Laser statistics and cavity length

Volume 22, number 3

OPTICS COMMUNICATIONS

September 1977

LASER STATISTICS AND CAVITY LENGTH* L. ALLEN *, C.-Y. HUANG and L. MANDEL Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA Received 4 July 1977

The effect on the optical field of changing the cavity length of a laser, while keeping the resonance frequency constant, is examined. It is shown that the fully quantized laser theory gives explicitexpressions for the length dependence of the light intensity distribution, that are not shared by the semiclassical theory, and are quite different above, at, and below threshold. Some possible experiments are suggested.

1. Introduction Mthough most of the predictions of the laser theory have now been subjected to quite searching experimental tests, the question how the properties of the emitted light depend on cavity size seems not to have been explored in detail. Yet this question happens to be particularly interesting, because it reflects an important difference between the fully quantized [1] and the semiclassical [2] laser theories. The difference is related to the fact that when the electromagnetic field is quantized in terms of the modes of the optical cavity, the coupling constant between atoms and field is explicitly dependent on cavity length, whereas no such explicit dependence appears in the semiclassical theory. As a result, the two theories do not necessarily make the same predictions for the effect of varying the cavity length. It is of course true that the augmented semiclassical theories [3], in which random noise sources are introduced in an ad hoc manner to simulate the effects of spontaneous emission, can be made almost equivalent to the quantum theory. But this is achieved by the somewhat arbitrary adjustment of a parameter, and does not follow from basic principles as in the fully

quantized laser theory. In the following we examine the predicted effects of varying the cavity length on the statistical properties of the light emitted by a laser. In order to illustrate the problem we consider the somewhat idealized model of a laser illustrated in fig. 1, in which the cavity mirrors M 1 and M 2 are flat, or almost flat, and the active medium S occupies only a small fraction of the available cavity volume. We suppose the laser is oscillating in a single longitudinal mode of frequency v. We now ask how the optical field changes in intensity and in its fluctuation properties when the mirror M 2 is displaced longitudinally to a new position M3, such that the oscillation frequency v remains unchanged. Superficially it might seem that neither the light intensity nor its fluctuations would be affected by such a displacement. In fact, both are expected to change, in general, as we shall show.

S * Work supported in part by the National Science Foundation and by the Air Force Office of Scientific Research. * Permanent address, School of Mathematical and Physical Sciences, University of Sussex, England.

MI

I

Mz

I I I

M3

Fig. 1. Outline of the laser system.

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September 1977

2. Semiclassical theory

3. Quantized field theory

We start with Lamb's semiclassical expression for the steady state light intensity in the laser cavity well above threshold. The square of the electric field E is given by the formula [2]

Tire fully quantized theory of the laser as given by Scully and Lamb [1 ] leads to the following expression for th~ probability distribution p(n) of the number of laser photons n, tl

-1/QAV+ (~ 2/eohku)Z i (p-co)

E2 =

C

. (1)

p(n) =~- p(0) l-I

(X/'~/8)(~4/eOh3 TaTbkU)[l+'),2abd~ (u cJ)l

(2)

in which k is the wave number of the light, l is the cavity length, and L is the fractional loss per cavity transit. The number AVis an average atomic excitation density and is defined by l

R - - l f N(z)dz,

(3)

0 where N(z) is the density of excited atoms at position z in the cavity, and the integral ;s to be taken over the entire cavity length. ~ is the transition atomic dipole moment, u is the root mean squared atomic velocity, 7a and 7b are decay rates for the atomic levels a and b with level spacing hco taking part in laser action and Tab is their mean..6? (u) is the Lorentzian function Z?(u)-- 1/(Ta2b + u2),

(4)

and Z i (u)is the imaginary part of the dimensionless plasma dispersion function [2]. Of the various terms in eq. (1) only Q and AYdepend on the cavity length L However, if the integral in eq. (3)is independent of/because the active atoms are confined to a small region o f the cavity, and if the loss L is dominated by reflection losses and is largely independent of cavity length, then it follows from eqs. (2) and (3) that the product QA7 is independent of cavity length. According to the Lamb theory, the light intensity should therefore be independent o f cavity length l also. Needless to say, this conclusion is restricted to the region well above the laser threshold, to which eq. (1) applies. The theory is silent on the question o f the corresponding light intensity fluctuations, and on the behavior at and below threshold.

