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Detection of mode-locked laser signals
Volume 5, number 5
OPTICS COMMUNICATIONS
DETECTION OF MODE-LOCKED
August 1972
LASER SIGNALS *
R.J. D'ORAZIO and N. GEORGE California Institute of Technology, Pasadena, California 91109, USA Received 31 January 1972
The passive Fabry-P6rot cavity is shown to be a good practical approach to the matched-filter optimization for the sensitive detection of a mode-locked laser signal. Doppler measurements of relative motion over a wide range of velocities are possible simply by measuring the cavity length for a peaked output.
1. Introduction
2. Elements of the theory
Multi-tone lasers of the mode-locked [ 1] and cavitydumped [2] types emit their energy in short pulses. Sensitive detection of these emissions for point-topoint communications or echo-ranging systems can be accomplished by using appropriate filtering at the optical frequencies before detection and radio-frequency amplification. In the present work we describe our approach to matched filtering for these signals. In the literature related prior studies of laser detection include scanning F a b r y - P 6 r o t cavities using a single pass-band of the passive cavity to analyze laser radiation [3], spatial filtering techniques [4], and various laser heterodyne techniques [ 5 - 7 ] . Our optical receiver for mode-locked gas laser signals consists of a passive laser cavity controlled in length and a photodetector with its associated electronics. The length of the passive Fabry-P~r0t cavity is chosen roughly equal to the cavity length of the transmitting laser, but with provision for fine fractional wavelength control of its length. In addition to the selective filtering characteristics of the passive cavity (passbands of unity transmission matching the frequencies of the multi-mode laser), a readout of the vernier length control, peaking the output, provides for an extremely wide range of velocity measurements with either an active or passive vehicle moving relative to the receiver.
For convenience we will assume an unmodulated modeqocked laser as our signal source. The idealized electric field amplitude at the output of a mode-locked laser with 2N + 1 modes is given by
* Research supported in part by the Air Force Office of Scientific Research.
N E(t) = ~ exp [i(~0+P~Oc)t ] p=-N
=
sin [(2N+l) ~Oct/2] sin(wct/2) exp (iw0t)
,
(1)
where 6o0 is the optical frequency at mode center, ~Oc=+c/h is the spacing between the unit amplitude modes, and h is the effective cavity length. Consider the passive cavity as shown in fig. I where h 0 is the it- . . . . . . . . . . . . . . . . . . . . . I I
/
I
I
I-
h
L+
-+
I~ t4
',lln
" T
.v li +,
/
ho~
/
i
[] +N
+:
No/[
I t. . . . . . . . . . . . . . . . . . . . . . .
•
Fig. 1. Passive cavity receiver. The components are: (T) laser transmitter; (M3, M4) laser mirrors; (h) laser cavity length; (E) signal; (v) velocity of laser relative to receiver; (No) noise; (F) coarse bandpass!filter; (M l, M2) passive cavity mirrors; (ho) passive cavity length; (D) detector; (A) detector electronics~ (C) mirror control. 407
Volume 5, number 5
OPTICS COMMUNICATIONS
cavity length, M 1 is the fixed cavity mirror, and M 2 is the movable cavity mirror. The transmissivity T(co) = It(co)12, where t(co) is the amplitude transmission function for the cavity, may be shown to be *"
T( co) -
Itlt212 exp (-2o.ho)
1
(2)
[1 - rlr 2 exp (-20h0)]2 [1 + P sin2(coho/c)] where 4rlr 2 exp ( - 2 a h 0 ) P -=
(3) [1 - rlr 2 exp (--2oh0) ] 2
'
t 1 and t 2 are the transmission functions for mirror 1 and mirror 2 respectively, r 1 (mirror 1 right side incidence) and r 2 (mirror 2 left side incidence) are the reflection coefficients, ~ is the cavity loss per unit length, and c is the speed of light. Idealizing to the no loss case, noting that the amplitude transmission and reflection cQefficients can be taken constant over the Doppler line width, and setting r 1 = r2, one can reduce eq. (2) to 1
