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Intensity correlation time of an optical field
Volume 36, number 2
OPTICS COMMUNICATIONS
15 January 1981
INTENSITY CORRELATION TIME OF AN OPTICAL FIELD * L. MANDEL Dept. of Physics and Astronomy Rochester, NY 14627, USA
and Institute of Optics, University of Rochester,
Received 3 October 1980
A measure of the time for which intensity correlations persist in a stationary optical field, suitable for both classical and quantum fields, is proposed, and illustrated by several examples.
The intensity correlation time of an optical field is a measure of the time interval over which correlations of the intensity fluctuations persist. For a thermal field, for which phase and amplitude fluctuations are coupled, the reciprocal bandwidth of the spectral distribution would appear to provide a convenient measure. But even in this simple case difficulties sometimes arise, because the bandwidth does not always have an unambiguous interpretation [ 11. Attempts have been made to define the coherent time and the bandwidth of terms of the second moments [2] of the functions ly(‘)12 and #2(~), where y(r) is the normalized second order correlation function of the field amplitude, and 4(w) is its Fourier transform, the normalized spectral density. Alternatively, consideration based on photoelectric counting measurements have led to the expression [3]
slr(r)12dr
--oo
for the coherence time of thermal light, and this may differ substantially from the second moment in some cases [ 11. However, when we come to non-thermal fields, such as those produced by lasers and by quantum sources, entirely new problems arise, because the correlation times associated with phase fluctuations and with intensity fluctuations may differ by orders * This work was supported in part by the National Science Foundation and by the Air Force Office of Scientific Research.
of magnitude. The second order correlation function y(r) is then inappropriate for determining the intensity correlation time, and the intensity correlation fiurction has to be used. In general, there are at least two different time scales that can be associated with the intensity fluctuations of an optical field. The first, that we may denote by T1 , is a measure of the shortest time interval in which the light intensity changes appreciably, while the second, to be denoted by T,, measures the longest time interval over which appreciable correlations persist. In some cases the two may be equal, but this is not true in general. In the following we introduce a simple expression for T,, that is applicable to any stationary, ergodic field, classical or quantized, and is closely related to measurements with a photodetector. In the case of thermal light, T1 and Tc coincide, and Tc goes over into the expression J~_l~(r)12dt proposed earlier [3]. For the sake of generality we shall describe the field in quantum mechanical terms, in which the light intensity I(t) is a Hilbert space operator *, and ( ) denotes the quantum expectation value. However, I(t) may be regarded as a randomly fluctuating c-number function whenever the field is describable classically, and ( > then denotes the average over the ensemble. For the definition of T, we start from the idea that intensity correlations of an optical field do not persist over time intervals much longer than the intensity * Hilbert space operators are distinguished by the caret “. 87
Volume 36, number 2
OPTICS COMMUNICATIONS
correlation time r,. If we define a normalized ty correlation function by X(r) = ( 7: A&&
t T) : ),
intensi-
(1)
then X(r) dies away to zero once 171appreciably exceeds T,. We have written the quantum mechanical correlation function in normal order and in time order (indicated by : : and T), because in this form it is most closely related to photoelectric measurements [4-61. For many kinds of optical field X(0) has a magnitude of order unity, although it can be larger or smaller in principle. A(r) is of course zero for a strictly coherent field in a coherent state [4,6]. The condition that X(r) becomes very small once r appreciably exceeds T, can now be expressed in the form
1
lh(r)ldrg
r
lh(QldT,
(2)
15 January 1981
w=+o
+P2)ly(7)12,
(5)
where P is the degree of polarization and y(r) is the normalized second order correlation function. h(r) is always non-negative, and it follows from eq. (4) that
T, =
s -co
Ir@>12dr,
(6)
in agreement with the formula given previously
[ 1,3] for thermal light. It has been demonstrated in photon counting measurements that bunching of the counts extends over time intervals of this order [8]. (b) A single-mode laser operating in the neighborhood of the threshold of oscillation has an intensity correlation function that can be expressed in the form c93
U
Tc
and T, is the shortest time for which this inequality is valid. It should be noted that we need to take the modulus of h(r) under the integral because h(r) may be negative or oscillatory. Hence
fc
(7) with V,,, 2 0 and Zm V,,, = 1. This is again a positive function, and when it is substituted in eq. (4) we obtain for the intensity correlation time
_T 7 I h(r)1dr. s IX(T)ldT= _m c If we approximate I A( by a triangular function in the range -Tc to T,, we can evaluate the left-hand integral and obtain
Tc _f lh(0)1(1- ld/T,)d~*
-Tc
J IA(r)I dr _m
or m
T, =
s
Ih(W@)I dr.
