- Email: [email protected]
Photon counting distributions for interacting laser modes
V,-~t~rr,,~. ~mb¢~ I
OPTICS COMMUNICATIONS
September 1973
PHOTON COUNTING DISTRIBUTIONS FOR I N T E R A C T ~ G LASER MODES W. WONNEBERGERand J. LEMPERT Fakult~t fiir PhysOt der Universit~t, ~'eiburg i. Br.. Germany Received 5 June 1973
T'he pholtm tour ling distributions obtahted from the beta-distribution for the intensity of interacting waves are ~ - , ~ _ , ~ to de~o'ibe tt, e inten~W fluctuations in a single laser mode which is intensity-coupled to other equivalent : ~ , ~ s c,f a rnultimoce laser. Numerical results are given for a twe-mode laser including dead-time corrections for ¢x = ~ . ~ counting-inter,,ai- dead-time ratios.
amplatude fluctu~tiom of a single laser mode setim~led from the output of a multhnode laser differ from tlhoR of a single mode laser (cf. [1 ]) due to interac~ bet~x.en the modes. In ref. 121 it was predicted and experimentally verified ~ a t for a nonresonant feedback laser an in~e~D mte~acti~ between the modes thermalizes ea
than one for a laser far above fllreshold. For N~n ,ON m ~, 1, Q(~,) factorizes into ( 1 +aX) -Nm. Therefore, the N m investigated modes appear thermal and independent in spite of the fact that all modes are statistically dependent. For N m ~ N m , the strong correlations between the modes enter in an analogous manner as described in [21 for N m ~, 1. Eq. (2), therefore, interpolates between these extremes. In the original derivation [3] 9f eq. (2), starting from a specific interaction model, one should have N m ~ 10. However, eq. (2), being associated with the beta-distribution, physically is a good ,oproxhnation for all values o f N m provided i'-.2, m~- ,aable to Fred at some instant of time the mean tot;' output intensity (/max) = Nma already in the N~ moies investigated. This situation is effectively ruled out as long as
((zX/)2>= N
m ( N m -- N m ) a 2
/(Nm +
I)
(l)
~--.|
'~ (/max-/) 2 = ( N ~here : , is the scaled intensity [5] of the lath mode, th* ~nerating funcuon Q(M = ( e x p ( - L 0 > may be eva~aared and is shown in eq. (2):
tF:
(2)
m -
Nm)2a 2
giving (Nm -
+ 1).
(3)
This condition restricts the possible combinations of N m and N m values. The precise form of the interaction
Here. 1F~ ~ the confluent hypergeometric function. The p~lmpmg parameter a [5] is usually much larger
associated with eq. (2) under such nonasymptotic conditions is not known at present. It is expected to be
OPTICS COMMUNICATIONS
Volume 9, number 1
some type of intensity interaction with "cut off" between equivalent modes. We propose to use eq. (2) for the description of the amplitude fluctuations in a sLngle laser mode of (say) a five-mode laser with normal multimode interaction which excludes mode locking. For a combination like ( N m = 1, N m = 5) eq. (2) predicts substantially different results compared to the thermal case as well as to the amplitude stabilized case. These differences are most easily seen in photon counting experiments [6-9]. Therefore, we have computed numerical results for the expected photon counting distribution of a single laser mode coupled to four other iasing modes. For N m = 1, the recurrence relation for the counting probability p(n, T < %) (counting interval T smaller than intensity correlation time ¢c) associated with eq. (2) reads (cf. [3]):
p(n_l)_Nm + n - 1 +Nm(n ) p(n) Nm(, >
n+ 1 Nm(n)P(n+ l). (4)
This equation permits convenient numerical evaluation ofp(n). For a realistic experimental situation, the counting interval T must be rather small, presumably T ~ 10 -7 see. Since the minimum dead time of the photon counter will be about 10 -8 see, dead-time corrections must be included in the evaluation of eq. (4). The theory of dead-time corrections for nonparalyzable counters in the case T < r e has been given by B6dard [ lO]. According to his work, the corrected distribution function p(n, T, r, (n)c) (r dead time, (n)c corrected mean number of counts) for n >_ l is given by
p(n, T, r,
f ., 0
For n = 0, P(/) is the with Q(~) defined in
#
"fin, ~ =/" ,ix x n- t e-X.
,~n-l)_7(n+l,~n) n!
)
(6)
o
The quantity l~n Is def'med in eq. (7): ~n = a T / ( l
(7)
-nr/T) - vnl.
