Optoelectronic super-gaussian mirrors based on the thermo-optical effect

Optics Communications 93 (1992) 163-168 North-Holland

OPTICS

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Optoelectronic super-gaussian mirrors based on the thermo-optical effect Antonello Cutolo, Giancarlo Calafiore Dipartimento Ing. Elettronica, Via Claudio 21, 80125 Naples, Italy

and Salvatore Solimeno Dipartimento Scienze Fisiche, Mostra d'Oltremare Pad. 16, 80125 Naples, Italy Received 20 March 1992; revised manuscript received 10 June 1992

Taking advantage of the thermo-optical effect we discuss the possibility to build tapered mirrors and windows, the optical parameters of which can be electrically modulated. The most relevant result is that these devices can be easily employed to realize super-gaussian mirrors both in the infrared and in the visible, instrumental for the construction of high efficiency and diffraction limited lasers.

1. Introduction

The idea of exploiting resonators with tapered mirrors in order to improve the laser beam quality is fairly old [ 1-6 ]. In recent years, Svelto and his coworkers [ 7-10 ] have demonstrated that the use of super-gaussian mirrors, i.e. characterized by a radial profile of the reflectivity described by R ( r ) =Ro exp [ - 2(r/w)"] ,

(1)

where Ro is the intensity peak reflectivity, r is the radial coordinate, w is the mirror spot size and n is the order of super-gaussianity, leads to the construction of diffraction-limited and high-efficiency lasers. They have removed most of the main disadvantages of resonators made up with gaussian mirrors ( n = 2 ) leading to a better optimization of the laser performance as well [9 ]. In particular, they have characterWork jointly supported by the "Progetto Finalizzato Tecnologie Elettro-Ottiche" of the "Consiglio Nazionale delle Ricerche", by the "Ministero deil'Universit~t e della Ricerca Scientifica e Tecnologica" and by the "Regione Campania" of Italy.

ized a pulsed Nd:YAG laser with the optical resonator made up with different tapered mirrors. They have mounted mirrors with n=2.8, 5, 9, 35 obtaining output pulses with energies equal to 280, 314, 364, 430 mJ and with focusing efficiencies 0.92, 0.89, 0.79, 0.64, respectively, where the focusing efficiency is defined as the fraction of the total power collected within a spot of assigned radius on the focal plane of a lens. From these experimental data, it is easy to understand that the use of super-gaussian mirrors increases the optical quality of the laser beams leading to almost diffraction limited laser beams [9 ]. The required super-gaussian profile of the reflectivity (cf. eq. ( 1 ) ) can be attained with a dielectric multilayer where one film has a variable thickness d(r) [ 10]. On the other hand, during the development of a project finalized to the analysis of new schemes for electrically driven optical modulators [ 11 ] our attention was caught by the high value of the thermooptical coefficient of both silicon and lithium niobate for infrared and visible applications, respectively (cf. table 1 ). With these considerations in mind, it is straightforward to attempt to investigate the pos-

0030-4018/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

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Table 1 Thermo-optical properties of some materials.

Z

Material

Refraction index

dn d-T ( × lOs)

silicon LiNbO3

3.55 2.200 2.865 2.262 1.512 1.805 1.618 1.517 2.5

18.5 5.3 -7.2 - 4.1 1.5 1.04 -0.35 0.29 > 10

TiO2

PbMoO4 soda-lime glass SF-6 PSK-02 BK-7 PLZT

sibility of making super-gaussian mirrors by exploiting the thermo-optical effect. Although the idea of exploiting the thermo-optical effect is not new, the novelty of our approach is the possibility of making tapered mirrors (in particular super-gaussian) which can be electrically controlled. The main goal of this paper is to show the feasibility of electrically driven super-gaussian mirrors based on the thermo-optical effect. In this line of argument, we limit ourselves to analyze the optical parameters of a disk made of a thermo-optical material (e.g. silicon). It is understood that the final device will be made with a multistrate dielectric mirror where one film has been made with a thermo-optical material. Encouraged by the results reported in this paper a more detailed analysis, both theoretical and experimental, is in progress.

