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Phase conjugation studies of a quasi-cw CO2 laser in liquid CS2
Volume 74, number 1,2
OPTICS COMMUNICATIONS
1 December 1989
P H A S E C O N J U G A T I O N S T U D I E S OF A Q U A S I - C W CO2 L A S E R I N L I Q U I D CS2 P.E. D Y E R and J.S. L E G G A T T Department of Applied Physics, School of Engineering and Computing, University of Hull, Hull HU6 7RX, UK Received 14 June 1989
Degenerate-four-wave-mixingof an RF excited, quasi-cw, CO2 laser in liquid CS2 has produced phase conjugate reflectivities up to 16% from thermally driven gratings. A novel technique has been used to confirm that the long-period grating dominates and to measure its lifetime. Convection appears to limit the maximum phase-conjugate return.
Phase conjugation studies of CO2 lasers have mainly been restricted to high-power pulsed devices [ 1-4 ], presumably because o f the lack o f suitable, efficient, non-linear media in the 9-11 Ixm region. For degenerate-four-wave-mixing ( D F W M ) experiments, thermally generated gratings can, in principle, be utilized [ 5-8 ] as a strong but slow time-response scatterer. As the long wavelength CO2 laser produces relatively large grating spacings, the grating decay through thermal dissipation effects can be long allowing large amplitudes to be built up [ 9 ]. Here we report phase conjugation studies of a quasi-cw CO2 laser using thermally driven D F W M in liquid CS2 with reflection efficiencies reaching up to 16%. This extends earlier work we reported using a TEA CO2 and liquid CS2 and in which it was suggested that improved efficiencies should be possible with longer duration, higher fluence, pulses [ 7 ]. The experiments used a standard D F W M configuration (fig. l ), with the liquid CS2 contained in a 20 m m long cell of internal diameter l0 or 50 ram. A 30 W R F excited CO2 laser (Laser Applications Limited W G 3 5 ) was employed to produce an output duration ranging from - 1 ms to true cw by electronic control of the R F power supply. Following an initial narrow switching spike, the laser power remained essentially constant until the beam was turned off (fig. 2a). The laser output was expanded and collimated using an afocal telescope to produce 8 m m diameter Present address: British Telecom Research Laboratories, Martlesham Heath, Ipswich, UK. 124
LONG PERIOD GRATING
++AROP P
I 1
_--==
A"+ARO PO+ : i
CHOPPER CS 2 CELL PHABE
~
CONJUG~E RETURN ~
PROBE
CHOPPER
Fig. 1. Experimental configuration showinglocations of the beam chopper. (e -1 irradiance) beams with p u m p and probe average powers of ~ 10 W and ~ 1 W respectively at the cell. Phase conjugate returns were recorded with an EEV pyroelectric vidicon camera for two-dimensional imaging, and a fast time-response gold-doped germanium detector for obtaining the temporal profile. Absolute power reflection coefficients were based on measurements made using a specially constructed pyroelectric device having a large area and long-duration hold-time, ~ 1 s [ 10 ]. The phase conjugate nature of the D F W M return for the quasi-cw laser was confirmed using a thermal imaging camera (fig. 3). Here results for a forward pump-probe angle of 0 = 0 . 9 ° and CO2 laser pulse duration o f 40 ms, are shown for both the conventional mirror and phase conjugate "mirror" return with and without an NaCl-flat phase aberrator in the probe beam. The distortion undoing properties are clearly demonstrated, although the general quality o f
0030-4018/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland)
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1 December 1989
OPTICS COMMUNICATIONS
Volume 74, number 1,2
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When only the probe was chopped the PCR fell during the blocked intervals, but not to zero (fig. 2d). Finally for a chopped forward-pump beam the pulse shape was identical to that for the chopped probe shown in fig. 2d. These findings can be interpreted consistently if the PCR is due to scattering of the backward-pump by the long-period grating formed between the forward-pump and probe beams. The slow decay of the return when either the forward pump or probe is interrupted indicates that the grating is relatively long lived and, as will be discussed below, this is consistent with a thermal decay mechanism. The absence of a significant contribution to the PCR from the short-period grating formed between the backward-pump and probe is also expected, as thermal conduction "smearing" limits its amplitude and the scattering efficiency to low values. To analyse the thermal gratings, convection is initially neglected and the one-dimensional heat conduction equation [ 11 ]
02T
otlo ( l + q c o s k x ) = p c O T