The coefficient a measures the quantum efficien~v of the detector. Note, that
(n)= vO(1)-=-(n)0 and in addition, we def'me
n = vn (I). One would like to expressp(n. T, ¢, (n) c) solely in terms of uncorrected distribution functions p(n, T, 0, (n)) for appropriately chosen mean values (n). This is accompit~ed by the following transformations:
~,(n, ~) = vn f dx x n - l e-~X 0 =p
Uging the generating function Q(~)= (exp(-M)), one obtains
p(n, T, r, (n)c) n-I
- v("n -- Il ) ! (_vn_, a ) Pn
d/P(/) 7_ " , (n-i)!
Sepmmber t973
l -Q(X) x ) [ ~=,,,,_~
rl
(5)
one has of course p(0, T, r) = p(0, T, 0). intensity distribution function associated and 7 is the incomplete gamma-function eq. (6):
The n-fold derivatives are evaluated as follows. With
(-valax),, f(x) - ~ . l one has (!/,~)[nl = vnn! x-Cn~I).
Volume 9, number 1
OPTICS COMMUNICATIONS
September 1973
Table 1 Photon counting distributions p (n) for a single mode of a five-mode laser with intensity mode coupling. The distributions p (n) are derived from the beta-distribution for the mode intensity and corrected for several ratios of short counting interval T to dead time r. (n >is the uneon-~ted number of mean counts. 4 .
.
.
.
.
.
5 .
6 -
7
8
_
_
9 .
.
.
.
.
.
10 .
.
.
.
.
.
.
.
.
0
0.1424
0.1697
0.1894
0.2040
0.2152
0.2242
0.2315
1
0.1957
0.2053
0.2106
0.2137
0.2157
0.2171
0.2180
2
0.2696
0.2391
0.2222
0.2113
0.2037
0.1981
0.1938
3
0.3042
0.2387
0.2053
0.1856
0.1728
0.1637
0.1570
4
0.0881
0.1354
0.1346
0.1271
0.1201
0.1144
0.1099
0.0118
0.0369
0.0514
0.0576
0.0601
0.0609
0.0010
0.0068
0.0140
0.0195
0.0234
0.0001
0.0009
0.0028
0,0051
0.0000
0.0001
0.0004
0.0000
0.0000
5 6 •7 8
9 10
0.0000
(n>
4.996
4.442
3.868
3.288
Using tS~ relation, the following identity is valid: M
(Q(]k)I~k)[nl= m= 0~ pmm, (~)
~k-(m+l' Q(~k)[n-ml.
According to Mandel's formula [ 11 ]:
[n-ml t = (n-m)! p(n-m, T, O, (n )n) ,
Q(~,) l~=pn
(8)
one, therefore, ends up with the relation r/
p(n, 7, r, (n)c) = m~0 p(n-m, T, O, (n)n) n-1
- ~_t p ( n - m - l , T , O , (n>n l)" (9) rrt---'0 With the definition 2~ l = 0, this relation is valid for all 0 ~
3.081
2.931
2.817
positive it must be replaced by zero and the associated uncorrected counting distribution is p(n) = 5n, O" Using eqs. (4) and (9), numerical results were computed for N m = 5, (n)c = 2000, and several values for the ratio T/¢ of counting interval T to dead time r. They are given in table 1. For comparison, the corresponding values for a thermal (or thermalized [6, 12]) mode are given in table 2. The Bose-Einstein counting distribution appropriate for this case was simulated by settingN m = IO s in eq. (4). It is seen, that the counting distribution of a single laser mode intensity.coupled to only four other equivalent modes is similar to the Bose-Einstein distribution in contrast to the case of a single "free" laser mode for which the appropriate counting distri. bution is similar to the Poisson distribution [ l ]. In order to use the X2 statistical test to discriminate experimentally between the two cases in tables 1 and 2, a sufficiently large number of samples is necessary. This number should be of the order 104.
Volume 9, number 1
OPTICS COMMUNICATIONS
September 1973
Table 2 IPhoton counting disUibutions p (n) of a thermal light mode during short counting interval T. The distributions p (n) include dea6 time corrections for several ratios of counting interval T to dead time ~-. (n) is the uncorrected number of mean counts. _
4
.
..
.
.
.
.
.
.
.
.
.
.
5
.
. .
.
.
.
.
.
6 .
.
.
.
.
.
.
.
.
.
.
.
.
.
..
7 .
.
.
~
.
.
.
.
.
.
.
.
.
.
.
.
9 ..
8 ,
10 .
.
.
.
.
.
.
.
.
.
.
.
