. In practice, this factor may reach lo2 and lo4, respectively.

Finally, and perhaps most important for an eventual success in technical applications, is the possibility to fabricate two-dimensional and one-dimensional optical guides by similar thin-film and photolitho- graphic processes as are used so widely today in semiconductor technology. Based on this possibility is the concept of integrated optics [14, 151 : The construction of reliable, inexpensive, miniature optical circuits, performing passive and active functions in optical communications systems. Theoretically best suited for this purpose are, of course, one-dimensional optical guides, e.g. in the form of strip guides, because they should permit the highest packing density and give best performance in active optical components. Such’ one-dimensional guides have been fabricated and operated in a number of laboratories [16-191. In that fabrication, however, it is a severe problem to maintain the required submicroscopic edge tole- rances ( - 0.05 pm) over the necessary macroscopic lengths ( - 5 mm) of these devices [18].

For this reason, the majority of all experiments relating to integrated optics up to the present time have employed two-dimensional, planar guides. Their fabrication requires control of only one dimension (the film thickness), which is much less of a problem. By the same argument, then, it IS not unreasonable to expect that planar light guides will also play a significant role in future technical applications of guided wave optics. This expectation is supported by the successful demonstration of a number of very efficient integrated optics modulators, all based on Bragg-reflection in planar optical guides [20-221. Another micro-optical component depending on a two-dimensional guide is the Fourier-transform device (fig. 2) proposed by Shubert and Harris [3]. It is a

promising active element for real-time analog-optical data processing systems. Furthermore, two-dimen- sional planar waveguides would probably soon find significant applications in optical communication systems if a simple and (in both directions) efficient method would be found [I51 for coupling a multi- mode optical fibre to a thin-film waveguide. To accommodate all modes of the fibre, the beam in the thin-film guide would have a width at least of the order of 1 mm.

FIG. 2 . - Proposed real-time Fourier-transform device based on planar optical guide. M is an array of modulators, impressing the input function f ( y ) as a spatial modulation on the parallel beam. The intensity distribution received by the deiector array D in the focal line of lens L is closely related to the Fourier transform of f(y).

If, in any of these applications, the path of light in the plane of the guide is to be controlled, it can be done by elements based on geometrical optics, like mirrors, prisms, lenses, and gratings. This type of control is obviously required for beams in a simple planar optical guide. The same type of control is required, however, also in systems employing very wide strip guides or strip guides of medium widths operating near cutoff (rib guide [23]), because in those guides the minimum permissible bending radii are excessively large. Any directional control of the beam in the plane of these guides, therefore, has to be done by geometrical-optical means. This control is the subject of the next section.

11.-CONTROL OF BEAMS IN THE PLANE OF THE GUIDE

In the approximations of geometrical optics; the propagation of light in a given mode of a planar optical guide depends on the two-dimensional dis- tribution N(x , y ) of the (( effective )) index in a comple- tely analogous way as the ordinary light propagation in a bulk optical medium depends on the three- dimensional distribution n(x, y , z ) of the ordinary refractive index [lo]. In both cases, the light rays are straight lines in regions of uniform index, but they bend where the ray passes through a region of non-

Nouv. Rev. Optique, 1975, t. 6, no 5

uniform index. The amount of bending is governed, in both cases, essentially by Snell’s law. By arranging on a planar guide properly shaped regions of diffe- rent effective indices, it is possible, therefore, to control the propagation of light in the plane of the guide similarly as we do in bulk optics : a triangular region of modified effective index, e.g., deflects the beam like a prism, and a lens-shaped region (as in fig. 2) focuses or defocuses a bundle of parallel guided rays like a positive or negative lens, respectively, depend- ing on the curvatures of the lens boundaries and on the relative magnitudes of the effective indices [lo]. To minimise losses by reflection or scattering, it must be ensured at the boundaries that the light incident in the mode under consideration is coupled with high efficiency into the same mode on the other side of the boundary. This can usually be achieved by suitably tapering the transition from one region to the other one.

Thus, the basic elements available for directional control in planar guides are essentially those known from bulk optics : totally reflecting mirrors, prisms, lenses, and gratings. In contrast to bulk optics, howe- ver, in two-dimensional optics various methods exist by which the required changes of the effective index

R. ULRICH 255

FIG. 3 . - Methods of locally varying the effective index of a planar optical guide. (a) change of core material; (b) variation of guide thickness; (e) combination of (a) and (6). Figure 3d) : Variation

of optical path length by distortion of lens into third dimension.