~5xZ+ # Fig. 2. (a) CO2laserpulse shape; 35 ms pulseduration. (b) Phaseconjugate return. (c) Phase-conjugatereturn with choppedbackward-pump beam. (d) Phase-conjugate return with chopped probe beam. the return is lower than for similar TEA C O 2 laser experiments [7]. This is probably because of the poorer beam quality of the RF laser and thermally induced pump beam distortion in the CS2. To investigate the origin and time-response of the phase conjugate return (PCR), the pump and probe beams were encoded using a simple amplitude modulator [Rofin Frequency Programmable Chopper 7500] located at one of the points shown in fig. 1. Without beam modulation the PCR was observed to build up somewhat faster than linearly in time as seen in fig. 2b, and for extended duration pulses reached a steady maximum. Sustained exposure, however, ultimately caused the return to fall or exhibit erratic oscillations probably because of convection currents in the heated CS2 cell. When the backward-pump beam was modulated the PCR had a peak envelope similar to that in fig. 2b but fell to zero during the periods that the backward pump was interrupted by the chopper, fig. 2c.
It 0t
(1)
used to describe the temperature rise, T. The source term corresponds to an interference pattern of infinite extent having a wavenumber k = 4n sin ( 0 / 2 ) / 2 where 0 is the angle between the probe and either the forward or backward probe beam. The parameter q is defined by q = 2 (If,j p )
x/2/Io,
where lr,b is either the forward (If) or backward (Ib) pump irradiance and Io=If, b+Ip with Ip being the probe irradiance. In eq. ( 1 ), p, c and/~ are the density, specific heat at constant pressure, and thermal conduction coefficient respectively, and c~ is the absorption coefficient ( 0 . 0 3 5 / m m ) at the laser wavelength for CS2. The solution to eq. (1) for constant irradiance beams incident at t = 0 is
T= otlot pc + ~qotlo [ 1 - e x p ( - t / r ) ]
cos k x ,
where the characteristic thermal time constant is z=pc/ltk 2. Neglecting the first term on the righthand-side, which describes spatially uniform heating, the amplitude of the index grating is 125
Volume 74, number 1,2
OPTICS COMMUNICATIONS
1 December 1989
Fig. 3. Beam profiles recorded using a pyroeleetric vidicon. (a) Conventional mirror reflection. (b) Phase conjugate reflection. (c) Conventional mirror reflection with aberrator. (d) Phase conjugate reflection with aberrator.
An = 2 (If.b/p) 1/27,~ /tk2 [1-exp(-t/z)
],
(2)
where y = d n / d T is the rate o f change of refractive index with temperature. Using the Lorenz-Lorentz law [121, 7 = f l ( n 2 - 1 ) ( n 2 + 2 ) / 6 n , where fl is the volume expansion coefficient and n the refractive index of CS2 giving ~ ~ 1.2 × 10- 3 K - ~. For the present experiments in which 0 is < 4 ° for the forward beam and near 180 ° for the backward beam, the index amplitude is much smaller for the short-period grating because o f the k 2 dividing term. Thus, as supported by the experimental results, only low angle Bragg scattering from the long-period grating need be considered which has a diffraction efficiency [ 13 ] r/= exp ( - o t L ) sin 2 (rtAnL/2,,) ,
(3)
where L = 2 0 m m is the interaction length and 2v the vacuum wavelength. Consideration o f eqs. (2) and (3) shows the P C R , / p c = qlb, will initially increase with time and. for small values o f the argument in eq. (3), ultimately reach a steady level when t >> z where z.~4-250 ms for the present conditions 126
(0~0.5--4 ° ). The early t i m e behaviour is qualitatively confirmed by the results in fig. 2b, although precise comparison is not possible because of the initial switching spike on the R F laser output. Saturation of the P C R was also observed for sufficiently long duration exposures. It follows from eqs. (2) and (3) that when one of the two beams forming the long-period grating is interrupted the P C R will decay with a characteristic rate ~ = 2 / z . This can be written as y = 8It2,ttO2/p£A 2 where 0~ is the (small) external pumpprobe angle [ 7 ]. Measurements of 7 were obtained from the rate o f fall o f the P C R during the periods that the chopped probe beam was blocked (fig. 3d) for various values o f the angle 0e. The results, in fig. 4, agree well with the expected variation for 0e >~ 1.5 o but for smaller angles there is a significant discrepancy, with the decay rate tending to a constant finite value rather than zero. This behaviour did not differ when a 10 m m diameter rather than 50 m m diameter cell was used (fig. 4) so that whole cell convection is not a likely cause. The fit to the data in fig. 4 uses a thermal conductivity o f 1.2 X 10 -4 W / r a m /
Volume 74, number 1,2
OPTICS COMMUNICATIONS
1 December 1989 I
300
•
50mm
~cell
•
10 m m
,,
/'
,,
200 7 ~9
Ip
a loo
I
I
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d
?