252

(5)

Here A, C, B are gain, loss and saturation parameters having the dimensions of frequency, which, on resonance, are given by

Here Q is the Q (quality factor) of the cavity, that may also be expressed in the form

Q = kl/L,

A_A~BC

-r=0 r + A/B "

A = 2 R a (g2/'),a3,b)

(6)

B = 8 Ra (g2/Ta'),b) (g2/')'a')'ab )

(7)

c = v/Q.

(8)

Here R a is the rate at which excited laser atoms in the upper state a are being produced, and g is the coupting constant between an atom and the field, which is inversely proportional to the square root of the quantization volume. More explicitly

g= ~(co/2~%IA) 1/2,

(9)

where A is the effective transverse area of the optical cavity. The loss parameter C equals the cavity bandwidth, and, with the help ofeq. (2), may also be expressed in the form

C = cL/27rl.

(10)

It follows from eqs. (6), (9) and (10) that the dimensionless ratio A/C, that is unity at threshold and greater or less than unity above or below threshold, respectively, is independent of cavity length l. However, the large dimensionless number A/B is proportional to l. At this stage it is convenient to introduce two new parameters. Thus we put

A/B = Ta3'ab/4g 2 -- N 2 ,

(11 )

where N O is a large number that will be seen to be of order of the mean photon number at threshold, and we introduce a pump parameter a by putting

A/C = exp (a/xf2No).

(12)

Evidently a will be zero at threshold and greater or less than zero above or below threshold, respectively.

Volume 22, number 3

OPTICS COMMUNICATIONS

In practice the gain and loss parameters A and C are not too different in magnitude, so that the ratio A/C is generally in the neighborhood of unity and lal < N 0. From the foregoing it then follows that N O cc X/7, and, since

(13)

A/C is independent of l, that

a cc ,v/[.

(14)

With the help of the new parameters a and NO, and by making use of the Stirling expansion for the factorial in eq. (5), with the assumption n < N 2, we may readily re-express eq. (5) in the analytically simpler form [4] t_

2

-

"

(15)

K 1 is a constant to ensure the normalization o f p ( n ) . When a = 0 the mean value o f n is approximately (2/rr)l/2No, which shows that N O is a measure of the average photon number at threshold. In order to investigate the dependence of the optical field on cavity length l, we observe that the light intensity I is proportional to the density of photons n/lA, or more precisely,

I = en/lA,

(16)

where intensity I is expressed in photons per unit area per unit time. Hence the corresponding probability distribution P(I) is given by P ( I ) = K 2 exp

r [

c 0" 7 U0 y1

L - ~ I - j ~ l A ) / 2 \ / - ~ - , _],

(17)

where K 2 is another normalizing constant. We may look on I as an almost continuous variable, and we observe that eq. (17)has the form of the well known intensity distribution that is given by the augmented semiclassical theories [3]. We now examine the implications of this formula. Well above threshold (a >> 1), eq. (17) has the form of a gaussian distribution in I, with mean and variance given by


(18)

and <(AI)2) =

(cNo/lA)2.

September 1977

exactly as predicted by the semiclassical theory in section 2. On the other hand, the root mean squared intensity fluctuation (> 1. In that case large values of the light intensity I are very improbable, and eq. ( 1 7 ) m a y be approximated by the expression

P(1) = K 3 exp ( - [ a l l A l / x / 2 c X o ) ,

in which K 3 is another normalizing constant. This is an exponential distribution in I, and is characteristic of thermal radiation [e.g. 5]. The mean light intensity and its variance are now given by

( I) = x/2cNo/ lallA,

From eqs. (13) and (14) it then follows that the mean light intensity
(21)

and <(AI)2> =

2(cNo/lallA)2"