FIG. 4. - Lens in planar optical guide (ZnSJilm on glass substrate). The guide thickness is approx. 2 400 A inside the lens region, and

700 A outside (focal length 2 mm). (After Ref. 10).

256 R. ULRICH Nouv. Rev. Optique, 1975, t. 6 , no 5

in a specified region can be realized (fig. 3). The effec- tive index depends jointly on the guide thickness and on the bulk indices of all materials from which the guide is combined [ l , 10, 241. Therefore, any one of these parameters can be used, singly or jointly with others, to adjust the effective index. This possibility was meant when the planar guide was called an (( artificial dielectric )).

The most direct method [3] to change the effective index would be to fabricate the waveguide inside the specified region from a different optical material than outside, equivalent to embedding a very thin lens into the guide (see fig. 3a). Another possibility is to vary simply the thickness of the guide [lo, 421 (see fig. 3b). For a given mode, a region of larger film thickness always has a higher effective index than a thinner region. Figure 4 shows, as an example, a planar guide with a lens [lo]. Inside the lens-shaped region the film is thicker than outside. A combination of the two methods of varying the effective index is to deposit [25, 261 a film of a second material, prefe- rably of a high index, in the specified region on the planar guide (see fig. 3c). In the specified area, then, the planar guide has a composite core. The advantage of this method is a greater freedom in the choice of materials.

Besides for the fabrication of prisms and lenses in planar guides, these same methods are also used for the fabrication of the B r a g reflectors employed as filters and retroreflectors in planar guides [27].

Another very important group of methods to modify the effective index, and thus to control the direction of light in a guide, is to subject the guide to any one of various externally applied fields : To electric or magnetic fields, or to the stress and strain fields of an acoustic wave. As these fields can be conveniently turned on and off, they are used, of course, for the construction of modulators and deflectors in planar waveguides [20-22, 281. Because they cause index variations only of the order of

... the configuration of these fields is usually arranged to produce a Bragg reflection grat- ing in the planar guide [29]. An example is the acousto- optic modulator [20] shown in figure 5. An electro-

Light beam

1

Quartz substrate + glass film

Diffracted beam

.ks .AC

FIG. 5. - Directional control of a light beam in a planar guide by Bragg rejlection at the periodic index variations associated with

an acoustic surface wave. (After Ref. 20).

optic analog to this deflector/modulator has also been demonstrated [21]. A related method that has been used to modulate the direction of a beam in a planar guide is the injection of free carriers [30-321. Their presence in the guide reduces the index, and a beam deflection results at the boundaries of the injection region.

So far, we have tacitly assumed that the two- dimensional guides are perfectly planar. This is not necessary, however. Compared to three- and one-dimensional propagation media, the two-dimen- sional guide is unique in that we can distort it into the third dimension (e.g. by an indention as in fig. 3 4 , and, in this way, create image forming geodesic optical elements [33]. Figure 6 shows as an example a geodesic lens. On a microscopic scale, the guide in

FIG. 6. - Schematic v i m of a focussing geodesic lens, formed by indention of the guide. The slab thickness W is uniform.

such elements is everywhere planar and uniform and the light rays are straight. On a global scale, however, the light rays follow geodesic lines as the shortest connections between two points. Such geodesic ele- ments have been known for some time in surface acoustics [37l and have been demonstrated in optics by Righini et af. [33]. In contrast to the previously mentioned methods of directional control, geodesic optics has the advantage of being nondispersive, operating also with thick, multimode guides. The focal length of a geodesic lens, for example, is the same for all modes of a thick guide. Therefore, geo- desic optical elements may perhaps find applications in conjunction with multimode optical fibres. They cannot, however, be fabricated by the usual photo- lithographic processes. Rather, one would probably press or cast the appropriate substrates and then apply the guide thereon by e.g., a diffusion process [35].

An interesting combination of a geodesic lens with a lens based on the variation of the film thickness was demonstrated recently by Harper and Spiller [36]. In their arrangement, the film thickness is increased slightly in the region of the indention. Therefore, the collimating action of the two lenses add, whereas

Nouv. Rev. Optique, 1975, t. 6, no 5 R. ULRICH 257

their spherical aberrations, having opposite signs, cancel almost completely. The result is a corrected lens.