2. Description of the device Although the final device will have a more involved structure, in order to reach our goal, in this paper, we analyze the optical properties of a silicon disk of radius R and thickness 2d (cf. fig. 1 ). Its border is kept at a temperature To above room temperature. Due to the free convection thermal exchange, the temperature increase in the middle of the disk is lower than To thus generating a temperature gradient inside the disk. In passing, we explicitly observe that the heating of the disk border might be easily achieved by letting a current flow along it. This solution, although very simple, presents the advantage 164

1 October 1992

t°1

Fig. 1. Schematic of the silicon disk, of radius R and thickness

2d, in its reference system. of permitting an easy real time control of the optical parameters of the tapered device. In addition, the required current intensity is smaller than 2 A. If the two surfaces are parallel and optically finished, the silicon disk can be considered as a FabryP6rot etalon. Now, due to the electrically induced thermal gradient, the optical path is radially inhomogeneous. As a consequence both the reflection and transmission coefficients of our device depend on the radial coordinate. In particular, as will be shown later, this functional dependence is super-gaussian (cf. eq. (1)). In this line of argument, the first step of our analysis is to solve the heat conduction equation in steady state conditions: V2T(r, z)---0,

(2)

with the following boundary conditions,

T(r, z)[r=R = T o ,

(3a)

K, fl-~ . . . . =-Th[T(r,z= +_d)-T~,] ,

(3b)

where K, is the thermal conductivity of the disk (e.g. K,=264 W / m for silicon), h is the free convection thermal exchange coefficient, To is the electrically driven temperature of the border of the disk and T~ is the room temperature. While the other coefficients are exactly known, the coefficient h strongly depends on the particular operating conditions. In practical cases its value is about 2.2 W / K m 2. Equations (2), (3) can be easily solved to give

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T(p, ()=ToF(p, C) = To (1 -2 ~

Jo(q,,P) cosh(q.(C) ), n=o q,J1 (q,) [cosh(q,~) +yq, sinh(q,~) ] (4)

wherep=r/R ( 0 < p < 1 ) and ~=z/d ( - 1 <~< 1 ) are the normalized coordinates, ~= d/ R, 7= Kt/ hR, do(x) and Jv I (X) are the Bessel function of the first kind and zeroth and first order, respectively, and qn is the nth solution of the equation Jo(x)=0. F(p, ~) can be considered as a geometrical shape function of the temperature, which depends on both the thermal coefficient 7 and the smallness parameter ~ and is independent of To (cf. fig. 2). In eq. (4) the room temperature (T~) has been put equal to zero. Figure 2 shows the radial profile of the function F(p, ~=0) for different values ofy and for ~=0.001. An inspection of this figure reveals that the temperature in the middle of the disk (cf. fig. 1 ) can be more than 10% lower than the border temperature. Due to the smallness of the parameter E( = d/R), the gradient along the axis of the disk can be neglected thus permitting one to assume the temperature practically independent of the axial coordinate z. In particular, several numerical simulations have indicated that the temperature (cf. eq. (4)) can be approximated by

T(p, z) ~-.T(p, 0) ~ To[ 1 - A * ( 1 _pZ) ] ,

interest, we always have e < 10- 3 thus indicating that eq. (5) holds with a completely negligible error ( < 10-4). The coefficient/t is a quite involved function of the disk parameters. Therefore, in fig. 3, we have plotted the coefficient A as a function of the input parameters E and To. 3. The apodized device According to our previous discussion, the disk shown in fig. 1 can be seen as an etalon, the reflecting surfaces of which are represented by the two air-silicon interfaces spaced by the thickness (2d) of the disk. From the standard theory of Fabry-P6rot interferometers it is easy to derive its intensity transmission (7"1) and reflection (RI) coefficients together with the phases of the transmitted (qbT) and reflected (q~R) wavefronts under the hypotheses of incident plane wave. Accordingly, we can write 16n 2

TI= ( n + 1 )4"1- ( n - -

1 )4-- 2(n2--

1 )2 c o s ( d )

'

(6a)

n2 + 1

q~T=arctan \---2-n--ntan(d)

)

(6b)

,

2(n z - 1)Z[ 1 - c o s ( d ) ] RI= ( n + 1)4+ ( n - 1 ) 4 - 2 ( n Z - 1)z cos(d) '

