Interferometric variable reflectivity mirror

July 1997

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Optical Materials 8 (1997) 157-160

Interferometric variable reflectivity mirror M. Keselbrener *, S. Ruschin Department of Electronical Engineering, Physical Electronics, Faculty of Engineering, Tel-Avic University, Tel-Aviu 69978, Israel Received 25 November 1996; revised 31 January 1997; accepted 1 May 1997

Abstract

We describe a variable reflectivity mirror, based on the frustrated total internal reflection effect in conjunction with a Fox-Smith interferometer. The device has a performance which is very sensitive to frequency in the sense that it changes significantly its radial reflectivity distribution as a function of the light wavelength. This property has potential applications in laser resonators. We calculate the theoretical reflectivity of this mirror, for the TE polarization, on the basis of the plane-wave approximation. The wavelength reflectivity dependence is shown and discussed in a simple case of the mirror implementation. Figures of merit of the reflectivity distributions are presented. © 1997 Published by Elsevier Science B.V.

Apodized filters and mirrors have been proposed and developed [1,2] based on the frustrated internal reflection effect. These elements consist of two glass prisms, almost in contact along one of their surfaces, which, instead of being planar, are optically polished at a suitable curvature. When the light impinges upon the interface, at an angle at which total internal reflection occurs, the light is transmitted with a transversely varying transmission profile. This profile mainly depends on the profile of the gap created at the interface between the two prisms. Quasi-gaussian transmission profiles can be obtained simply using spherical surfaces. We propose here the insertion of such a device within an interferometer such as the F o x - S m i t h type. The structure can be simply implemented, if the second prism has two 100% reflecting faces (see Fig. la, b). It was found that such a structure has the ability to change its spatial

* Corresponding author. Fax: + 972 3 6429540.

d

t (a)

tout (b)

h

Fig. 1. Schematics of the devices.

reflectivity distribution in a way that is very sensitive to the radiation frequency. The back reflectivity function rin can be obtained, for the case of uniform spacing between the two prisms (see Fig. la), from the superposition of plane wave reflections, between the two 100% mirrors and the prism hypothenuse. It is given by: t t exp(i 6 ) rin = 1 - r2 e x p ( i 6 ) '

00925-3467/97/$17.00 © 1997 Published by Elsevier Science B.V. All rights reserved. PII S 0 9 2 5 - 3 4 6 7 ( 9 7 ) 0 0 0 4 1 - 4

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M. Keselbrener, S. Ruschin / Optical Materials 8 (1997) 157-160

: 1I l[l ! ~

T h e y can be obtained from the surface conditions and are of the form [3]:

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=

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i-

'

...... i ............ i............. !°"I •= 0.2 I/

(3) 1 + r12 r23 e x p ( 2 i / 3 ) '

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t12 t23 e x p ( i / 3 )

iz

1 + r12r23 e x p ( 2 i / 3 ) '

t=

i

h [Hoo-meterI

(4)

rl2 , r23 are the well k n o w n reflection Fresnel formulae of the two interfaces respectively, and t12, t23 are the corresponding transmissivity. The phase /3 is given by

h [Hero-meter]

Fig. 2. (a) Homogeneous film power reflcctivity (R), transmissivity (T) and (b) phase reflectivity in function of h with 01 = ~-/4, n I = 1.5, n 2 = 1.0, n 3 = 1.5 at )t = 1.06 ~m, for the TE polarization.

/3

27r ho n 2 h

=

--

cos

02

(5)

,

with where the phase shift 6 is: (2) cos 0 z = i Here d is the prism d i m e n s i o n , n 3 is the refraction index o f the prism and A0 is the light w a v e l e n g t h in v a c u u m , r and t are the reflectivity and transmissivity respectively, at total internal reflection of the thin film, created by the two h y p o t h e n u s e prism faces.

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and

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nl2 = n 2 / n l ,

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-0.4

-o.a

-o.z

-0.1

o wl

o.1

o.z

o.3

0.4

[Angs~om]

Fig. 3. Power reflectivity as a function of the wavelength for different widths h. The wavelength unit is .A.. The zero corresponds to A = 1.06 /zm. The prism side dimension is d = 1.0 cm, corresponding to a free spectral range of 10 GHz.