Finally, a new method of directional control in the plane of the guide should be possible by adapt- ing to a thin, two-dimensional guide the self-focussing or self-imaging principles which will be explained.

III. - SELF-FOeUSSING AND SELF-IMAGING IN THICK PLANAR GUIDES

The planar optical systems considered above can produce images of one-dimensional objects in which the information is arranged along a line parallel to the guide. We now consider (fig. 7) a thick planar guide and a one-dimensional object in which the information is arranged in a direction normal to the plane of the guide. The simplest possible object is a narrow slit P, parallel to the guide and illuminated from the left. It is well known that real images Q , , Q 2 , . . . of such an object can be produced by a (( self- focussing )) planar guide, whose refractive index varies along the z direction according to a square law [6]. What is to be shown here, however, is the fact that such images can be produced also with planar guides whose index is uniform over the thickness W of the guide [37, 381. Such a step-index planar guide is technically simpler, of course, than the graded index guide.

Theory To explain the formation of images in any type

of thick, multimode optical guide, we represent the amplitude distribution V(x, z ) in the object plane (x = 0) as a superposition of the mode functions Fm(z) of the guide with certain amplitudes a, :

v(0, Z ) = 2 a m Fm(z) . m

(1)

The modes propagate independently along the guide, each one with a different propagation constant P,. Suppressing the time dependence exp( - iot), we obtain, therefore, the field distribution across a distant plane, x = L, as

(2)

J’(L, Z ) = exp(iP0 L ) a m Fm(Z) exp[i(Pm - P o ) L ] . m

In this expression, the phase of the fundamental mode (m = 0) has been taken as a common factor out of the sum. This field distribution Eq. (2) would be an exact image of the object V(0, z ) if a plane, x = L, could be found, so that all phases under the sum of Eq. (2) were unity :

( 3 )

The alternative (- 1)”’ given here corresponds to an inverted image if we assume that the guide has a symmetric index profile, so that the mode functions are altematingly even and odd : Choosing the coor- dinate system so that z = 0 is the midplane of the guide, we have

(4) Fm(- z ) = (- 1)“ Fm(z) .

The meaning of the Eq. (3) is that the phase change suffered by the various modes along the guide should differ by exactly integer multiples of 2 n or of n, respectively. Hence, the existence of self-images depends on the spectrum of propagation constants /I, of the guide.

Planes with that property, Eq. (3), are known to exist in a self-focussing )) planar guides [6 ] . The transverse distribution of their refractive index has the form n(z) = no - n, z2. The propagation cons- tants in this guide are spaced equidistantly

( P m - P m + 1) = (2 n,/’nd1’’ 2

and self-images are formed at the positions (h = 1, 2, 3 . . . )

( 5 ) Lh = hn(n0/2 n1)”2 .

In a planar guide of homogeneous refractive index, equivalent image planes do not exist in a strict sense. We may, however, slightly relax the imaging condi- tions, Eq. (3), by permitting small deviations, nA,, of the phases from exactly integer multiples of 2 n or n. As long as I A,,, 1 < 1, the resulting image distor- tions can be expected to be small. A closer investiga- tion [37l of the spectrum of propagation constants Pm of the planar step-index guide shows then that solu-

I ’X x = o X=L,

FIG. I . - Cross section of planar optical guide with object P and images Q,, Q2, Q, ...

258 R. ULRICH N o w . Rev. Optique, 1975, t. 6, no 5

tions of the relaxed Eq. (3) do indeed exist for not too thin guides :

(6) Lh = 4 hnW;/J. . Here, h = 1, 2, 3. .. is a small integer, n is the refrac- tive index of the slab material, and 2 is the wavelength in free space. The quantity W, is an (( active )) thickness guide. It is equal to the physical thickness W of the slab, corrected very slightly for the penetration of the fields into the cladding materials of index n, on both sides

(7) W, = w + (;L/z) (n,/n)2u (n2 - nf)-' '2 . The exponent 0 here is r~ = 0 for TE polarisation and 0 = 1 for TM.