_m
We shall take eq. (4) as the definition of the intensity correlation time T, for any kind of stationary, ergodie optical field, classical or quantized. Of course if h(r) = 0 for all 7, as for a completely coherent state of the field, the integral is not defined, but the concept of intensity correlation time then becomes meaningless. We now examine some special classes of fields, in order to illustrate the significance of the definition. (a) For thermal light that obeys the cross-spectral purity condition, we have 173 88
This is an average of the decay times l/X, weighted by the appropriate factors V,,, , and is therefore a reasonable measure. (c) For a laser that is oscillating well above threshold in two modes with frequencies WI and w2 independently, it may be shown that [lo], to a good approximation, h(7) = e+r
cos (w2 - w1)7,
(9)
where /3 is the effective linewidth of each mode. Evidently intensity correlations persist for a time interval of order l/p in this case, and the correlations change signs periodically. If we substitute this form of h(r) in the definition (4) we may readily show that 2
T, =
P2 + (a2 - wI)2
x
[
B+(W2-W1)c”sech2(oziiP,,,.
1
(10)
Volume 36, number 2
OPTICS COMMUNICATIONS
This is always of order l//3, whether ]w2 - wl] is larger than, equal to, or smaller than 0, and is therefore a reasonable measure of the correlation time. It should be noted that, had we omitted the modulus signs in the definition (4) of T,, we would have been led to the value 2fi/[p2 t (w2 - w~)~], which is much smaller than l/P when lo2 .- w1 I % /3, and is therefore not an appropriate measure of correlation time. (d) Finally, we consider the correlation properties of resonance fluorescence radiation, which has unmistakable quantum features that cannot be described classically. It can be shown that for the fluorescence emitted by a single atom in the presence of a coherent field on resonance, h(r) has the form [ 1 l]
X [cos(R’flr) t (3/2Sl’)sin(R’/Ir)], ST = (s22//?
- l/4)“?
(11)
Here 2/3 is the Einstein A-coefficient for the atomic transition, and s2 is the atomic Rabi frequency in the presence of the exciting field. It will be seen that A(O) = -1, that X(T) remains negative for sufficiently small values of T, and that it rises from its initial value --I. It would be quite impossible for any classical optical field to exhibit such features. However, irrespective of these non-classical properties, it is clear that correlations persist for times of order 1/p, no matter how strong or weak the driving field may be. If we use eq. (11) in the definition (4) we readily find for the correlation time Tc = 8/37$
when CL//33 1,
= 310
when a//3 < 1,
02)
which is always of order l/o and isan intuitively reasonable measure. Again it is worth noting that, had we omitted the modulus signs under the integral in eq. (4) we would have arrived at the answer (3/2/I)/
15 January 1981
@J2/f12 t l), which can be much shorter than l/p when LX//3>> 1. These examples confirm that the definition of T, we have given is appropriate for a wide class of optical fields. This work was supported in part by the National Science Foundation and by the Air Force Office of Scientific Research.