T(CO)=
(4)
1 + P sin2(coh0/e ) Now if the signal from a mode-locked laser is coupled into the passive laser cavity, the power spectral density of the output electric field, Sg(co), is related to the power spectral density of the input, SE(co), by eq. (4) as 1
Sg(co) =
1 + P sin2(coho/e )-'"
SE(co)
(5)
and for eq. (1) N SE(co) = ~ (27r) '/28[co - (coO +Pcoc)]
p=-N
where 6(t) is the Dirac delta function. In physical terms a larger output results if T(co) peaks at large values of SE(co ). Thus, a reasonable choice is to set h 0 --h in order that T(co 0 + pcoc) = 1 for p = 0, + 1 . . . . . + N. The coarse filter F blocks frequencies outside of this band. * Eq. (2) is a generalization, to include frequency variations of q, t2, r I and r2, of the well known expression for cavity transmissivity in a Fabry-Perot, see, e.g., ref. [8]. 408
August 1972
To study this system further, we use the matchedfilter criterion resulting from the maximization of the peak value of the signal power to the mean-squarednoise power. This criterion specifies that the amplitude transmission function be tm(co ) = A F*(E(t))/Sn(CO), where F is the Fourier transform operator, Sn(co ) is the power spectral density of the additive input noise, the asterisk denotes the complex conjugate, and A is any nonzero complex colastant [9]. For an actual laser signal, writing F(E(t)) for the multi-tone laser with finite linewidths, Acol, will yield an expression, as an alternative to the transform of eq. (1), which is comparable to the amplitude transmission function t(co) for the cavity. Since T-- tt*, we can also make this comparison on a mode-by-mode basis by expanding eq. (4) near the zeros, cop, of sin2(coho/c ). This gives the approximate lineshape factor 1/[1 + (Aco/Acoo)2], where Ace = w-COp and
Atop = c/hoPV2 ; The F a b r y - P e r o t cavity is probably as close a physical realization to a matched-filter for the multitoned laser as can be attained in a passive system. Even so, gain narrowing invariably results in AceI < Atop [10], thereby limiting the observed improvement in signal-to-nois~ ratio from its optimal value. For high gain lasers with cavities of low finesse, the receiver can be made close to ideal, while greater departures are to be expected in the case of low gain. We note, too, that larger bandwidths, Atop, are called for with information modulated lasers and cavity-dumped lasers where mode-locking may not have been employed. Now, suppose that the mode-locked laser is moving toward our receiver with a velocity o. For TEM waves, an emitted frequency co' will be observed up shifted to co given by co = 3'(1 + o/c)co' in which 7 = [ 1 - (o/c) 2] -'/2, [ 11 ]. Assuming normal incidence, by eq. ( I ) the input signal, i.e., the Doppler shifted electric field, is readily expressed as follows: sin [(2N+ 1) coc'y (1 + o/c)t/2 ] e(t) =
sin [cocT(1 + o/c)t/2]
X exp [ico0-Y(1 + o/c)t]
(6)
Thus, in the case where there is relative motion, optimal detection of the mode-locked laser signal requires
Volume 5, number 5
OPTICS COMMUNICATIONS
a receiver with a cavity length h 0, given by
nO
rrc Wc3'(l +
o/c)
h 3,(1 + o/c)
o
(7)
Similarly if the mode-locked laser and the passive cavity were on a common platform, then the echo from a vehicle moving toward this platform with velocity o would be shifted to 6o = (1 + 20/c)~', where we have set ~ = 1. So by vernier adjustments (PZT driven mirror) of the passive cavity we can read a large range of approach velocities, with a resolution independent of the velocity o, i.e., 8h/h ~- 8o/c for o/c ,~ 1. Thus with 8h/h = 3.3 X l0 - 9 we find a resolution of 80 ~ lm/sec.