According to Eq. (6) the first self-image forms at a distance Ll = 4 n W f / 2 from the object. This image is inverted with respect to the midplane of the guide. The first image may act as an object for further self- images, produced by the same process. The second self-image, therefore, is erect and is located at L, = 2 L,. Further self-images exist at all multiples of the first imaging length, Lh = hLl. They are alter- natingly erect (even h) and inverted (odd h). The relation, Eq. (6), between the positions L, of the self-images and the active width W, of the guide is represented in figure 8, assuming n = 1.5 and ;L = 0.6 pm.

200 n

I E 100 U g E 20 0 3 10

5 0.5 I 2 5 lo 20 50 loo zoo 500 1000

LENGTH L Cmml

FIG. 8. - Relation between the positions LA of self-images in a slab waveguide and the active thickness of the slab.

Experiment For an experimental demonstration of the self-

imaging effect, a guide corresponding approximately to the point marked in figure 8 is used. The light- guiding slab (fig. 9) is a high-index liquid (n = 1.635), held by capillary forces between two optically flat plates of fused quartz (n, = 1.458). The length of the guide ( L = 25 mm) and the wavelength (A = 0.633 pm) are fixed. The imaging condition, Eq. (5) is adjusted by varying the thickness W of the liquid film, using spacers and mechanical pressure. The one-dimensional object is a narrow slit in a chromium mask on glass. It is pressed against one end of the guide and is illuminated from the rear with the linearly polarised light of a HeNe laser. The other end of the guide is covered with a thin glass. The self-images forming

there are observed or photographed through a high-power microscope.

FUSED QUARTZ

OBJECTIVE 16x

FIG. 9. - Experimental arrangement for demonstration of self- imaging in planar optical guide.

Without proper adjustment of the thickness W, generally a large number of irregular bright and dark lines are observed. When the imaging condition Eq. ( 5 ) is approached ( W x 50 pm), however, all these lines coalesce into a single sharp image of the slit (fig. 10). This image (h = 1) is inverted, as can be concluded from its relative movement when the object is moved up or down. During such movement, the image does neither change its shape nor its bright- ness.

By reducing the thickness of the guide to approxi- mately 35 pm, 29 pm, or 25 pm, the self-images of higher orders h = 2, 3 ,4 are observed.

They look essentially identical to the h = 1 image, except that their relative direction of motion is inverted when h = 2 or h = 4. The spatial resolution in the h = 1 image is estimated visually from the sharpness of the edges as 1 . ... 2 pm, depending cri- tically upon proper illumination of the object with a wide aperture condensor. The position of the image along the guide shows a weak polarisation dependence. For a fixed thickness W of the guide, the imaging distance for TE polarised light is found to be noticeably longer than for TM polarisation. This observation is in agreement with Eq. (6) and (7) and may be interpreted as a manifestation of the Goos-Hahnchen effect [7].

When the thickness of the guide is not uniform but tapered along the direction of propagation, the self-images are magnified or demagnified depending on the direction of the taper. If Wu(x) denotes the local (( active )) thickness according to Eq. (6), a self- image at position x = L is magnified by the factor

(8) CC = Wu(L)/wa(O) when the object is at x = 0. Thus, neglecting the small correction of Eq. (7), the image size increases or decreases in direct proportion to the local thickness of the guide. This theoretical result, Eq. (8), is based on the assumption that for a sufficiently slender and smooth taper the modes of the guide follow adiaba- tically any variation of the thickness. It can further be shown that in the case of a linear taper, the imaging condition, Eq. (6), has to be replaced by

(9) L, = 4 hnW,(O) wa(Lh)/i .

Nouv. Rev. Optique, 1975, t. 6, no 5 R. ULRICH 259

FIG. 10. - First self-image (h = 1) of a 3 pm wide slit. The thickness of the guide is uniform, W N 50 pm.