References [l] L. Mandel and E. Wolf, Proc. Phys. Sot. (London) 80 (1962) 894. [2] E. Wolf, Proc. Phys. Sot. (London) 71 (1958) 257. (31 L. Mandel, Proc. Phys. Sot. (London) 74 (1959) 233; see also Progress in optics, Vol. 2, ed. E. Wolf (North-Holland Amsterdam, 1963) p. 181. [4] R.J. Glauber, Phys. Rev. 130 (1963) 2529; 131 (1963) 2766; see also Quantum optics and electronics, eds. C. deWitt, A. Blandin and C. Cohen-Tannoudji (Gordon and Breach, New York, 1965) p. 63. [5] P.L. Kelley and W .H. Kleiner, Phys. Rev. 136 (1963) A316. I61 L. Mandel and E. Wolf, Rev. Mod. Phys. 37 (1965) 231. 171 E. Wolf, Proc. Phys. Sot. (London) 76 (1960) 424; L. Mandel and E. Wolf, Phys. Rev. 124 (1961) 1696. 181 B.L. Morgan and L. Mandel, Phys. Rev. Lett. 16 (1966) 1012; D.B. Scarl, Phys. Rev. Lett. 17 (1966) 663; D.T. Phillips, H. Kleiman and S.P. Davis, Phys. Rev. 153 (1967) 113. [91 H. Risken and H.D. Vollmer, Zeits. f. Phys. 201 (1967) 323; H. Risken, in: Progress in optics, Vol. 8, ed. E. Wolf (North-Holland, Amsterdam, 1970) p. 239; R.D. Hempstead and M. Lax, Phys. Rev. 161 (1967) 350. [lOI L. Mandel, in: Modern optics, ed. J. Fox (Polytechnic Press, Brooklyn, N.Y., 1967) P. 143. illI H.J. Carmichael and D.F. Walls, J. Phys. B 9 (1976) L43; 9 (1976) 1199; H.J. Kimble and L. Mandel, Phys. Rev. A 13 (1976) 2123; M. Dagenais and L. Mandel, Phys. Rev. A 18 (1978) 2217.
89
OPTICS COMMUNICATIONS
15 January 1981
INTENSITY CORRELATION TIME OF AN OPTICAL FIELD * L. MANDEL Dept. of Physics and Astronomy Rochester, NY 14627, USA
and Institute of Optics, University of Rochester,
Received 3 October 1980
A measure of the time for which intensity correlations persist in a stationary optical field, suitable for both classical and quantum fields, is proposed, and illustrated by several examples.
The intensity correlation time of an optical field is a measure of the time interval over which correlations of the intensity fluctuations persist. For a thermal field, for which phase and amplitude fluctuations are coupled, the reciprocal bandwidth of the spectral distribution would appear to provide a convenient measure. But even in this simple case difficulties sometimes arise, because the bandwidth does not always have an unambiguous interpretation [ 11. Attempts have been made to define the coherent time and the bandwidth of terms of the second moments [2] of the functions ly(‘)12 and #2(~), where y(r) is the normalized second order correlation function of the field amplitude, and 4(w) is its Fourier transform, the normalized spectral density. Alternatively, consideration based on photoelectric counting measurements have led to the expression [3]
slr(r)12dr
--oo
for the coherence time of thermal light, and this may differ substantially from the second moment in some cases [ 11. However, when we come to non-thermal fields, such as those produced by lasers and by quantum sources, entirely new problems arise, because the correlation times associated with phase fluctuations and with intensity fluctuations may differ by orders * This work was supported in part by the National Science Foundation and by the Air Force Office of Scientific Research.
