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Temporal response of stimulated Brillouin scattering phase conjugation
Optics Communications North-Holland
94 (1992)
346-352
OPTICS COMMUNICATIONS
Temporal response of stimulated Brillouin scattering phase conjugation M.R. Osborne and M.A. O’Key Laser Technology Centre, AEA Technology, Culham Laboratory, Abingdon, Oxon, OX14 3DB. UK Received
I April 1992
The response of stimulated Brillouin scattering phase conjugate reflectors to rapidly changing input beam profiles is investigated experimentally. Efficient, high fidelity response on timescales several times shorter than the acoustic decay time, ra, is demonstrated. The data obtained agree with a simple, semi-analytical theory which can be used to scale the results to other wavelengths, intensities and media.
1. Introduction
2. Theoretical background
Stimulated Brillouin scattering (SBS) has been extensively investigated as a technique for phase conjugating laser radiation [l-4]. Correction for a wide range of aberrations has been demonstrated in double pass configurations. Although, in principle, capable of compensating highly dynamic aberrations, most work has been carried out using static or slowly varying aberrators. This is because the majority of effort has been directed towards thermally generated or atmospheric aberrations, which vary only relatively slowly. However, the recent increased interest in using SBS both to correct for very rapidly varying aberrations (such as those that may be produced in plasmas or discharge excited lasers [ 5,6] ) and at longer wavelengths (e.g. for 3-5 pm HL laser emission [ 71) where the response is inherently slower, have made an investigation of the speed of response of the SBS process to temporally varying optical inputs of great importance. After a brief description of the theoretical background, the present paper presents the results of an experimental investigation of the speed and fidelity of response of several SBS media, and shows how these results may be interpreted through a semi analytical treatment.
In SBS, the input wave can be considered as being backscattered from a moving acoustic wave within the nonlinear medium. The Doppler shifted backscatter then beats with the incoming wave, and this beat frequency enhances the acoustic wave through electrostriction, establishing a positive feedback loop. The hypersonic acoustic wave assumes a phase structure which mimics that of the input beam, resulting in the backscatter approximating to the phase conjugate of the input. However, if the phasefront of the input wave were to change instantaneously, the existing acoustic wave would have the wrong phase structure, and non-conjugate backscatter would occur until the SBS medium has had time to respond to the change in input, and adjust the acoustic wavefront accordingly. It has been widely qualitatively accepted that this “response time” is closely related to the time taken for the pre-existing acoustic wave to decay, although this argument can only readily be made quantitative for the most simple of cases as described below. The acoustic decay time, fg, is a well documented figure for several materials [8,9] and has been determined both spectroscopically (as the inverse of the spontaneous Brillouin scattering linewidth) [lo] and directly by a probing technique [ 111. The decay time is a function of the wavelength of the acoustic grating, Izg, which, in backward scat-
346
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tering, closely approximates to half of the optical wavelength, jlL. Considering the classical acoustic damping mechanisms of bulk and shear viscosity, and thermal conductivity, one obtains the relationship [1,9,111 z,aA;ul~.
(1)
This has been shown to apply accurately for nominally monatomic gasses e.g. xenon, but requires some modification in polyatomic gases due to the vibrational and rotational degrees of freedom [ 111. Although there is little experimental evidence, eq. ( 1) is presumed to be reasonably accurate in liquids [ 11 where the relatively high viscosity dominates. With no driving term and taking the plane wave approximation, an acoustic wave, Q (where Q is proportional to the density change, Ap, in the medium) will decay with time as
Q(O=Q(o> exp(--U&J
,
(2)
where rr, is the intensity acoustic damping time. The reflectivity, R, of the acoustic grating will vary in proportion to the square of the modulation of the refractive index [ 12 1, and hence as &* and Q*. It will therefore decay according to R(t)=R(O)
exp( -t/q.,)
.
