Correlations of intensity fluctuations for threshold laser light with finite counting time

Chang, R. F.

Detenbeck, R. W.

Korenman, V.

English

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)
PHOTON STATISTICS FOR THRESHOLD LASER
LINT WITH FINITE COUNTING TIME
by
R. F. Chang
R. W. Detenbeck
V. Korenman
TECHNICAL REPORT NO. 788
January 1968
UNIVERSITY OF MARYLAND
DEPARTMENT OF PHYSICS AND ASTRONOMY
COLLEGE PARK, MARYLAND
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PHOTON STATISTICS FOR THRESHOLD LASER
LIGHT WITH FINITE COUNTING TIME*
by
R. F. Chang
R. W. Detenbeckt
V. Korenman
Department of Physics and Astronomy
University of Maryland, College Park, Maryland
*Work supported by NASA (grant NGR 21-002-022), ARPA (Contract SC-101),
and the U.S. Army Research Office (contract DAHC04 67 C 0023, under
Project Defender), and the University of Maryland Computer Science
Center under NASA grant NsG-398.
Department of Physics, University of Vermont, Burlington, Vermont.
Abstract
The 2nd, 3rd and 4th cumulants of the distribution
of light intensity from a He-Ne laser operating just
below threshold have been measured as counting time is
increased. The results agree with predictions com-
puted under the assumption that the dependence of the
correlation functions on the time variables is the same
as for Gaussian light.
PHOTON STATISTICS FOR THRESHOLD LASER LIGHT
WITH FINITE COUNTING TIMES*
Early measurements of correlations ofintensity fluctuations in
well stabilized He-Ne lasers operating near threshold reported only the
correlations of fluctuations all at the same time [1,2]. These determined
the probability density of instantaneous samples of the intensity (I.P.D.).
Recent studies [3,4] have provided the time dependence of the two-time cor-
relation function in the threshold region. All these results have been con-
sistent with current laser neories based on the Fokker-Planck equation with
Markoffian noise sources [5-10].
We add here measurements of the time dependence of the first four
normalized cumulants for counting times from 3 to 1000 microseconds, for a
laser operating slightly below threshold where the fluctuation time is about
100 microseconds. What is measured is an average over each of the time
variables of the correlation of intensity fluctuations at different times,
as a function of the averaging time, T. This can be considered as defining
an effective I.P.D. which is a function of T.
The T dependence of this effective I.P.D. is not determined by
its T - 0 limit, the usual I.P.D. In what follows we compare our results
with a time dependence computed under the assumption that the dependence
of the correlation functions on their time variables is that of Gaussian
light. The time dependence of higher correlations can also be computed
from the Fokker-Planck equation with Markoffian noise [5-10]. Unfortunately
all the coefficients needed for such a computation have not been calculated
*
Work supported by NASA (grant NGR 21-002-022), ARPA (Contract SD-101),
and the U. S. Army Research Office (contract DAHC04 67 C 0023, under
Project Defender), and the University of Maryland Computer Science Center
under NASA grant NsG-398.
2in the literature, but a preliminary study implies that for our conditions
the result would not be noticeably different from the simple assumption
above.
For Gaussian light the nth cumulant of the effective I.P.D. is [11,12]
J
T
O
Kn(T) = Kn(0) 	 Y(tl - t 2 Wt 2  - t3 ). . . Y(tn - t1 ) dtl .	 .dtn (1)
where Y is the normalized field auto-correlation function. The spectrum of
Y is Lorentzian in our region of operation, chosen to be below threshold in
order that Q3 and Q4 be sensibly different from zero. t For a Lorentzian
spectrum, following eq. (1), the cumulants are [13]
Q2 (T)/Q2 (0) = 2Z-1 + 2(e Z - 1)Z-2 ,	 (2a)
Q3 (T)/Q3 (0) = 6(e-Z + 1)Z
-2
 + 12(e-Z - 1)Z-3 ,	 (2b)
Q4 (T)/Q4 (0) - 8e-ZZ-2 + 20(2e Z + 1)Z
-3
 + 2(e-2Z + 28e-Z - 29)Z-4 , (20
where Z - rT, t is the half width at half maximum of the intensity power
spectrum and Qn(T) is the nth normalized cumulant [1].
Our measurements were made with the equipment and technique
reported previously [1], recording photoelectron distributions with various
counting times for laser intensities of 17% and 42% of the threshold value.
A least squares fit was made of the data Q2 (T) to (the logarithm
of) Eq. (2) using Q2 (0) and T as parameters. The same value of P was then
used for the Q3 (T) and Q4 (T) data, adjusting Q3 (0) and Q4 (0) for the best
fit. Fig. 1 shows data for Qn (T)/Qn (0) along with eq. (2).
tTheory [5,8] and experiment [3,4] agree that below threshold the spectrum
of Y is quite accurately a Lorentzian.
3The good fit to our simple ansatz above seems to imply that in
the region under study where the static intensity correlations are already
quite different from their Gaussian limitih the dynamic correlations can
still be described by the simple one-decay-time approximation appropriate
to Gaussian light. At higher intensities, closer to and above the threshold,
we expect deviations from this simple result.
41. R. F. Chang, R. W. Detenbeck, V, Korenman, C. 0. Alley and U. Hochuli,
Phys. Letters 25A (1967) 272.
2. F. T. Arecchi, G. S. Rodari and A. Sona, Phys. Letters 25A (1967) 59.
3. F. T. Arecchi, M. Giglio, and A. Sona, Phys. Lett. 25A (1967) 341.
4. F. Davidson and L. Mandel, Phys. Lett. 25A (1967) 700.
5. R. D. Hempstead and M. Lax, Phys. Rev. 161 (1967) 350.
6. M. Lax, Phys. Rev. 160 (1967) 290.
7. M. Lax and W. H. Louisell, IEEE J. Quantum Electronics 3 (1967) 47.
8. H. Risken, Z. Physik 186 (1965) 85 and 191 (1966) 302; see also V. Arzt,
H. Haken, H. Risken, H. Sauermann, Ch. Schmid, and W. Weidlich, Z. Physik
197 (1966) 207; and H. Risken and H. D. Vollmer, Z. Physik 201 (1967) 323.
9. M. Scully and W. E. Lamb, Phys. Rev. Letters 16 (1966) 853.
10. J. A. Fleck, Phys. Rev. 149 (1966) 309 and 322.
11. D. Slepian, Bell Syst. Tech. J. 27 (1958) 163.
12. L. Mandel, Progress in Optics II, et. E. Wolf (North-Holland Publ. Co.,
Amsterdam, 1963) p. 229.
13. G. Bddard, Phys. Rev 151 (1966) 1038.
5Fig. 1 Open circles are values of Q (T)/Q 
n 
(0) at 17%
n 
of threshold. Solid circles are at 42% of
threshold. Statistical standard deviations are
shown when larger than dot size. The curves
are the predictions of Eq. (2).
6Q 2 (1
0 2 (C
IC
I	 Q 3 (T
10
	 Q, (C
10
I	 Q4(T,
100 Q 4 (0
IT
lo-
10-
0
IT
10 -1	 1.0
	 10	 102
r 
Fi g. 1
0  C
02(c
1(
1 03(l
10 0 0((
10
I	 Q4(T
100 Q 4
 (C
10'
IT
10'
10
10'
10''	 1.0
	 10	 102
r 


Photon statistics for threshold laser light with finite counting time

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