of magnitude. The second order correlation function y(r) is then inappropriate for determining the intensity correlation time, and the intensity correlation fiurction has to be used. In general, there are at least two different time scales that can be associated with the intensity fluctuations of an optical field. The first, that we may denote by T1 , is a measure of the shortest time interval in which the light intensity changes appreciably, while the second, to be denoted by T,, measures the longest time interval over which appreciable correlations persist. In some cases the two may be equal, but this is not true in general. In the following we introduce a simple expression for T,, that is applicable to any stationary, ergodic field, classical or quantized, and is closely related to measurements with a photodetector. In the case of thermal light, T1 and Tc coincide, and Tc goes over into the expression J~_l~(r)12dt proposed earlier [3]. For the sake of generality we shall describe the field in quantum mechanical terms, in which the light intensity I(t) is a Hilbert space operator *, and ( ) denotes the quantum expectation value. However, I(t) may be regarded as a randomly fluctuating c-number function whenever the field is describable classically, and ( > then denotes the average over the ensemble. For the definition of T, we start from the idea that intensity correlations of an optical field do not persist over time intervals much longer than the intensity * Hilbert space operators are distinguished by the caret “. 87
Volume 36, number 2
OPTICS COMMUNICATIONS
correlation time r,. If we define a normalized ty correlation function by X(r) = ( 7: A&&
t T) : ),
intensi-
(1)
then X(r) dies away to zero once 171appreciably exceeds T,. We have written the quantum mechanical correlation function in normal order and in time order (indicated by : : and T), because in this form it is most closely related to photoelectric measurements [4-61. For many kinds of optical field X(0) has a magnitude of order unity, although it can be larger or smaller in principle. A(r) is of course zero for a strictly coherent field in a coherent state [4,6]. The condition that X(r) becomes very small once r appreciably exceeds T, can now be expressed in the form
1
lh(r)ldrg
r
lh(QldT,
(2)
15 January 1981
w=+o
+P2)ly(7)12,
(5)
where P is the degree of polarization and y(r) is the normalized second order correlation function. h(r) is always non-negative, and it follows from eq. (4) that
T, =
s -co
Ir@>12dr,
(6)
in agreement with the formula given previously
[ 1,3] for thermal light. It has been demonstrated in photon counting measurements that bunching of the counts extends over time intervals of this order [8]. (b) A single-mode laser operating in the neighborhood of the threshold of oscillation has an intensity correlation function that can be expressed in the form c93
U
Tc
and T, is the shortest time for which this inequality is valid. It should be noted that we need to take the modulus of h(r) under the integral because h(r) may be negative or oscillatory. Hence
fc
(7) with V,,, 2 0 and Zm V,,, = 1. This is again a positive function, and when it is substituted in eq. (4) we obtain for the intensity correlation time
_T 7 I h(r)1dr. s IX(T)ldT= _m c If we approximate I A( by a triangular function in the range -Tc to T,, we can evaluate the left-hand integral and obtain
Tc _f lh(0)1(1- ld/T,)d~*
-Tc
J IA(r)I dr _m
or m
T, =
s
Ih(W@)I dr.
_m
We shall take eq. (4) as the definition of the intensity correlation time T, for any kind of stationary, ergodie optical field, classical or quantized. Of course if h(r) = 0 for all 7, as for a completely coherent state of the field, the integral is not defined, but the concept of intensity correlation time then becomes meaningless. We now examine some special classes of fields, in order to illustrate the significance of the definition. (a) For thermal light that obeys the cross-spectral purity condition, we have 173 88
This is an average of the decay times l/X, weighted by the appropriate factors V,,, , and is therefore a reasonable measure. (c) For a laser that is oscillating well above threshold in two modes with frequencies WI and w2 independently, it may be shown that [lo], to a good approximation, h(7) = e+r
cos (w2 - w1)7,
(9)
where /3 is the effective linewidth of each mode. Evidently intensity correlations persist for a time interval of order l/p in this case, and the correlations change signs periodically. If we substitute this form of h(r) in the definition (4) we may readily show that 2
T, =
P2 + (a2 - wI)2
x
[
B+(W2-W1)c”sech2(oziiP,,,.