3. Summary It has been shown that the passive laser cavity is a good approximation to the matched filter for modelocked laser signals. Further study of the use of the passive cavity in contrast to no cavity shows that the signal-to-noise ratio improves approximately by the finesse, which is typically 100 to 200 for high quality multilayers spaced by 1 meter and operating at 6328 A. This comparison is made for a uniform noise background. Also the peak value of the temporally varying signal-to-noise ratio improves with the number of laser modes as (2N + 1) 2, i.e., as the peak power of the
August 1972
mode-locked laser. Further improvement of the S/N above that obtained with the passive cavity may be accomplished with subsequent processing of the detector output, such as boxcar-integration or time-sampled displays. In applications of the receiver to information modulated multi-tone lasers, the effective bandwidth of the passive Fabry-P~rot can be controlled by appropriate choices o f r 1 and r 2 in eq. (2).
References [ 1] L.E. Hargrove, R.L. Fork and M.A. Pollack, Appl. Phys. Letters 5 (1964) 4. [2] W.H. Steier, Proc. IEEE 54 (1966) 1604. [3] R.L. Fork, D.R. Herriott and H. Kogelnik, Appl. Opt. 3 (1964) 1471. [4] H. Kogelnik and A. Yariv, Proc. IEEE 52 (1964) 165. [5] R.W. Uhlhorn and D.F. Holshouser, IEEE J. Quantum Electron. QE-6 (1970} 775. [6] C.M. Sonnenschien and F.A. Horrigan, Appl. Opt. 10 (1971) 1600. [7] M.J. Rudd, J. Phys. E: Sci. Instr. 2 (1969) 55. [8] M. Born and E. Wolf, Principles of optics, 4th Ed. (Permagon Press, London, 1970) p. 62, 327. [9] W.M. Brown, Analysis of the linear time-invariant systems (McGraw-Hill, New York, 1963) p. 245. [ 10] A. Yariv, Quantum electronics (Wiley, New York, 1967) p. 409. [ 11 ] C.H. Papas, Theory of electromagnetic wave propagation (McGraw-Hill, New York, 1965) p. 225.
409
OPTICS COMMUNICATIONS
DETECTION OF MODE-LOCKED
August 1972
LASER SIGNALS *
R.J. D'ORAZIO and N. GEORGE California Institute of Technology, Pasadena, California 91109, USA Received 31 January 1972
The passive Fabry-P6rot cavity is shown to be a good practical approach to the matched-filter optimization for the sensitive detection of a mode-locked laser signal. Doppler measurements of relative motion over a wide range of velocities are possible simply by measuring the cavity length for a peaked output.
1. Introduction
2. Elements of the theory
Multi-tone lasers of the mode-locked [ 1] and cavitydumped [2] types emit their energy in short pulses. Sensitive detection of these emissions for point-topoint communications or echo-ranging systems can be accomplished by using appropriate filtering at the optical frequencies before detection and radio-frequency amplification. In the present work we describe our approach to matched filtering for these signals. In the literature related prior studies of laser detection include scanning F a b r y - P 6 r o t cavities using a single pass-band of the passive cavity to analyze laser radiation [3], spatial filtering techniques [4], and various laser heterodyne techniques [ 5 - 7 ] . Our optical receiver for mode-locked gas laser signals consists of a passive laser cavity controlled in length and a photodetector with its associated electronics. The length of the passive Fabry-P~r0t cavity is chosen roughly equal to the cavity length of the transmitting laser, but with provision for fine fractional wavelength control of its length. In addition to the selective filtering characteristics of the passive cavity (passbands of unity transmission matching the frequencies of the multi-mode laser), a readout of the vernier length control, peaking the output, provides for an extremely wide range of velocity measurements with either an active or passive vehicle moving relative to the receiver.
For convenience we will assume an unmodulated modeqocked laser as our signal source. The idealized electric field amplitude at the output of a mode-locked laser with 2N + 1 modes is given by
* Research supported in part by the Air Force Office of Scientific Research.