(3)
If a low power (i.e. below SBS threshold) probe beam with the same spatial profile and colinear with the input is then applied to this acoustic wave, the reflectivity may be probed directly. This has been reported in ref. [ 111. If the below threshold probe has a different spatial profile, or is not colinear with the original beam, a more complex situation applies. The simplest case is that of a plane probe, displaced at an angle A6 from the original wave. In this case, it can be shown [ 12,13 ] that relatively efficient (i.e. 50% of the value for colinear incidence) specular reflection will occur for angles A0 in the range 805
-tJm,
(4)
where b is the effective depth of the grating, which will be between one and five [ 14 ] times the Rayleigh range in the focused geometry, assuming this to be shorter than both the cell length and three times the coherence length. The Rayleigh range for a beam with a divergence which is m * times diffraction limited is given by
b=m*Af */nd2,
1 December
1992
(5)
wherefis the focal length of the lens and d the beam diameter. Substituting eq. (5) in eq. (4) and neglecting the factor of Jn!2 which is of order unity, gives AOsd/m*f.
(6)
Inequality ( 6 ) implies that efficient reflection should occur for angles within the focussing cone of the lens. Hence, significant non-conjugate (near) backscatter may be expected to occur for rapid variations in spatial profile which may be considered as a perturbation of the profile within a fixed beam diameter. The reflectivity will be slightly reduced from, but have the same temporal form as eq. (3 ). If the probe beam is of high power (i.e. above SBS threshold) it will eventually generate its own acoustic wave, producing a phase conjugate return. However, this will not occur instantaneously, being determined by the transient SBS threshold. In the case of a single input beam with backscatter starting from thermal noise phonons, the transient threshold may be expressed as
[151, (g,l j IL dt/rie)ii2=25,
(7)
-cc where gB is the Brillouin gain coefficient, Zis the interaction length and IL the intensity of the incident laser. It is difficult to quantify how relevant eq. (7) is to the present situation of rapidly changing small variations of the input spatial profile. Several factors may significantly alter eq. ( 7 ) . Firstly, for times prior to the attainment of the transient threshold, a fraction of the incident radiation may be Bragg reflected from the existing (decaying) grating. This will give rise to non-conjugate backscatter as described previously. It will also modify the expected value and distribution of I,_ in the focal region. Perhaps more significantly, the input and Bragg reflected waves will be perfectly matched in angle and frequency to enhance the existing grating by non 180’ SBS [ 161. In conventional, single beam SBS, non 180’ scattering is suppressed by a combination of increased transient threshold, reduced interaction length due to beam walk-off, and, in some cases, increased laser bandwidth restrictions. In the present 347
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case, however, the transient threshold does not apply as the required grating is already in existence. Also, the greater bandwidth restrictions are of no significance at the small angular deviation from 180’ considered. Only beam walk-off remains as a detriment to producing SBS at the Bragg diffraction angle from the original acoustic grating. Secondly, even assuming that the true phase conjugate return of the probe beam eventually dominates, the question arises as to whether the relevant acoustic wave is generated from noise, as eq. (5 ) presumes, or is produced as some form of mutation of the old acoustic wave which will be many orders of magnitude (perhaps 1O-l 3) above the thermal noise background. Finally, the existence of more than two beams in the interaction region, as could be produced by the existence of both a Bragg reflected and a phase conjugate reflected version of the new input pulse could allow four-wave mixing processes producing a beam counter-propagating to the specular reflection. Any temporal overlap between the original and the new input (or Stokes) beams, will further complicate the interaction. The precise effects of all of the above processes will depend critically on the exact form of the spatiotemporal variation of the input radiation - a single general case cannot be identified. Therefore, an experimental investigation was performed to determine the ability of SBS conjugators to respond to rapid variations, and which factors control this ability.
3. Experimental Given that the effects to be investigated were expected to scale as the square of the laser wavelength, eq. ( I), a tunable source of radiation was favoured. For ease of diagnostics, the photographic region of the spectrum was preferred. The relatively narrow bandwidth and high peak powers necessary to reach SBS threshold, together with the above factors, lead to the choice of an excimer-pumped tunable dye laser as the source of radiation. Moreover, previous experience had indicated that it was relatively easy to produce severely aberrated beams directly from such systems. The kind of structure which can be produced is shown graphically in the upper part of fig. 348
I December
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time
2mm
Input
I
Phase conjugate return
’
Fig.1
1Ons
’ 710nm
n-hexane
Fig. 1. Streak of input and backscatter of 7 10 nm radiation using n-hexane (~~-0.76 ns) as the SBS medium. Good spatial and temporal fidelity is observed.