n-
I
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(1) n0.1
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Pump
2
Probe
:Angle/Degrees
t
i
t
Q.J
1
2
3
4
Pump Probe : A n g l e / D e g r e e s
Fig. 4. Grating decay-rate as a function of the forward pumpprobe angle, 0e. The solid line is calculated from the conduction loss model.
Fig. 5. Phase-conjugate power-reflectioncoefficient as a function of forward pump-probe angle 0c. The solid line is calculated from eq. (4).
K for C S 2 which can be compared with an accepted value o f 1.6X 10 -4 W / m m / K [14]. The effective power reflection coefficient for the PCR, measured for sufficiently long duration pulses that a steady-state was approximately obtained, is shown in fig. 5 as a function o f 0e. A m a x i m u m reflectivity of ~ 16% was obtained for 0e,~ 1-1.5 °. From eqs. (2) and (3) the power reflection coefficient is found by integrating over the beam spatial profiles,
and by expanding the sin 2 term to third order. For 0 > I. 5 ° the data show a similar variation to that predicted by eq. (4) but with a somewhat lower magnitude for the refection coefficient (fig. 5 ). This discrepancy can probably be accounted for by the approximation involved in using a gaussian distribution to describe the spatial profiles o f the interacting beams. The limited data obtained also suggest that the reflectivity flattens-off below ~ 1.5 ° in contrast to the theoretical expectation. The departure o f the experimental decay-rate and reflectivity from that expected at small angles (figs. 4 and 5) suggest that the assumption o f conduction dominated loss breaks down in this region and that convection acts to dissipate the index gratings. For gratings formed in the vertical plane, the convective velocity, V, will be approximately [ 15 ]
R = f I b e x p ( - c t L ) sin2{(Iflp) 1/2 ~'v 0
× 7 [ 1 - e x p ( -otL)/21t#02]} r d r OO
--1
V=d2 p B g A T / v , where attenuation of the forward p u m p and the probe beam is now included. This equation was evaluated assuming the same gaussian profiles for Ib, If and Ip,
where d is the half-spacing o f the gratings, g is the acceleration due to gravity, A T is the temperature difference between the points of m a x i m u m and min127
Volume 74, number 1,2
OPTICS COMMUNICATIONS
i m u m irradiance, and v is the fluid viscosity. As a rough criterion it can be expected that the gratings will be able to build-up to their m a x i m u m a m p l i t u d e if the thermal c o n d u c t i o n time-constant z is significantly less than the t i m e for convective flow to occur over the b e a m radius r. This requires z
or
1 December 1989
also has an intrinsically high non-linear response [6,9]. The authors acknowledge the technical assistance o f BL Tait a n d helpful discussion with Ms. J. Coates o f Lasers Applications L i m i t e d ( H u l l ) . JL acknowledges the receipt o f a Science and Engineering research Council CASE Studentship with British Aerospace.
6 < ( 1 6 7 t 2 r v l t / p 2 f l A T g c ) t/4 .