0
0.1546
0.1841
0.2056
0.2217
0.2343
0~244 3
0.2524
1
0.1978
0.2070
0.2120
0.2149
0.2167
0.2179
0.2188
2
0.2530
0.2244
0.2083
0.1979
0.1906
0.1~52
0.1811
3
0.2821
0.2168
0.1853
0.1671
0.1552
0.1468
0.1406
4
0.1125
0.1446
0.1322
0.1201
0.1113
0.1048
0.0998
0.0231
0.0530
0.0625
0.0640
0.0633
0.0621
0.0036
0.0153
0.0242
0.0290
0.0314
0.0005
0.0037
0.0079
0.0114
0.0000
0.0008
0.0023
0.0000
0.0001
5 6 7 8
9 0 (n )
, .
,
,
_
0.0000 5.504
4.086
3.868
Acknowledgements The authors are indebted to Dr. J. Herrington and Dr. W. Wettling for helpful discussions.
References [ 1] M. Sargent !I1 and M.O. Scully, in: Laser Handbook, F.T. Arecchi and E.O. Schulz-DuBois, eds. (NorthHolland, Amsterdam, 1972), oh. A2, p. 45, H. Haken, ibid., ch. A3, p. 115; F.T. Arecchi and V. Degiorgio, ibid., ch. AS, p. 191. [2] R.V. Ambartsumyan, P.G. K~yukov, V.S. Letokhov, and Yu.A. Matveets, Zh. Eksp. Teor. Fiz. 53 (1967) 1955 (JETP 26 (1968) 1109).
3.512
3.269
3.094
2.962
[3] W. Wonneberger and J. Lempert, Phys. Letters 43A (1973) 75 and Z. Naturforsch. 28a (1973) 762. [4] W. Brunner and H. Paul, Ann. Physik 23 (1969) 152 and 384. tSl H. Risken, Z, Phys. 186 (1965) 85. [61 F.T. Arecchi, Phys. Re,,,. Letters 15 (1965) 912. [7] C. Freed and H.A. Haus, Phys. Rev. Letters 15 (1965) 943. [Sl W.A. Smith and J.A. Armstrong, Phys. Rev. Letters 16 (1966) 1169. 191 F.T. Azecchi and V. Degiorgio in ref. [ 1 ]. [to] G. B~dard, Proc. Phys. Soc. 90 (1967) 131. 111] L. Mandel, Proc. Phys. Soc. 72 (1958) 1037. 112] W. Martienssen and E. Spiller, Am. J. Phys. 32 (1964) 919.
OPTICS COMMUNICATIONS
September 1973
PHOTON COUNTING DISTRIBUTIONS FOR I N T E R A C T ~ G LASER MODES W. WONNEBERGERand J. LEMPERT Fakult~t fiir PhysOt der Universit~t, ~'eiburg i. Br.. Germany Received 5 June 1973
T'he pholtm tour ling distributions obtahted from the beta-distribution for the intensity of interacting waves are ~ - , ~ _ , ~ to de~o'ibe tt, e inten~W fluctuations in a single laser mode which is intensity-coupled to other equivalent : ~ , ~ s c,f a rnultimoce laser. Numerical results are given for a twe-mode laser including dead-time corrections for ¢x = ~ . ~ counting-inter,,ai- dead-time ratios.
amplatude fluctu~tiom of a single laser mode setim~led from the output of a multhnode laser differ from tlhoR of a single mode laser (cf. [1 ]) due to interac~ bet~x.en the modes. In ref. 121 it was predicted and experimentally verified ~ a t for a nonresonant feedback laser an in~e~D mte~acti~ between the modes thermalizes ea
than one for a laser far above fllreshold. For N~n ,ON m ~, 1, Q(~,) factorizes into ( 1 +aX) -Nm. Therefore, the N m investigated modes appear thermal and independent in spite of the fact that all modes are statistically dependent. For N m ~ N m , the strong correlations between the modes enter in an analogous manner as described in [21 for N m ~, 1. Eq. (2), therefore, interpolates between these extremes. In the original derivation [3] 9f eq. (2), starting from a specific interaction model, one should have N m ~ 10. However, eq. (2), being associated with the beta-distribution, physically is a good ,oproxhnation for all values o f N m provided i'-.2, m~- ,aable to Fred at some instant of time the mean tot;' output intensity (/max) = Nma already in the N~ moies investigated. This situation is effectively ruled out as long as
((zX/)2>= N
m ( N m -- N m ) a 2
/(Nm +
I)
(l)
~--.|
'~ (/max-/) 2 = ( N ~here : , is the scaled intensity [5] of the lath mode, th* ~nerating funcuon Q(M = ( e x p ( - L 0 > may be eva~aared and is shown in eq. (2):
tF:
(2)
m -
Nm)2a 2
giving (Nm -
+ 1).