1
(10)
Volume 36, number 2
OPTICS COMMUNICATIONS
This is always of order l//3, whether ]w2 - wl] is larger than, equal to, or smaller than 0, and is therefore a reasonable measure of the correlation time. It should be noted that, had we omitted the modulus signs in the definition (4) of T,, we would have been led to the value 2fi/[p2 t (w2 - w~)~], which is much smaller than l/P when lo2 .- w1 I % /3, and is therefore not an appropriate measure of correlation time. (d) Finally, we consider the correlation properties of resonance fluorescence radiation, which has unmistakable quantum features that cannot be described classically. It can be shown that for the fluorescence emitted by a single atom in the presence of a coherent field on resonance, h(r) has the form [ 1 l]
X [cos(R’flr) t (3/2Sl’)sin(R’/Ir)], ST = (s22//?
- l/4)“?
(11)
Here 2/3 is the Einstein A-coefficient for the atomic transition, and s2 is the atomic Rabi frequency in the presence of the exciting field. It will be seen that A(O) = -1, that X(T) remains negative for sufficiently small values of T, and that it rises from its initial value --I. It would be quite impossible for any classical optical field to exhibit such features. However, irrespective of these non-classical properties, it is clear that correlations persist for times of order 1/p, no matter how strong or weak the driving field may be. If we use eq. (11) in the definition (4) we readily find for the correlation time Tc = 8/37$
when CL//33 1,
= 310
when a//3 < 1,
02)
which is always of order l/o and isan intuitively reasonable measure. Again it is worth noting that, had we omitted the modulus signs under the integral in eq. (4) we would have arrived at the answer (3/2/I)/
15 January 1981
@J2/f12 t l), which can be much shorter than l/p when LX//3>> 1. These examples confirm that the definition of T, we have given is appropriate for a wide class of optical fields. This work was supported in part by the National Science Foundation and by the Air Force Office of Scientific Research.
References [l] L. Mandel and E. Wolf, Proc. Phys. Sot. (London) 80 (1962) 894. [2] E. Wolf, Proc. Phys. Sot. (London) 71 (1958) 257. (31 L. Mandel, Proc. Phys. Sot. (London) 74 (1959) 233; see also Progress in optics, Vol. 2, ed. E. Wolf (North-Holland Amsterdam, 1963) p. 181. [4] R.J. Glauber, Phys. Rev. 130 (1963) 2529; 131 (1963) 2766; see also Quantum optics and electronics, eds. C. deWitt, A. Blandin and C. Cohen-Tannoudji (Gordon and Breach, New York, 1965) p. 63. [5] P.L. Kelley and W .H. Kleiner, Phys. Rev. 136 (1963) A316. I61 L. Mandel and E. Wolf, Rev. Mod. Phys. 37 (1965) 231. 171 E. Wolf, Proc. Phys. Sot. (London) 76 (1960) 424; L. Mandel and E. Wolf, Phys. Rev. 124 (1961) 1696. 181 B.L. Morgan and L. Mandel, Phys. Rev. Lett. 16 (1966) 1012; D.B. Scarl, Phys. Rev. Lett. 17 (1966) 663; D.T. Phillips, H. Kleiman and S.P. Davis, Phys. Rev. 153 (1967) 113. [91 H. Risken and H.D. Vollmer, Zeits. f. Phys. 201 (1967) 323; H. Risken, in: Progress in optics, Vol. 8, ed. E. Wolf (North-Holland, Amsterdam, 1970) p. 239; R.D. Hempstead and M. Lax, Phys. Rev. 161 (1967) 350. [lOI L. Mandel, in: Modern optics, ed. J. Fox (Polytechnic Press, Brooklyn, N.Y., 1967) P. 143. illI H.J. Carmichael and D.F. Walls, J. Phys. B 9 (1976) L43; 9 (1976) 1199; H.J. Kimble and L. Mandel, Phys. Rev. A 13 (1976) 2123; M. Dagenais and L. Mandel, Phys. Rev. A 18 (1978) 2217.
89