(22)

From eqs. (13) and (14) we now find that both the average light intensity and the root mean squared intensity fluctuation (((AI)2))1/2 vary inversely with the cavity length l, so that their ratio is independent of/. Of course below threshold spontaneous emission dominates over stimulated emission, and the falling off of the light intensity with cavity size may be regard. ed as a reflection of the fact that the phase space available for spontaneous emission into the cavity mode decreases with increasing/. Finally, we examine the intermediate case in which the laser is precisely at threshold (a = 0). From eq. (17) we then find

p(l) = K 2 exp ( - I212A2/2c2N2),

(23)

from which it follows that = (2/7r) 1/2 (cNo/IA)

(19)

(20)

(24)

and ((51) 2 > = ( 1 - 2 H r )

(cNo/lA)2.

(25) 253

Volume 22, number 3 I

OPTICS COMMUNICATIONS I

I

V

September 1977 :

O~)

I

v A

a:4

<~ v V "6 g g

I

O~OO -

O=0 o=1 0=2

"5

g o5-

o~

Y

IO

I

I

125

1.5

I 175

2.Ù

IO

Relative Change of Length

I

I

125

15

t 175

20

Relative Change of Length

Fig. 2. The variation of the light intensity with cavity length for various initial working points of the laser.

Fig. 3. The variation of the light intensity fluctuations with cavity length for various initial working points of the laser.

According to eqs. (13) and (14), both the mean light intensity and its root mean squared fluctuation are now proportional to 1/x/~, so that their ratio is again independent of cavity length l. Thus, at the threshold, the mean light intensity varies with l in a manner that is intermediate between the above and below threshold situations, but the intensity fluctuations depend on l as they do above threshold. From eq. (17) we may readily obtain expressions for the mean light intensity ( I ) and its moments in general, as has been shown by Risken [6]. We find

sumption throughout is that the cavity resonance frequency does not change and that all laser atoms are exposed to essentially the same laser field. It will be seen that there is a gradual change in the length dependence o f the optical field from below to above threshold. These conclusions should lend themselves to detailed experimental test. By operating a laser in the neighborhood of its threshold with the help of a feedback arrangement, and measuring the probability distribution of the photoelectric pulses registered by a photodetector for various lengths, we should be able to test the validity of eqs. (26) and (27).

•

cN 0

(17 =~A-

F

Lo

+

2 exp ( - ~ 1a Z2) 7 (26) x/~[ 1+ eft½ a ] J ' References

and ( ( A / ) 2) = I ×

(cNo/lA)2 a e x p ( _ ~1 a2)

1 - - -/ ~ [ l + e r f l a- ]

2 e x p ( - 2! a 2 ).~, lr[l+erf½a]2 ]

(27)

with acc M'/-and N Occ ~ Needless to say, by varying l we shall generally change the working point of the laser also, although the laser will remain above or below threshold, as the case may be. Figs. 2 and 3 show the predicted effect on the mean light intensity and on the intensity fluctuations o f varying the cavity length from an initial value l to a final value 21, for various initial pump parameters a. The as-

254

[1] M.O. Scully and W.E. Lamb Jr., Phys. Rev. 159 (1967) 208. [2] W.E. Lamb Jr., Phys. Rev, 134 (1964) A1429. [3] H. Risken, Zeits. F. Physik 191 (1965) 186; H. Risken and H.D. Vollmer, Zeits F. Physik 201 (1967) 323; R.D. Hempstead and M. Lax, Phys. Rev. 161 (1967) 350. [4] J.A. Abate, H.J. Kimble and L. Mandel, Phys. Rev. A14 (1976) 788. [5] L. Mandel and E. Wolf, Rev. Mad. Phys. 37 (1965) 231. [6] H. Risken, in: Progress in optics, Vol. 8, ed. E. Wolf (North-Holland, Amsterdam, 1970) p. 239; H. Haken, Laser theory, from Handbuch der Physik, ed. S. Fliigge (Springer, Heidelberg, New York, 1970).