N E(t) = ~ exp [i(~0+P~Oc)t ] p=-N
=
sin [(2N+l) ~Oct/2] sin(wct/2) exp (iw0t)
,
(1)
where 6o0 is the optical frequency at mode center, ~Oc=+c/h is the spacing between the unit amplitude modes, and h is the effective cavity length. Consider the passive cavity as shown in fig. I where h 0 is the it- . . . . . . . . . . . . . . . . . . . . . I I
/
I
I
I-
h
L+
-+
I~ t4
',lln
" T
.v li +,
/
ho~
/
i
[] +N
+:
No/[
I t. . . . . . . . . . . . . . . . . . . . . . .
•
Fig. 1. Passive cavity receiver. The components are: (T) laser transmitter; (M3, M4) laser mirrors; (h) laser cavity length; (E) signal; (v) velocity of laser relative to receiver; (No) noise; (F) coarse bandpass!filter; (M l, M2) passive cavity mirrors; (ho) passive cavity length; (D) detector; (A) detector electronics~ (C) mirror control. 407
Volume 5, number 5
OPTICS COMMUNICATIONS
cavity length, M 1 is the fixed cavity mirror, and M 2 is the movable cavity mirror. The transmissivity T(co) = It(co)12, where t(co) is the amplitude transmission function for the cavity, may be shown to be *"
T( co) -
Itlt212 exp (-2o.ho)
1
(2)
[1 - rlr 2 exp (-20h0)]2 [1 + P sin2(coho/c)] where 4rlr 2 exp ( - 2 a h 0 ) P -=
(3) [1 - rlr 2 exp (--2oh0) ] 2
'
t 1 and t 2 are the transmission functions for mirror 1 and mirror 2 respectively, r 1 (mirror 1 right side incidence) and r 2 (mirror 2 left side incidence) are the reflection coefficients, ~ is the cavity loss per unit length, and c is the speed of light. Idealizing to the no loss case, noting that the amplitude transmission and reflection cQefficients can be taken constant over the Doppler line width, and setting r 1 = r2, one can reduce eq. (2) to 1
T(CO)=
(4)
1 + P sin2(coh0/e ) Now if the signal from a mode-locked laser is coupled into the passive laser cavity, the power spectral density of the output electric field, Sg(co), is related to the power spectral density of the input, SE(co), by eq. (4) as 1
Sg(co) =
1 + P sin2(coho/e )-'"
SE(co)
(5)
and for eq. (1) N SE(co) = ~ (27r) '/28[co - (coO +Pcoc)]
p=-N
where 6(t) is the Dirac delta function. In physical terms a larger output results if T(co) peaks at large values of SE(co ). Thus, a reasonable choice is to set h 0 --h in order that T(co 0 + pcoc) = 1 for p = 0, + 1 . . . . . + N. The coarse filter F blocks frequencies outside of this band. * Eq. (2) is a generalization, to include frequency variations of q, t2, r I and r2, of the well known expression for cavity transmissivity in a Fabry-Perot, see, e.g., ref. [8]. 408
August 1972
To study this system further, we use the matchedfilter criterion resulting from the maximization of the peak value of the signal power to the mean-squarednoise power. This criterion specifies that the amplitude transmission function be tm(co ) = A F*(E(t))/Sn(CO), where F is the Fourier transform operator, Sn(co ) is the power spectral density of the additive input noise, the asterisk denotes the complex conjugate, and A is any nonzero complex colastant [9]. For an actual laser signal, writing F(E(t)) for the multi-tone laser with finite linewidths, Acol, will yield an expression, as an alternative to the transform of eq. (1), which is comparable to the amplitude transmission function t(co) for the cavity. Since T-- tt*, we can also make this comparison on a mode-by-mode basis by expanding eq. (4) near the zeros, cop, of sin2(coho/c ). This gives the approximate lineshape factor 1/[1 + (Aco/Acoo)2], where Ace = w-COp and