Magnified self-images have been observed in full through the microscope onto a slit/photomultiplier agreement with the above theory. Figure 11 is a combination and slowly scanned. The result is given photograph of the first (h = 1) magnified self-image in figure 12. One abscissa relates to the position of the same slit as figure 10, but produced here by a (x = L ) of the image, the other one to the position guide whose width is tapered to yield a magnification (x = 0) of the object. The object here is a 6 pm wide of p zz 5. Higher magnifications up to ,u = 15 have slit. The spatial resolution of the scanning system also been realized. By inverting the direction of the was approximately 2 ... 3 pm, referred to the image taper it is possible, of course, to produce demagnified plane x = L, and may be neglected here. We see self-images. from the slopes of the image in figure 12 that the

A quantitative number for the resolution of self- l0/90 % width of the step response function is approxi- images is obtained from a measurement of the inten- mately 1.7 pm when referred to the object plane. sity profile across a magnified image like figure 11. It may be expected that the resolution of the non- For that purpose, the image of x = L is projected magnified self-images (p = 1) is even better.

FIG. 11. ~ Magnzjied self-image (h = 1 ) of a 3 pm wide slit, formed by a guide whose thickness is tapered from W(0) N 22 pm

to W(L,) N 105 pm.

260 R. ULRICH Nouv. Rev. Optique, 1975, t. 6, no 5

1 I I 1 I 1 ,Z 0 50 lOOpm (x=L)

F ~ G . 12. - Measured intensizy profile across a magnij?ed self image like figure 11. The object is a 6 pm wide slit.

The formation of self-images is not restricted to monochromatic, coherent light. Self-images have also been observed with light from a tungsten lamp, filtered through a 50 A interference filter. Their resolution is somewhat inferior, however. A theoreti- cal estimate [37] shows that the o tical bandwidth should be limited to Aj./i < I/hM . Here, M is the maximum number of modes contributing to the image, i.e. the highest mode number up to which the phase errors in Eq. (4) are small, I A , I < f for all m < M . In the present example, M % 30, so that an optical bandwidth of 7 A would be tolerable here at h = 1 without seriously reducing the resolution. A similar tolerance must be observed for the thickness, AWjW < h M 2 . This tolerance is 300 A here at h = 1. It applies, however,. only to the average thickness of the guide, because a smooth taper does not seriously affect the image quality, as had been shown.

4

Discussion As we have seen, the self-imaging property of a

guide is linked to a characteristic feature of its spec- trum of propagation constants B,,, : Self-imaging exist if and only if all differences (8, - B,) are very nearly the integer multiples of some fundamental difference A/?. The first self-imaging distance is then nIAP or 2 n1A.B.

For this reason, the step-index, round optical fibre cannot be expected to show self-imaging. In its spectrum of j?, no regularity of the required type can be detected.

It is interesting also to consider the formation of self-images from a ray-optical point of view. It is found then, that a self-image is formed by the constructive interference of a finite number of rays, whose path lengths differ by integer multiples of the wavelength. This process differs fundamentally from the image formation by a lens or by a self-focusing guide, wherein the image results from the interference of a continuous set of rays having (ideally) no path differences. In this respect, the self-image corresponds to the images formed by a Fresnel zone plate [9].

A close relationship exists also between the self- images discussed here and the self-images which are known to form at certain distances behind a periodic object [38-421. A virtual, periodic object is formed

here by the repeated reflections of the object in both walls of the guide which act as mirrors [38]. The erect (even h) self-images in the guide may be derived formally as the Fourier self-images [41] of the men- tioned periodic object, whereas the inverted (odd h ) self-images in the guide correspond to the so-called Fresnel self-images [42].

The principle of self-imaging has been discussed here for thick planar guides, supporting many modes. They produce one-dimensional self-images only. Self- images with two-dimensional resolution would require an imaging mechanism also in the direction parallel to the plane of the guide. Such imaging could possibly by realized by either distorting the self-imaging planar guide so as to form a geodesic lens, or by choosing a guide of very nearly square cross-section. In the latter case, two conditions like Eq. (6 ) have be satis- fied independently, one for each transverse dimension. Because a field that is TE polarised for one dimension is TM with respect to the other one, a small but finite difference in the two transverse dimensions of the guide would be necessary, so that the two correspond- ing (( active )) thicknesses became equal.

Applications of the described self-imaging effect may be expected in two different configurations. In the first one, thick multimode slab guides may be used in a similar fashion as in the experiment of figure 9. It should be possible, for example, to image in this way the output from the long, thin junction of a GaAs laser into the edge of a thinfilm guide.

The other configuration would be obtained by applying the principle of self-imaging to the trans-

FIG. 13. - Proposed self-imaging optical strip guide, acting as a simple cross-over of two narrower strip guides 1 and 2.