1 which is a streak record of the dye laser output. The resolution in this figure is approximately 150 km in the vertical dimension (cf beam diameter of - 1.5 mm) and approximately 200 ps in the horizontal dimension (cf pulse length of - 15 ns). The modulation observed is caused by a number of factors. The excimer laser pump beam naturally shows two components of different divergence, which have different temporal evolutions and can be angularly separated by adjustment of the excimer cavity, so that they are focussed in different places by the cylindrical lens which concentrates the radiation onto the dye oscillator cell. The focal regions are also separated in the direction of beam propagation. This complex pump beam behaviour added to the longitudinal mode beating of the dye oscillator produces the highly modulated beam which, after one stage of unsaturated amplification is shown in fig. 1. Operated in this mode, the output from the Littman type [ 17 ] dye laser has a temporally averaged bandwidth of typically 4-6 GHz, as measured with a Fabry-Ptrot etalon (corresponding to 4-6 longitudinal modes) and a beam divergence of typically 2 mrad (full angle), being approximately three times the diffraction limit. The phase of this beam is clearly complex and difficult to define precisely, but will certainly vary on a timescale not slower than the observed intensity fluctuations. The input (reference) and phase conjugate beams were recorded with
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the experimental arrangement shown in fig. 2. Mirror M is slightly misaligned vertically to separate the two beams, and the distances x and z were made equal to produce temporally synchronised beams on the diffusing screen thereby greatly simplifying the recording of the streaks. However, in this arrangement the input and backscattered waves are not recorded in conjugate planes, the input beam effectively propagating a distance 2x+ 2y further than the conjugated beam before reaching the screen. The effects of this mismatch were minimised by a combination of two procedures. Firstly, the distances x and y were minimised, giving 2x+2y=40 cm, and secondly the input beam was propagated a considerable distance ( > 6 m) from any source of aberration before reaching the beamsplitter in fig. 2. At such distances, the change in spatial profile with propagation is slow, and is indeed found to be negligible over the 40 cm path difference. Several gaseous and liquid SBS media of differing acoustic decay times were investigated (see table 1) . The acoustic decay time was varied in two ways. Firstly by tuning the dye laser, and using eq. ( 1)) or one of its modified forms [ 111. This method keeps several SBS material parameters constant (steady state gain coefficient, pressure, density, viscosity,
temperature) but does vary the transient threshold, as described by eq. (7 ). To obtain some other decay times, the pressure of the gaseous media was varied, although this has the disadvantage of altering the parameters kept constant above. In all, a range of r, of approximately two orders of magnitude from 0.14 to 16 ns was covered.
4. Results and discussion Under some conditions both the spatial and temporal fidelity of the conjugation were good, as can be seen from the comparison of input and SBS return beams shown in fig. 1. This result was obtained using n-hexane as SBS medium with an input wavelength of 7 10 nm. As shown in table 1, the acoustic decay time in this case of -0.76 ns is comparable to the timescale on which gross changes of the input beam occur. Also, in this case, it can be seen that there is very little delay to the initiation of the backscatter. In fig. 3, the same wavelength (7 10 nm) is used with 30 bar of CClF3 as the SBS medium. In this case the acoustic decay time is -8.8 ns. However, after an onset delay of approximately 3 ns, high fidelity response on a nanosecond timescale is still demon-
From Dye Laser
SBS Cell
t-z-
\Diffusing Screen
Fig. 2. Experimental arrangement for the recording of input and backscatter streaks. The distances x, y and z are discussed in the text.
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Table 1 Material
gn (cmMW-I)
rB (ns) 710nm
490 nm
455 nm
390 nm
308 nm
Xe a) (Sob) (25b) CClF, b, (30b) CZFeC) (23b) CS2 d, (liq)
0.09 0.015 0.056 0.016 0.15
16.1 8.1 8.8 4.9 2.83
11.1 5.6 7.5 3.4 1.95
6.6 3.3 5.6 2.0 1.16
4.9 2.5 4.6 1.5 0.85 ”
3.0 1.5 3.3 0.92 _ e)
n-hex d, (liq)
0.023
0.76
0.52
0.31
0.23
0.14
a) Assuming d, Assuming
b, ri’ =4.78X 109/& ideal L* variation. ‘) Absorbs. viscosity high, :.1’ variation.
