Decay time of optical resonators

Volume

23. number

3

December

OPTICS COMMUNICATIONS

1977

DECAY TIME OF OPTICAL RESONATORS W. MOLLER and H. WEBER Fachbereich Received

Physik, Universitri’t Kaiserslautern,

21 September

Germany

1977

The decay time T of an optical resonator depends on the special type of resonator (active, passive, homogeneous, inhomogeneous). Starting from a general expression for decay times, some special cases are derived and discussed in detail. It is shown, that the commonly used formulars give only lower or upper limits. These various relations are relevant for high gain laser with low reflectivities.

The resonator decay time r is an essential parameter used in the laser rate equations. Threshold pumping rate, optimal output coupling and frequency of spiking systems depend on this parameter. The decay time characterizes the losses of the system due to reflectivities R 1, R, of the mirrors and internal losses V of the resonator. Two expressions are generally used [l]

sonator; P the total power dissipation due to output coupling and internal losses; E the energy stored inside the resonator. In any cavity of Fabry-Perot type, between the two mirrors a standing wave is built up. It is a superposition of two travelling waves J’(x) in opposite direction. A straightforward calculation from eq. (3) results in

(1) r=

sL

J+(x) tr(x) c(x)

0

*L

r=

(2)

l-&?&V

with tL = L/c = resonator single pass time. Eq. (2) is an approximation of eq. (1) for small losses, i.e. 1-&&I/ < 1, but eq. (1) cannot be valid in any case. If && I/< l/e (losses 63%), the decay time becomes smaller than the single pass time tL, which is impossible for empty resonators. It will be shown that the decay time depends on the type of resonator (passive, active, homogeneous or inhomogeneous) and that eqs. (1) (2) are lower or upper limits respectively for the decay time. The decay time may be defined as [4] : q

_E

T=-dqldt-P

’

with q the number of photons, stored inside the re440

X (l-R&f+(L)

dx

+ (1 -R1)J-(0)

L •t

_I-

(Y(x)[J+(x)+J-(x)]dX

0

-1

1

.

(4)

The first two terms in the denominator indicate the mirror losses, the third term takes into account the resonator losses o(x). Intensities J’(x), phase velocity c(x) and loss constant o(x) may be functions of x, if the cavity includes inhomogeneous elements. Eq. (4) holds for any type of resonator, active and passive, assuming steady state, e.g. dq/dt < tL. Otherwise the cavity cannot be treated as a resonator [5]. In the following section the decay time given by eq. (4) will be evaluated for some special resonators, Passive resonators. The intensities J’ inside the resonator are related to the input intensity Jo by

Volume

23, number

December

OPTICS COMMUNICATIONS

3

1977

WL b

RI -

IO

Jo

0 /

x 6 Fig. 1. The general type of an optical resonator with amplifier G and two mirrors R 1, R2. s(x), J-(x): Intensities of the two travelling waves.

J+(x) --~ JO

_ Tl

J-(~)

e-“X

N

’

_= Jo

T,

V2R2 eaX N

0

including distributed losses V = exp(-cuL). TI is the mirror transmission and N the well known resonance term

Fig. 2. The resonator

0.5

for a homogene-

l+VR, V-l --rCTL ln V l_ V2RIR2’

(a) active resonator

decay

time r as a function of reflectivity to the single pass time fL = L/c. with I = L, (b) passive resonator.

(6)

Only in the low loss region 1 - V2R,R2 Q 1, eq. (5) can be approximated by eq. (2). Two points are remarkable: - The relation is not symmetric in the reflectivities R 1, R2, as the input intensity is coupled in an asymmetric way. - The decay time of the empty resonator (V = 1) is never below the single pass time tL. A reasonable result in contrast to eq. (1). Active, homogeneous resonators. More interesting is the decay time of active systems with an amplifying medium of length I and gain factor G = exp (gl). We assume (Y= 0, V = 1 and stationarity G2R,R2

= J+(O) R, G2 e-G. we ob-

tL

00)

lnm

1 +R2 l-RIR,’

J-(x)

Together with the condition of stationarity, tain from eq. (4) for the decay time .T=-

which reduces for the empty resonator (V = 1) to r=tL

RI

RI for V = Ra = 1, normalized

J+(x) = J+(O) es, From eq. (4) we obtain immediately ous resonator

I

The decay time of an active system can become much smaller than the single pass time. This is a consequence of the intensity J’ which increases exponentially with x and has its maximum at the mirror R 1 or R, respectively. Eq. (10) holds for high gain lasers with low reflectivities, e.g. direct coated Nd-YAG lasers or diode lasers [2,3]. Active, inhomogeneous resonator I< L. The relation for the decay time becomes very complex, if the cavity length L is not equal the length of the amplifier. But for a typical solid state laser system, I
= R, G J+(O),

J- (0) = R2 G2 J+(O), neglecting internal

= 1.

The gain constant may not depend on x. If the system is homogeneous, i.e. cavity length L equal amplifier length Z, the intensities are

J-(L)

7 = tL

losses (V = 1). Eq. (4) gives:

(a/L)~2(1+R1)+(b/L)tlR7(1+R2) G(l-RI)+fi(l-R2)

.

(“)

The decay time now depends on the position of the 441

OPTICS COMMUNICATIONS

Volume 23, number 3 active medium. is approximately

7 = tL

(2a/L)~&

Usually,

the reflectivity

of one mirror

100% and eq. (11) result in

(l-R.21

*

(12)

a=O, e.g. amplifier near the high reflecting mirror, has a maximum. A symmetric resonator with RI = R2 = R gives

442

limit R --f 1 can be ap-

+ (b/L) (1 +R2)

It is easily shown, that the decay for the position

___1 +R ‘= tL 2(] -R)’

which in the high reflectivity proximated by eq. (2).

December 1977

(13)

References [I] A. Yariv, Quantum electronics (Wiley, New York, 1975) Second Edition, p. 142. [2] Y.J. Kaufmann and V.P. Oppenheimer, Appl. Optics 13 (1974) 374. [ 3 ] W.W. Rigrod, Appl. Physics 36 (1965) 2487. [4] A.F. Harvey, Coherent light (Wiley Interscience 1970) p. 201. [5] A.A. Griitter and H. Weber, Z.f. angew. Mathem. u. Physik 22 (1971) 465.