Atop = c/hoPV2 ; The F a b r y - P e r o t cavity is probably as close a physical realization to a matched-filter for the multitoned laser as can be attained in a passive system. Even so, gain narrowing invariably results in AceI < Atop [10], thereby limiting the observed improvement in signal-to-nois~ ratio from its optimal value. For high gain lasers with cavities of low finesse, the receiver can be made close to ideal, while greater departures are to be expected in the case of low gain. We note, too, that larger bandwidths, Atop, are called for with information modulated lasers and cavity-dumped lasers where mode-locking may not have been employed. Now, suppose that the mode-locked laser is moving toward our receiver with a velocity o. For TEM waves, an emitted frequency co' will be observed up shifted to co given by co = 3'(1 + o/c)co' in which 7 = [ 1 - (o/c) 2] -'/2, [ 11 ]. Assuming normal incidence, by eq. ( I ) the input signal, i.e., the Doppler shifted electric field, is readily expressed as follows: sin [(2N+ 1) coc'y (1 + o/c)t/2 ] e(t) =
sin [cocT(1 + o/c)t/2]
X exp [ico0-Y(1 + o/c)t]
(6)
Thus, in the case where there is relative motion, optimal detection of the mode-locked laser signal requires
Volume 5, number 5
OPTICS COMMUNICATIONS
a receiver with a cavity length h 0, given by
nO
rrc Wc3'(l +
o/c)
h 3,(1 + o/c)
o
(7)
Similarly if the mode-locked laser and the passive cavity were on a common platform, then the echo from a vehicle moving toward this platform with velocity o would be shifted to 6o = (1 + 20/c)~', where we have set ~ = 1. So by vernier adjustments (PZT driven mirror) of the passive cavity we can read a large range of approach velocities, with a resolution independent of the velocity o, i.e., 8h/h ~- 8o/c for o/c ,~ 1. Thus with 8h/h = 3.3 X l0 - 9 we find a resolution of 80 ~ lm/sec.
3. Summary It has been shown that the passive laser cavity is a good approximation to the matched filter for modelocked laser signals. Further study of the use of the passive cavity in contrast to no cavity shows that the signal-to-noise ratio improves approximately by the finesse, which is typically 100 to 200 for high quality multilayers spaced by 1 meter and operating at 6328 A. This comparison is made for a uniform noise background. Also the peak value of the temporally varying signal-to-noise ratio improves with the number of laser modes as (2N + 1) 2, i.e., as the peak power of the
August 1972
mode-locked laser. Further improvement of the S/N above that obtained with the passive cavity may be accomplished with subsequent processing of the detector output, such as boxcar-integration or time-sampled displays. In applications of the receiver to information modulated multi-tone lasers, the effective bandwidth of the passive Fabry-P~rot can be controlled by appropriate choices o f r 1 and r 2 in eq. (2).
References [ 1] L.E. Hargrove, R.L. Fork and M.A. Pollack, Appl. Phys. Letters 5 (1964) 4. [2] W.H. Steier, Proc. IEEE 54 (1966) 1604. [3] R.L. Fork, D.R. Herriott and H. Kogelnik, Appl. Opt. 3 (1964) 1471. [4] H. Kogelnik and A. Yariv, Proc. IEEE 52 (1964) 165. [5] R.W. Uhlhorn and D.F. Holshouser, IEEE J. Quantum Electron. QE-6 (1970} 775. [6] C.M. Sonnenschien and F.A. Horrigan, Appl. Opt. 10 (1971) 1600. [7] M.J. Rudd, J. Phys. E: Sci. Instr. 2 (1969) 55. [8] M. Born and E. Wolf, Principles of optics, 4th Ed. (Permagon Press, London, 1970) p. 62, 327. [9] W.M. Brown, Analysis of the linear time-invariant systems (McGraw-Hill, New York, 1963) p. 245. [ 10] A. Yariv, Quantum electronics (Wiley, New York, 1967) p. 409. [ 11 ] C.H. Papas, Theory of electromagnetic wave propagation (McGraw-Hill, New York, 1965) p. 225.
409