References
Evaluating this expression using the a p p r o p r i a t e values gives 6 < 0 . 5 m m or 0 > 0 . 6 °. This is close to the observed region in which the conduction d o m i n a t e d m o d e l appears to b r e a k down a n d gives credence to the suggestion that this is due to the onset o f convective effects. T h e r m a l defocusing o f the p u m p b e a m s will also contribute to a loss o f grating efficiency through smearing effects. Estimates show that a p p r o x i m a t e l y one wavelength o f aberration will be introduced into the p u m p b e a m s through thermal lensing in the liquid at a fluence o f ~ 104j m -2 or at a time o f about 50 ms in the present experiments. This m a y also lead to a reduction in the phase c o n jugate fidelity as m e n t i o n e d earlier, but unlike convection does not a p p e a r to p r o v i d e an explanation o f the enhanced grating decay rate. It is finally noted that o p e r a t i o n at small p u m p probe angles with reduced convective effects and imp r o v e d phase conjugate efficiency, is possible using a m o r e viscous fluid such as a liquid crystal which
[ 1] D.G. Steel, R.C. Lind and J.F. Lam, Phys. Rev. 23 ( 1981 ) 2513. [2] A. Elei, D. Rogovin, D. Depatie and D. Haueisen, J. Opt. Soc. Am. 70 (1980) 990. [ 3 ] R.A. Fisher and B.J. Feldman, Optics Lett. 4 (1979) 140. [4] E.E. Bergmann, I.J. Bigio, B.J. Feldman and R.A. Fisher, Optics Lett. 3 (1978) 82. [ 5 ] R.K. Jain and D.G. Steel, Optics Comm. 43 (1982) 72. [ 6 ] I.C. Khoo, P.Y. Yan, G.M. Finn, T.H. Liu and R.R. Michael, J. Opt. Soe. Am. B 5 (1988) 2020. [ 7 ] P.E. Dyer and J.S. Leggatt, Optics Lett. 7 (1988) 583. [8] H.J. Hoffman, J. Opt. Soc. Am. B3 (1986) 253. [9 ] L. Richard, J. Maurin and J.P. Huignard, Optics Comm. 57 (1986) 365. [ 10] J.S. Leggatt (unpublished). [ 11 ] H. Eichler and H. Stahl, Optics Comm. 6 ( 1973 ) 639. [ 12 ] R.S. Longhurst, Geometrical and physical optics (Longman, 3rd Ed., London, 1973). [ t 3 ] H.M. Smith, Principles of holography (Wiley 2nd Ed. New York, 1975). [14] K. Raznjevic, Handbook of thermodynamic tables and charts (Hemisphere Publishing Corp., Washington, 1976). [15]D.J. Tritton, Physical fluid dynamics (Van Nostrand Reinhold, London, 1977 ).
128
OPTICS COMMUNICATIONS
1 December 1989
P H A S E C O N J U G A T I O N S T U D I E S OF A Q U A S I - C W CO2 L A S E R I N L I Q U I D CS2 P.E. D Y E R and J.S. L E G G A T T Department of Applied Physics, School of Engineering and Computing, University of Hull, Hull HU6 7RX, UK Received 14 June 1989
Degenerate-four-wave-mixingof an RF excited, quasi-cw, CO2 laser in liquid CS2 has produced phase conjugate reflectivities up to 16% from thermally driven gratings. A novel technique has been used to confirm that the long-period grating dominates and to measure its lifetime. Convection appears to limit the maximum phase-conjugate return.