(3)
This condition restricts the possible combinations of N m and N m values. The precise form of the interaction
Here. 1F~ ~ the confluent hypergeometric function. The p~lmpmg parameter a [5] is usually much larger
associated with eq. (2) under such nonasymptotic conditions is not known at present. It is expected to be
OPTICS COMMUNICATIONS
Volume 9, number 1
some type of intensity interaction with "cut off" between equivalent modes. We propose to use eq. (2) for the description of the amplitude fluctuations in a sLngle laser mode of (say) a five-mode laser with normal multimode interaction which excludes mode locking. For a combination like ( N m = 1, N m = 5) eq. (2) predicts substantially different results compared to the thermal case as well as to the amplitude stabilized case. These differences are most easily seen in photon counting experiments [6-9]. Therefore, we have computed numerical results for the expected photon counting distribution of a single laser mode coupled to four other iasing modes. For N m = 1, the recurrence relation for the counting probability p(n, T < %) (counting interval T smaller than intensity correlation time ¢c) associated with eq. (2) reads (cf. [3]):
p(n_l)_Nm + n - 1 +Nm(n ) p(n) Nm(, >
n+ 1 Nm(n)P(n+ l). (4)
This equation permits convenient numerical evaluation ofp(n). For a realistic experimental situation, the counting interval T must be rather small, presumably T ~ 10 -7 see. Since the minimum dead time of the photon counter will be about 10 -8 see, dead-time corrections must be included in the evaluation of eq. (4). The theory of dead-time corrections for nonparalyzable counters in the case T < r e has been given by B6dard [ lO]. According to his work, the corrected distribution function p(n, T, r, (n)c) (r dead time, (n)c corrected mean number of counts) for n >_ l is given by
p(n, T, r,
f ., 0
For n = 0, P(/) is the with Q(~) defined in
#
"fin, ~ =/" ,ix x n- t e-X.
,~n-l)_7(n+l,~n) n!
)
(6)
o
The quantity l~n Is def'med in eq. (7): ~n = a T / ( l
(7)
-nr/T) - vnl.
The coefficient a measures the quantum efficien~v of the detector. Note, that
(n)= vO(1)-=-(n)0 and in addition, we def'me
~,(n, ~) = vn f dx x n - l e-~X 0 =p
Uging the generating function Q(~)= (exp(-M)), one obtains
p(n, T, r, (n)c) n-I
- v("n -- Il ) ! (_vn_, a ) Pn
d/P(/) 7_ " , (n-i)!
Sepmmber t973
l -Q(X) x ) [ ~=,,,,_~
rl
(5)
one has of course p(0, T, r) = p(0, T, 0). intensity distribution function associated and 7 is the incomplete gamma-function eq. (6):
The n-fold derivatives are evaluated as follows. With
(-valax),, f(x) - ~ . l one has (!/,~)[nl = vnn! x-Cn~I).
Volume 9, number 1
OPTICS COMMUNICATIONS
September 1973
Table 1 Photon counting distributions p (n) for a single mode of a five-mode laser with intensity mode coupling. The distributions p (n) are derived from the beta-distribution for the mode intensity and corrected for several ratios of short counting interval T to dead time r. (n >is the uneon-~ted number of mean counts. 4 .
.
.
.
.
.
5 .
6 -
7
8
_
_
9 .
.
.
.
.
.
10 .
.
.
.
.
.
.
.
.
0
0.1424
0.1697
0.1894
0.2040
0.2152
0.2242
0.2315
1
0.1957
0.2053
0.2106
0.2137
0.2157
0.2171
0.2180
2
0.2696
0.2391
0.2222
0.2113
0.2037
0.1981
0.1938
3
0.3042
0.2387
0.2053
0.1856
0.1728
0.1637
0.1570
4
0.0881
0.1354
0.1346
0.1271
0.1201
0.1144
0.1099
0.0118
0.0369
0.0514
0.0576
0.0601
0.0609
0.0010
0.0068
0.0140
0.0195
0.0234
0.0001
0.0009
0.0028
0,0051
0.0000
0.0001
0.0004
0.0000
0.0000
5 6 •7 8
9 10
0.0000
(n>
4.996
4.442
3.868
3.288
Using tS~ relation, the following identity is valid: M
(Q(]k)I~k)[nl= m= 0~ pmm, (~)
~k-(m+l' Q(~k)[n-ml.