+ 3.25 x 10’~
Input
Return
Fig. 3. Streak of input and backscatter of 7 10 nm radiation using CCIF, (30 bar) as the SBS medium. Temporal and spatial scales are as fig. 1. After an onset delay the fidelity remains good.
Input
Return
Fig. 4. As fig. 3, except reduced increased, and fidelity reduced.
input intensity.
Onset delay is
strated. At lower input intensities, however, a different behaviour is observed, as shown in fig. 4, which is for the same wavelength and material as fig. 3. The delay to the onset of backscatter is increased, and both the temporal and spatial profiles fail to follow the rapid variations of the input pulse. The above results may be explained by a simple qualitative argument based on the familiar transient threshold condition, eq. (7). This equation describes the growth of the Stokes wave by scattering from thermal noise acoustic phonons, and hence is 350
[ 111.
‘) Assuming
ideal at pressures
$23 bar
[ 111.
applicable to the leading edge of the input. The response of the medium to variations in the spatial or temporal profile of the input intensity which occur during the pulse may be expected to obey a similar formula. In this case, however, the acoustic wave is unlikely to grow from noise, as waves some lo- 13 orders of magnitude greater than this already exist (albeit decaying) in the medium, Accordingly, we may introduce a phenomenological factor A on the RHS of eq. (7) where A varies according to the magnitude of the change in the input profile (being unity for 100% change, i.e. the onset or cesation of the input, reducing towards zero for smaller changes). If the input laser intensity can be taken as the temporally averaged value, 4 over the response time, t’, one obtains the response time from eq. (7 ) as t’=25=A2rB/~gBl.
(8)
Recalling that steady-state Brillouin threshold can be written as g,ZZ= 30 [ 191, eq. (8) can be recast as t’x7JN where N represents the number of times above the steady-state threshold. The apparent linear relationship between response time and acoustic decay time, is tempered by the fact that g, depends on 7, via [9] gB =
yhk.rB 2n2c2VQo
’
where ye is the electrostrictive coefficient, n is the medium refractive index, V, the acoustic velocity and p0 the density of the SBS medium. Substituting eq. (9) in eq. (8) yields
(10)
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Recalling that rBaL2 (eq. ( 1) ), the linear relationship between t’ and 7B remains, and this clearly implies faster response at shorter wavelengths. This dependence has been confirmed experimentally in the present work. Media with large Brillouin nonlinearities, large ye, will also result in faster response. Equation ( 10) also indicates faster response as the interaction length is increased, as a result of the greater Brillouin gain. However, this trend must be tempered by the realisation that both reflectivity and, in particular, fidelity will suffer once the optical (double) transit time of the interaction region becomes comparable to the predicted response time, i.e. t’ cannot reasonably be lower than 2nllc. Perhaps of most interest is the inverse dependence of the response time, t’, on the laser intensity, I. This effect is shown quite graphically in figs. 3 and 4 where both the initial and intra-pulse response times are seen to vary with overall intensity. Of greatest practical use is the demonstration that, at sufficiently high intensities, the effective response time can be much faster than zB. Figure 3 shows intra-pulse response almost an order of magnitude faster than tB of 8.8 ns. Response times better than r,/20 have been achieved using high pressure xenon as the SBS medium. For variations of the type and magnitude of the present work, the fidelity appears to remain high during the majority of the pulse once the backscatter is initiated. The limit to the response time which can be achieved in a given medium by the simple expedient of increasing the focal intensity is governed by the onset of parasitic processes, and in particular breakdown. The effects on fidelity can be quite severe. Figure 5 shows the results of an experiment under the same conditions as gave the good conjugation
IflplJt
Return
Fig. 5. As fig. 1, except increased input intensity. Breakdown related phenomena greatly reduce the spatial fidelity.