M. Keselbrener, S. Ruschin / Optical Materials 8 (1997) 157-160

n I is the refraction index of the first prism, and n 2 is, in our case, equal to 1, the refraction index of the air thin film. 01 is the incident angle at the first prism hypothenuse face, when h is defined as the distance between the two prisms. The basic function of the device is best described by means of Fig. 3, where the back reflectivity function rin (see Fig. l a) is plotted as a function of the wavelength, for the case of uniform spacing between the two prisms (denoted by h in Fig. la). One observes the characteristic resonant behavior of a F o x - S m i t h interferometer. The reflectivity function is periodic in frequency, and the finesse can be controlled by changing the width h of the thin film created by the two prisms. A unique feature of the device appears, when the distance h is changed. In addition to the finesse variation due to the change of the hypothenuse reflectivity (see Fig. 2a), there is a shift in the frequency corresponding to the peak reflectivity. This shift is a result of the inherent phase shift of the reflected light in the frustrated total internal reflection process [3] (see Fig. 2b). The frequency shift is the crucial effect in the behavior of the entire reflecting structure, and explains the large sensitivity of the reflector to changes in frequency, when the distance parameter h varies with radial position. The example of Fig. 3 corresponds to a structure designed to work at a wavelength of 1.06 /zm and has a prism side length of d = 10 mm. Taking n I = n 3 = 1 . 5 , the figure shows a shift of 2.67 G H z for a variation in the gap thickness of 0.6 ~m. When the interface of one prism is replaced by a spherical surface, as depicted in Fig. lb, a spatially varying reflectivity function is obtained, having a pronounced dependence on frequency. This can be seen in Fig. 4, where transverse reflectivity profiles are plotted for different wavelengths. Consecutive patterns in the series differ by a value of about 300 M H z in the frequency of the incident light. Remarkable is the fact that some of the patterns are not monotonically descending as a function of the radial coordinate, in spite of the fact that for a spherical convex surface shape, the gap distance increases with the radius. This can be explained as follows: For a zero gap distance, the back reflectivity will be obviously one for all wavelengths. As the gap is increased, the reflectivity will decrease in general,

159

tl, 7-I t<'Jl !,2

Fig. 4. Power reflectivity as a function of radial co-ordinate at different wavelengths. Here the prism interface is spherical, with curvature radius of 5.0 m and is placed at a distance of 0.2 /xm from the second prism. The wavelength shift between two plots is 0.1 ~, or 0.27 GHz. The full horizontal axis scale is 9.0 mm, and the full vertical axis scale is 1.

due to the finite finesse of the interferometer, unless a resonance condition is eventually attained for a given frequency, in that case (see e.g., Fig. 4(3)-(8)) the reflectivity recovers the value of one, for certain distances. We conclude with some remarks about the applicability of the device presented. The reflectivity or transmission shapes that can be obtained with this arrangement are not limited only to those studied here. Indeed, if the shape of the interface between the two prisms is made non spherical, additional possibilities of spatial modulation arise. The insertion of the prisms ensemble in an interferometer, allows agile control of the shapes by means of frequency modulation within a practical range. Another attractive possibility is the insertion of the ensemble discussed as an output mirror in a laser cavity. The gain bandwidth available in many practical laser amplifiers, covers well the range of frequencies necessary in order to realize the spatial modulation effect. It is expected therefore that the device will act as a self adapting reflector, and will lock to a frequency that chooses the shape that will minimize losses and enhance the laser efficiency. We indend to concentrate on that application in following studies.

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M. Keselbrener, S. Ruschin / Optical Materials 8 (1997) 157-160

References [1] E. Armandillo, G. Giuliani, Opt. Lett. 9 (1985) 445. [2] S.G. Lukishova, S.A. Kovtonuk, A.A. Ermakov, P.P. Pashinin, E.E. Plavtov, A.S. Svakhin, A.A. Golubsky, Dielectric films

deposision with cross-section variable thickness for amplitude filters on the basis of frustrated total internal reflection, Proc. SPIE, vol. 1270, 1990, p. 260. [3] M. Born, E. Wolf, Principles of Optics, Pergamon, New York, 1959.