Nouv. Rev. Optique, 1975, t. 6, no 5 R. ULRICH 26 1

verse dimension of a thin planar optical guide. This is indicated schematically in figure 13. There is shown a substrate with a thin film optical guide, supporting e.g. only a single mode along its thin dimension. If the thin film were extending infinitely wide, this mode would be characterised by an effective index Ne,,. If we now cut a strip-guide of sufficiently large width W from that guide, we may expect that guide to show self-imaging. The reason is the direct correspondence, discussed earlier, between three- dimensional bulk optics and two-dimensional optics in a planar guide. According to that correspondence, the strip-guide should self-image its narrow end- faces upon each other, provided its length L and width W satisfy the general condition Eq. (6) for self-imaging, but with the bulk index n, occurring in that formula, replaced by Ne,,. If the dimensions are chosen so that h is an odd integer, the self-image is inverted. The arrangement of figure 13 can act then as a simple cross-over for two narrower strip- lines 1, 1' and 2, 2', which are connected to diagonally opposite positions at the end-faces. It is obvious, how this principle can be extended to a cross-over of more than two guides.

Another possible application of the arrangement of figure 13 should be as a filter for the separation (or combination) of light of two wavelengths ,I1 and A2 whose ratio is a rational number, &/Iu2 = a/b, with small integers a and b having no common divisor. As an illustration, we consider A 1 / i 2 = 2, and a guide like figure 13, designed to operate at an odd order for ,I2 (for example h(1,) = 1). Neglecting dispersion, then, that guide will operate at an even order for A1, because according to Eq. (6) the order h is propor- tional to the wavelength. Therefore, the image at A 2 is inverted, and at A1 is erect. When light of both wavelengths is fed into the narrow guide 1, the wave- lengths are separated in the outputs 1' and 2'. The generalisation to other ratios a/b is straightforward. This application would correspond directly to Ray- leighs experiment on the self-imaging effect of a grating [39].

IV. - CONCLUSIONS

We have reviewed some properties of planar waveguides relating to imaging. The planar guide was viewed as an artificial, two-dimensional propaga- tion medium for light. In the plane of the guide, pro- pagation is free and follows the well known laws of geometrical optics with a local, effective index Nerr(x,y) of refraction. In the direction normal to the plane of the guide, the light is confined. The details of this confinement determine the effective index and can be adjusted by various means to produce a desirable distribution of Nerf(x, y). Self- imaging, a new phenomenon in thick planar guides, was described theoretically and experimentally. Self- images result from the interference of a number of modes which propagate at different velocities, and whose phases all coincide in the image. The self- images can be erect or inverted and magnified or

demagnified, and they may find applications in integrated optics.

* * *

REFERENCES

[ 11 KANE (J.), OSTERBERG (H.). - J. Opt. Soc. Amer., 1967, 54,

[ 21 SCHINELLER (E. R.), FLAM (R. F.), WILMOT (D. W.). - J .

[ 31 SHUBERT (R.), HARRIS (J. H.). - ZEEE Trans., 1968, MTT-16,

[ 41 TIEN (P. K.), ULRICH (R.), MARTIN (R. J.). - Appl. Phys.

[ 51 TIEN (P. K.). - Appl. Opt., 1971, 10, 2395. [ 61 MARCUSE (D.). - Light Transmission Optics, Van Nostrand

[ 71 KAPANY (N. S.), BURKE (J. J.). - Optical Waveguides,

[ 81 MARCUSE (D.). - Theory of Dielectric Optical Waveguides,

[ 91 BORN (M.), WOLF (E.). - Principles of Optics, (3rd ed.),

[lo] ULRICH (R.), MARTIN (R. J.). - Appl. Opt., 1971, IO, 2077. [Il l ANDERSON (D. B.), BOYD (J. T.), MCMULLEN (J. D.). -

Proceedings of the Symposium on Submillimeter Waves, (MRI Series 20), Polytechnic Press, New York, 1971,

347.

Opt. Soc., 1968, 58, 1171.

1048.

Lett., 1969, 14, 291.

Reinhold, New York, 1972.

Academic Press, New York, 1972.

Academic Press, New York, 1974.