Phase conjugation studies of CO2 lasers have mainly been restricted to high-power pulsed devices [ 1-4 ], presumably because o f the lack o f suitable, efficient, non-linear media in the 9-11 Ixm region. For degenerate-four-wave-mixing ( D F W M ) experiments, thermally generated gratings can, in principle, be utilized [ 5-8 ] as a strong but slow time-response scatterer. As the long wavelength CO2 laser produces relatively large grating spacings, the grating decay through thermal dissipation effects can be long allowing large amplitudes to be built up [ 9 ]. Here we report phase conjugation studies of a quasi-cw CO2 laser using thermally driven D F W M in liquid CS2 with reflection efficiencies reaching up to 16%. This extends earlier work we reported using a TEA CO2 and liquid CS2 and in which it was suggested that improved efficiencies should be possible with longer duration, higher fluence, pulses [ 7 ]. The experiments used a standard D F W M configuration (fig. l ), with the liquid CS2 contained in a 20 m m long cell of internal diameter l0 or 50 ram. A 30 W R F excited CO2 laser (Laser Applications Limited W G 3 5 ) was employed to produce an output duration ranging from - 1 ms to true cw by electronic control of the R F power supply. Following an initial narrow switching spike, the laser power remained essentially constant until the beam was turned off (fig. 2a). The laser output was expanded and collimated using an afocal telescope to produce 8 m m diameter Present address: British Telecom Research Laboratories, Martlesham Heath, Ipswich, UK. 124
LONG PERIOD GRATING
++AROP P
I 1
_--==
A"+ARO PO+ : i
CHOPPER CS 2 CELL PHABE
~
CONJUG~E RETURN ~
PROBE
CHOPPER
Fig. 1. Experimental configuration showinglocations of the beam chopper. (e -1 irradiance) beams with p u m p and probe average powers of ~ 10 W and ~ 1 W respectively at the cell. Phase conjugate returns were recorded with an EEV pyroelectric vidicon camera for two-dimensional imaging, and a fast time-response gold-doped germanium detector for obtaining the temporal profile. Absolute power reflection coefficients were based on measurements made using a specially constructed pyroelectric device having a large area and long-duration hold-time, ~ 1 s [ 10 ]. The phase conjugate nature of the D F W M return for the quasi-cw laser was confirmed using a thermal imaging camera (fig. 3). Here results for a forward pump-probe angle of 0 = 0 . 9 ° and CO2 laser pulse duration o f 40 ms, are shown for both the conventional mirror and phase conjugate "mirror" return with and without an NaCl-flat phase aberrator in the probe beam. The distortion undoing properties are clearly demonstrated, although the general quality o f
0030-4018/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland)
-
-
--
5 m s L.-J
I
1
L_
i(b). ~1 --
5 m 5
-
/
:
./
I
_/'-/ Co:) --Sinsi i
[q
/i t/i /1111 [ .
1 December 1989
OPTICS COMMUNICATIONS
Volume 74, number 1,2
-5m5L-.J r-,..,
/'-/
/\
) I
When only the probe was chopped the PCR fell during the blocked intervals, but not to zero (fig. 2d). Finally for a chopped forward-pump beam the pulse shape was identical to that for the chopped probe shown in fig. 2d. These findings can be interpreted consistently if the PCR is due to scattering of the backward-pump by the long-period grating formed between the forward-pump and probe beams. The slow decay of the return when either the forward pump or probe is interrupted indicates that the grating is relatively long lived and, as will be discussed below, this is consistent with a thermal decay mechanism. The absence of a significant contribution to the PCR from the short-period grating formed between the backward-pump and probe is also expected, as thermal conduction "smearing" limits its amplitude and the scattering efficiency to low values. To analyse the thermal gratings, convection is initially neglected and the one-dimensional heat conduction equation [ 11 ]
02T
otlo ( l + q c o s k x ) = p c O T
~5xZ+ # Fig. 2. (a) CO2laserpulse shape; 35 ms pulseduration. (b) Phaseconjugate return. (c) Phase-conjugatereturn with choppedbackward-pump beam. (d) Phase-conjugate return with chopped probe beam. the return is lower than for similar TEA C O 2 laser experiments [7]. This is probably because of the poorer beam quality of the RF laser and thermally induced pump beam distortion in the CS2. To investigate the origin and time-response of the phase conjugate return (PCR), the pump and probe beams were encoded using a simple amplitude modulator [Rofin Frequency Programmable Chopper 7500] located at one of the points shown in fig. 1. Without beam modulation the PCR was observed to build up somewhat faster than linearly in time as seen in fig. 2b, and for extended duration pulses reached a steady maximum. Sustained exposure, however, ultimately caused the return to fall or exhibit erratic oscillations probably because of convection currents in the heated CS2 cell. When the backward-pump beam was modulated the PCR had a peak envelope similar to that in fig. 2b but fell to zero during the periods that the backward pump was interrupted by the chopper, fig. 2c.