(5)

with a relative error smaller than 10 -3 if ~<0.01. We explicitly underline that, in all cases of practical

(6c) ( 2n sin(d) ) OR =arctan \ ( n 2 + 1 ) [cos(d) - 1 ] '

(6d)

F(O,¢=o)

1.o 1.0

A \

\

0.9 0.8

0.5

0.7

o.o --- 7=1000

o'.s --

?=1500

t(

o.(

o.6ool

o.ool

..... 7=2000

Fig. 2. The radial profile of the temperature shape function ( = 0 ) (cf. eq. ( 4 ) ) for ¢=0.001 and for ( . . . . ) y=1000, ( 7 = 1500 and ( - - - ) y = 2000.

F(p, )

--- 1,=1000

-

-

y=1500

o.ol

dl

..... y=2000

Fig. 3. Plot of the constant A (cf. eq. ( 5 ) ) as a function of ( = 2d/R for different values of the thermal constant 7=Kt/hR.

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where n is the refractive index of silicon and O= 8nnd/2. In eqs. (6) the tapering effect comes from the radial dependence of the refractive index of silicon (n) according to (7a)

dn r=r~ T(r, z) , n=n(r, z ) = n o + d-~

1 O c t o b e r 1992

T I (o) 1.0

A

"

0.5

\, \,

,/

~\, //

from which 8rid dn T , O=q/o+ --~---~--~ (p)

(7b)

where ~Uo=8ndno/2, To~ is the room temperature, 2 is the wavelength in vacuum and no is the refractive index of silicon at T = To~. In eq. (7b), the functional dependence of the temperature on the axial coordinate (z) has been neglected in view of the previous discussion (cf. eq. ( 5 ) ) . An inspection of eqs. (6) might suggest that the tapering effect is accompanied by a focusing effect too. Indeed, many numerical simulations have shown that this effect is completely negligible. In fact, the device can be considered like a lens whose focal length is always longer than one kilometer in all practical cases. This result agrees with the phase distortion of the reflected wavefront calculated in ref. [9]. Accordingly, in the forthcoming analysis, we neglect any focusing effect. Figures 4 and 5 show the radial profiles of both the transmission (T~) and reflection coefficients (R~) for a variety of practical cases. These figures show that the maximum values of the transmission and of 1.0

R=[~'I

\, \

0.5

,/

'k 0.0

/ ,

0.5

O0

---7=1000

To=200

\\./

i

\I1.0

0.5

-- 7=1000

To=200

1.0 0

r=0001

Fig. 5. Radial profile of the transmittivity for ~=0.001, T0=200 K, 7= 1000 and ~o= 0. the reflection coefficients are always 1 and 0.7, respectively. A careful analysis of figs. 4, 5 suggests that both T~ and R~ might be approximated by a super-gaussian profile of the kind

--CTp a)

,

(8a)

RI ~RM exp(--CRp p) ,

(8b)

forp

(8c)

T 1 -.~ T M e x p (

where TM and RM are the maximum values reached in the middle of the disk, a and fl are the order of super-gaussianity and the constants Ca- and CR are related to the radius of the high-transmission and high-reflection zone of the disk, respectively (cf. eq. ( 1 ) ). Pma, indicates the range of the disk where the approximated relations (8a), (8b) can be considered well satisfied. Indeed, many numerical simulations have demonstrated that eqs. (8) are exact within a relative error smaller than 10-3 in a large variety of practical cases for Pr~axeven larger than 0.7. Figure 6 shows the parameters Cx and t~ (cf. eq. ( 8 ) ) as a function of the border temperature To. A similar plot is shown in fig. 7 for CR and ft.

4. The problem of the thermal stability 0

~=0001

Fig. 4. Radial profile of the reflectivityfor ~= 0.001, To= 200 K, y= 1000 and ~Uo=n. 166

0.0 0.0

In spite of its simplicity our device has shown the possibility of making electrically driven tapered mirrors. In addition, our model can easily provide useful information about the problem of the thermal

Volume 93, number 3,4

OPTICS COMMUNICATIONS

40

18C

f1"

CR

cT

////w// /// 20

/

(a)

12C /

.....-"" ..,"

///

!