Pergamon Press, New York, 1964.

pp. 191-210. [I31 KAMINOW (I. P.). - IEEE Trans, 1975, MTT-23, 57. . [14] MILLER (S. E.). - Bell Syst. Techn. J., 1969, 48, 2059. [I51 KOGELNIK (H.). - ZEEE Trans., 1975, MTT-23, 2. [I61 GOELL (J. E.), STANDLEY (R. D.). -Bell Syst. Techn. J.,

[17] GOELL (J. E.). - Appl. Opt., 1973, 12, 729. [18] OSTROWSKY (D. B.), PAPOUCHON (M.), ROY (A. M.), TRO-

[19] EVTUHOV (V.), YARIV (A.). - ZEEE Trans., 1975, MTT-23,44. [20] K m (L.), DAKS (M. L.), HEIDRICH (P. F.), SCOTT (B. A.).

[21] HAMMER (J. M.), PHILLIPS (W.). - Appl. Phys. Lett., 1974,

[22] SCHMIDT (R. V.), KAMINOW (I. P.). - ZEEE J. Quantum

[23] GOELL (J. E.). - Appl. Opt., 1973, 12, 2797. [24] KUESTER (E. F.), CHANG (D. C.). -ZEEE Trans., 1975,

[25] SHUBERT (R.), HARRIS (J. H.). - J. Opt. Soc. Amer., 1971,

[26] TIEN (P. K.), RIVA-SANSEVERINO (S.), MARTIN (R. J.), SMO-

[27] FLANDERS (D. C.), K~GELNIK (H.), SCHMIDT (R. V.), SHANK (C.

[28] TIEN (P. K.), MARTIN (R. J.), WOLFE (R.), LE CRAW (R. C.),

[29] LEDGER (J. F. S.), ASH (E. A.). - Electron. Lett., 1968, 4, 99. [30] CHEO (P. K.). - Appl. Phys. Lett., 1973, 22, 241. [31] CHANG (W. S. C.). - IEEE Trans., 1975, MTT-23, 31. [32] McFm (J. H.), NAHORY (R. E.), POLLACK (M. A.),

LOGAN (R. A.). - Appl. Phys. Lett., 1973, 23, 571.

1969, 48, 3445.

TEL (J.). - Appl. Opt., 1974, 13, 636.

- Appl. Phys. Lett., 1970, 17, 265.

24, 545.

Electron., 1975, QE-11, 57.

MTT-23, 98.

61, 154.

LINSKY (G.). - Appl. Phys. Lett., 1974, 24, 547.

V.). - Appl. Phys. Lett., 1974, 24, 194.

BLANK (S. L.). - Appl. Phys. Lett., 1972, 21, 394.

[33] RIGHINI (G. C.), RUSSO (V.), SATTINI (S.), TORALDO DI FRAN- CIA (G.). - Appl. Opt., 1972, 11, 1442.

[34] VAN DUZER (T.). - Proc. ZEEE, 1970, 58, 1230. [35]. - NOMURA (H.), OKADA (T.), OIKAWA (S.), SHIMADA (J.).

[36] HARPER (J. S.), SPILLER (E.). - Topical Meeting on Integrated Optics, New Orleans, 21-24 Jan. 1974, Digest of Techn. Papers, WB 1 1.

- Appl. Opt., 1975, 14, 586.

262 R. ULRICH Nouv. Rev. Optique, 1975, t. 6, no 5

[37] ULRICH (R.). - Opt. Comm., 1975, 13, 259. [38] BRYNGDAHL (0.). - J. Opt. Soc. Amer., 1973, 63, 416. [39] TALEOT (H.). - Phil. Mag., 1836, 9, 401. [40] LORD RAYLEIGH. - Phil. Mag., 1881, 11, 196. [41] COWLEY (J. M.), MOODIE (A. F.). - Proc. Roy. Soc. (London),

[42] WIKTHROP (J. T.), WORTHINGTO~ (C. R.). - J . Opt. Soc. ACKNOWLEDGMENT : The author wishes to thank G. ANKELE

1431 TSENG (C. C.), TSANG (W. T.), WANG (S.). - Opt. Comm., 1975, 13, 342.

1957, 870, 486-513.

Amer., 1965, 55, 373. for his assistance in the experiments.

(Manuserit repu le 20 mai 1975.)