It 0t
(1)
used to describe the temperature rise, T. The source term corresponds to an interference pattern of infinite extent having a wavenumber k = 4n sin ( 0 / 2 ) / 2 where 0 is the angle between the probe and either the forward or backward probe beam. The parameter q is defined by q = 2 (If,j p )
x/2/Io,
where lr,b is either the forward (If) or backward (Ib) pump irradiance and Io=If, b+Ip with Ip being the probe irradiance. In eq. ( 1 ), p, c and/~ are the density, specific heat at constant pressure, and thermal conduction coefficient respectively, and c~ is the absorption coefficient ( 0 . 0 3 5 / m m ) at the laser wavelength for CS2. The solution to eq. (1) for constant irradiance beams incident at t = 0 is
T= otlot pc + ~qotlo [ 1 - e x p ( - t / r ) ]
cos k x ,
where the characteristic thermal time constant is z=pc/ltk 2. Neglecting the first term on the righthand-side, which describes spatially uniform heating, the amplitude of the index grating is 125
Volume 74, number 1,2
OPTICS COMMUNICATIONS
1 December 1989
Fig. 3. Beam profiles recorded using a pyroeleetric vidicon. (a) Conventional mirror reflection. (b) Phase conjugate reflection. (c) Conventional mirror reflection with aberrator. (d) Phase conjugate reflection with aberrator.
An = 2 (If.b/p) 1/27,~ /tk2 [1-exp(-t/z)
],
(2)
where y = d n / d T is the rate o f change of refractive index with temperature. Using the Lorenz-Lorentz law [121, 7 = f l ( n 2 - 1 ) ( n 2 + 2 ) / 6 n , where fl is the volume expansion coefficient and n the refractive index of CS2 giving ~ ~ 1.2 × 10- 3 K - ~. For the present experiments in which 0 is < 4 ° for the forward beam and near 180 ° for the backward beam, the index amplitude is much smaller for the short-period grating because o f the k 2 dividing term. Thus, as supported by the experimental results, only low angle Bragg scattering from the long-period grating need be considered which has a diffraction efficiency [ 13 ] r/= exp ( - o t L ) sin 2 (rtAnL/2,,) ,
(3)
where L = 2 0 m m is the interaction length and 2v the vacuum wavelength. Consideration o f eqs. (2) and (3) shows the P C R , / p c = qlb, will initially increase with time and. for small values o f the argument in eq. (3), ultimately reach a steady level when t >> z where z.~4-250 ms for the present conditions 126
(0~0.5--4 ° ). The early t i m e behaviour is qualitatively confirmed by the results in fig. 2b, although precise comparison is not possible because of the initial switching spike on the R F laser output. Saturation of the P C R was also observed for sufficiently long duration exposures. It follows from eqs. (2) and (3) that when one of the two beams forming the long-period grating is interrupted the P C R will decay with a characteristic rate ~ = 2 / z . This can be written as y = 8It2,ttO2/p£A 2 where 0~ is the (small) external pumpprobe angle [ 7 ]. Measurements of 7 were obtained from the rate o f fall o f the P C R during the periods that the chopped probe beam was blocked (fig. 3d) for various values o f the angle 0e. The results, in fig. 4, agree well with the expected variation for 0e >~ 1.5 o but for smaller angles there is a significant discrepancy, with the decay rate tending to a constant finite value rather than zero. This behaviour did not differ when a 10 m m diameter rather than 50 m m diameter cell was used (fig. 4) so that whole cell convection is not a likely cause. The fit to the data in fig. 4 uses a thermal conductivity o f 1.2 X 10 -4 W / r a m /
Volume 74, number 1,2
OPTICS COMMUNICATIONS
1 December 1989 I
300
•
50mm
~cell
•
10 m m
,,
/'
,,
200 7 ~9
Ip
a loo
I
I
P
d
?
n-
I
10
>, >
(1) n0.1
•-'•y4 - 1
Pump
2
Probe
:Angle/Degrees
t
i
t
Q.J
1
2
3
4
Pump Probe : A n g l e / D e g r e e s
Fig. 4. Grating decay-rate as a function of the forward pumpprobe angle, 0e. The solid line is calculated from the conduction loss model.