According to Mandel's formula [ 11 ]:
[n-ml t = (n-m)! p(n-m, T, O, (n )n) ,
Q(~,) l~=pn
(8)
one, therefore, ends up with the relation r/
p(n, 7, r, (n)c) = m~0 p(n-m, T, O, (n)n) n-1
- ~_t p ( n - m - l , T , O , (n>n l)" (9) rrt---'0 With the definition 2~ l = 0, this relation is valid for all 0 ~
3.081
2.931
2.817
positive it must be replaced by zero and the associated uncorrected counting distribution is p(n) = 5n, O" Using eqs. (4) and (9), numerical results were computed for N m = 5, (n)c = 2000, and several values for the ratio T/¢ of counting interval T to dead time r. They are given in table 1. For comparison, the corresponding values for a thermal (or thermalized [6, 12]) mode are given in table 2. The Bose-Einstein counting distribution appropriate for this case was simulated by settingN m = IO s in eq. (4). It is seen, that the counting distribution of a single laser mode intensity.coupled to only four other equivalent modes is similar to the Bose-Einstein distribution in contrast to the case of a single "free" laser mode for which the appropriate counting distri. bution is similar to the Poisson distribution [ l ]. In order to use the X2 statistical test to discriminate experimentally between the two cases in tables 1 and 2, a sufficiently large number of samples is necessary. This number should be of the order 104.
Volume 9, number 1
OPTICS COMMUNICATIONS
September 1973
Table 2 IPhoton counting disUibutions p (n) of a thermal light mode during short counting interval T. The distributions p (n) include dea6 time corrections for several ratios of counting interval T to dead time ~-. (n) is the uncorrected number of mean counts. _
4
.
..
.
.
.
.
.
.
.
.
.
.
5
.
. .
.
.
.
.
.
6 .
.
.
.
.
.
.
.
.
.
.
.
.
.
..
7 .
.
.
~
.
.
.
.
.
.
.
.
.
.
.
.
9 ..
8 ,
10 .
.
.
.
.
.
.
.
.
.
.
.
0
0.1546
0.1841
0.2056
0.2217
0.2343
0~244 3
0.2524
1
0.1978
0.2070
0.2120
0.2149
0.2167
0.2179
0.2188
2
0.2530
0.2244
0.2083
0.1979
0.1906
0.1~52
0.1811
3
0.2821
0.2168
0.1853
0.1671
0.1552
0.1468
0.1406
4
0.1125
0.1446
0.1322
0.1201
0.1113
0.1048
0.0998
0.0231
0.0530
0.0625
0.0640
0.0633
0.0621
0.0036
0.0153
0.0242
0.0290
0.0314
0.0005
0.0037
0.0079
0.0114
0.0000
0.0008
0.0023
0.0000
0.0001
5 6 7 8
9 0 (n )
, .
,
,
_
0.0000 5.504
4.086
3.868
Acknowledgements The authors are indebted to Dr. J. Herrington and Dr. W. Wettling for helpful discussions.
References [ 1] M. Sargent !I1 and M.O. Scully, in: Laser Handbook, F.T. Arecchi and E.O. Schulz-DuBois, eds. (NorthHolland, Amsterdam, 1972), oh. A2, p. 45, H. Haken, ibid., ch. A3, p. 115; F.T. Arecchi and V. Degiorgio, ibid., ch. AS, p. 191. [2] R.V. Ambartsumyan, P.G. K~yukov, V.S. Letokhov, and Yu.A. Matveets, Zh. Eksp. Teor. Fiz. 53 (1967) 1955 (JETP 26 (1968) 1109).
3.512
3.269
3.094
2.962
[3] W. Wonneberger and J. Lempert, Phys. Letters 43A (1973) 75 and Z. Naturforsch. 28a (1973) 762. [4] W. Brunner and H. Paul, Ann. Physik 23 (1969) 152 and 384. tSl H. Risken, Z, Phys. 186 (1965) 85. [61 F.T. Arecchi, Phys. Re,,,. Letters 15 (1965) 912. [7] C. Freed and H.A. Haus, Phys. Rev. Letters 15 (1965) 943. [Sl W.A. Smith and J.A. Armstrong, Phys. Rev. Letters 16 (1966) 1169. 191 F.T. Azecchi and V. Degiorgio in ref. [ 1 ]. [to] G. B~dard, Proc. Phys. Soc. 90 (1967) 131. 111] L. Mandel, Proc. Phys. Soc. 72 (1958) 1037. 112] W. Martienssen and E. Spiller, Am. J. Phys. 32 (1964) 919.