(a)

/ /

/

/

/

/

.II

i

6(3

y.°" °/

////

1 October 1992

t"

To 160 --- 7=1000

--

260

7=1500

300

..... 7=2000

---

4.6 (1

200

100 7=1000

--

7=1500

3oo .....

~'=2000

6.5

/3

"-..

./I /

4.0

i//

5.0

(b)

//

(b) 3.4

35

T --- 7=1000

100

200

--

r=1500

T

300 ---

..... r=2000

Fig. 6. Plot of the super-gaus~ianity parameters CT and a (cf. eqs. (8) ) as a function of the border temperature To and the thermal constant 7for VJo=0 and ( = 0 . 0 0 1 .

stability. In fact, both a comparison of fig. 5 with fig. 6 and an inspection of eqs. (6) reveal that it is possible to switch the device from the high-reflection configuration to the high-transmission configuration, simply by changing the value of V/o from 0 + 2 n n to ~ + 2n re. This property, although useful for intracavity modulation applications, can generate some minor problem for a stable operation of the laser if no precaution is taken. With these considerations in mind we want to analyze the tolerance on the room temperature of the device. Taking in mind that V/o = 8 n d n o / 2 ,

(9)

16o

7=1000

--

7=1500

..... }'=2000

Fig. 7. Plot of the super-gaussianity parameters Ca and a (cf. eqs. (8) ) as a function of the border temperature To and the thermal constant 7 for ~ o = n and c=0.001.

T u(OI

1.G

0.5

o.q 0.0

with no = n ( T = Too), it is clear that the correct use of the tapered device implies an appropriate stabilization of he room temperature (Too). In this line of argument, an inspection of fig. 8 indicates that a

abo To

260

---~=0

0.2 --¥'~/4

..... W'~/2

0.4 --W-3~

--W-~

Fig. 8. The radial profile of the transmittivity for different values of the constant ~'o, 7 = 1000, ¢=0.001 and 7"o=300 K.

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Volume 93, number 3,4

OPTICS COMMUNICATIONS

change o f ~0 from 0 to i n reduces the t r a n s m i t t i v i t y by a factor equal to a b o u t 0.25. A linearization o f the functional d e p e n d e n c e o f the transmission coefficient decrease (ATI) from the phase ~o around ~/o= 0 gives the a p p r o x i m a t e d relation A T, .~ ~Uo/n ,

(10)

which, plugged into eq. ( 9 ) a n d taking in m i n d the variation o f the refractive index with temperature, gives d dn AT, = 8 ~-d--~AT.

(11)

Now, assuming d = 10 ~tm (the thickness o f the disk is 20 ~tm), 2 = 1 ~tm a n d the disk m a d e o f silicon (dn/dT=l.5XlO -4 K -a, cf, table 1), we get A TI = 0.01 AT. This result implies that the r o o m temperature must be controlled within half a degree to a v o i d a change o f the transmission coefficient higher than 1%. In o r d e r to get rid o f this problem, we observe that, instead o f controlling the r o o m temperature, we can simply control the b o r d e r t e m p e r a t u r e (To). In fact, small changes o f To (a few degrees ) do not change the a p o d i z a t i o n p a r a m e t e r s too much but are enough to keep the device locked in the hightransmission or high-reflection configuration according to the specific case. Although the discussion has been carried out only for silicon its extension to other materials is straightforward. In p a r t i c u l a r we notice that, due to its smaller t h e r m o - o p t i c a l coefficient, lithium niobate has m i n o r t e m p e r a t u r e control problems.

5. Conclusions and future trends In this p a p e r we have shown the feasibility o f elec-

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1 October 1992

trically driven tapered devices by taking advantage on the t h e r m o - o p t i c a l effect in silicon. O u r model is simplified with respect to the final configuration but it p e r m i t t e d us to reach our actual goal: to show the feasibility o f this device. Encouraged by these resuits, work is now in progress to analyze, both theoretically and experimentally, dielectric multistrate mirrors where one layer is m a d e with a t h e r m o - o p tical material (e.g. silicon or lithium n i o b a t e ) .

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