Fig. 5. Phase-conjugate power-reflectioncoefficient as a function of forward pump-probe angle 0c. The solid line is calculated from eq. (4).
K for C S 2 which can be compared with an accepted value o f 1.6X 10 -4 W / m m / K [14]. The effective power reflection coefficient for the PCR, measured for sufficiently long duration pulses that a steady-state was approximately obtained, is shown in fig. 5 as a function o f 0e. A m a x i m u m reflectivity of ~ 16% was obtained for 0e,~ 1-1.5 °. From eqs. (2) and (3) the power reflection coefficient is found by integrating over the beam spatial profiles,
and by expanding the sin 2 term to third order. For 0 > I. 5 ° the data show a similar variation to that predicted by eq. (4) but with a somewhat lower magnitude for the refection coefficient (fig. 5 ). This discrepancy can probably be accounted for by the approximation involved in using a gaussian distribution to describe the spatial profiles o f the interacting beams. The limited data obtained also suggest that the reflectivity flattens-off below ~ 1.5 ° in contrast to the theoretical expectation. The departure o f the experimental decay-rate and reflectivity from that expected at small angles (figs. 4 and 5) suggest that the assumption o f conduction dominated loss breaks down in this region and that convection acts to dissipate the index gratings. For gratings formed in the vertical plane, the convective velocity, V, will be approximately [ 15 ]
R = f I b e x p ( - c t L ) sin2{(Iflp) 1/2 ~'v 0
× 7 [ 1 - e x p ( -otL)/21t#02]} r d r OO
--1
V=d2 p B g A T / v , where attenuation of the forward p u m p and the probe beam is now included. This equation was evaluated assuming the same gaussian profiles for Ib, If and Ip,
where d is the half-spacing o f the gratings, g is the acceleration due to gravity, A T is the temperature difference between the points of m a x i m u m and min127
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OPTICS COMMUNICATIONS
i m u m irradiance, and v is the fluid viscosity. As a rough criterion it can be expected that the gratings will be able to build-up to their m a x i m u m a m p l i t u d e if the thermal c o n d u c t i o n time-constant z is significantly less than the t i m e for convective flow to occur over the b e a m radius r. This requires z
or
1 December 1989
also has an intrinsically high non-linear response [6,9]. The authors acknowledge the technical assistance o f BL Tait a n d helpful discussion with Ms. J. Coates o f Lasers Applications L i m i t e d ( H u l l ) . JL acknowledges the receipt o f a Science and Engineering research Council CASE Studentship with British Aerospace.
6 < ( 1 6 7 t 2 r v l t / p 2 f l A T g c ) t/4 .
References
Evaluating this expression using the a p p r o p r i a t e values gives 6 < 0 . 5 m m or 0 > 0 . 6 °. This is close to the observed region in which the conduction d o m i n a t e d m o d e l appears to b r e a k down a n d gives credence to the suggestion that this is due to the onset o f convective effects. T h e r m a l defocusing o f the p u m p b e a m s will also contribute to a loss o f grating efficiency through smearing effects. Estimates show that a p p r o x i m a t e l y one wavelength o f aberration will be introduced into the p u m p b e a m s through thermal lensing in the liquid at a fluence o f ~ 104j m -2 or at a time o f about 50 ms in the present experiments. This m a y also lead to a reduction in the phase c o n jugate fidelity as m e n t i o n e d earlier, but unlike convection does not a p p e a r to p r o v i d e an explanation o f the enhanced grating decay rate. It is finally noted that o p e r a t i o n at small p u m p probe angles with reduced convective effects and imp r o v e d phase conjugate efficiency, is possible using a m o r e viscous fluid such as a liquid crystal which
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