1 December 1992
shown in fig. 1, but with a higher intensity. The spatial profile is severely degraded, although the overall temporal profile shows fast response and high fidelity. The optical breakdown intensity for a given medium is a complex function of wavelength, pressure, pulse duration, focal spot size and medium purity [ 18,19 1. Both magnetic [ 201 and electric [ 2 1] fields can also be used to change the breakdown threshold. Only the most basic trends are considered in the present work. At laser wavelengths longer than approximately 1 urn, cascade ionisation processes dominate resulting in a breakdown threshold, IaD, which scales as ZBDa 1/A’ [ 18,191. In the limit of I= ZBD,eq. ( 10) then predicts that the response time is given as t&,,aA4 (or 7d2) indicating significant practical difficulties in obtaining rapid response at longer wavelengths. Rather conversely, for wavelength less than approximately 500 nm, multi photon ionisation becomes the dominant breakdown mechanism, and for n photon ionisation, this process has a breakdown threshold which scales roughly as Z,,aA”. This leads to t& aA2-” (or zB/An) which, as n is usually greater than 2, leads to a slower optimimal response time as the wavelength becomes shorter. In the intermediate optical breakdown region which incorporates the important spectral region ( 1 urn 212 500 nm) where both cascade and multiphoton ionisation effects are important, the breakdown intensity has a maximum and varies only relatively slowly with wavelength [ 22 1. It will be in this wavelength range where many media display their optimum, breakdown limited response time, GD.
5. Conclusions Efficient, high fidelity conjugation has been demonstrated on timescales much faster than the acoustic decay time, and the predictions of a simple semianalytical description of the response time have been qualitatively verified. The effects on spatial and temporal fidelity of using both too low an intensity, leading to sluggish response, and too high an intensity, leading to optical breakdown, have been graphically demonstrated. Using the present beam profiles, specular reflections [ 231 were not observed. The semi-analytical description allows the effects ob351
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served in the present work at high intensities and short pulses to be scaled to slower timescales, longer pulses and longer wavelengths. The areas of greatest applicability are likely to include phase conjugate resonators, where optical transit times of the aberrator (discharge, for example) are relatively short, and infra red conjugation, where the response is inherently slow.
Acknowledgements The authors wish to acknowledge the financial support of AEA Technology Corporate Research for this work.
References [ 1 ] B.Ya. Zeldovich, N.F. Pilipetskii and V.V. Shkunov, Principles of phase conjugation (Springer, Berlin 1985 ) [2] N.G. Basov, ed., Phase conjugation of laser emmission, (Nova Science, New York, 1988). [ 31 R.A Fisher, ed., Optical phase conjugation (Academic, New York, 1983). [4] R.W. Hellwarth, Opt. Eng. 21 (1982) 257. [ 5 ] M. Sugii, 0. Sugihara, M. Ando and K. Sasaki, J. Appl. Phys, 62 (1987) 3480. [ 61 M.R. Osborne, W.A. Schroeder, M.J. Damzen and M.H.R. Hutchinson, Apppl. Phys. B 48 ( 1989) 35 1.
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[ 71 M.T. Duignan, B.J. Feldman and W.T. Whitney, Optics Lett. 12 (1987) 111. [8] V.S. Starunov and I.L. Fabelinski, Sov. Phys. Uspk. 12 (1970) 463. [9] W. Kaiser and M. Maier, in: Laser handbook, Vol II, eds. Arecchi and Dubois (North-Holland, Amsterdam, 1972). [lo] R.Y. Chiao and P.A. Fleury, in: Physics of quantum electronics, eds. Kelley, Lax and Tannerwald (McGraw-Hill, New York, 1966). [ 111 M.J. Damzen, M.H.R. Hutchinson and W.A. Schroeder, IEEE J. Quantum Electron. QE23 (1987) 328. [ 121 H. Kogelnik, Bell Syst. Techn. J. 48 (1969) 2909. [ 13lM.R. Osborne, M.A. G’Key and A.J.B. Travis, Culham International Report, APN 547 (unpublished). [ 141 J. Munch, R.F. Wuerker and M.J. LeFebvre, Appl. Optics 28 (1989) 3099. [ 151 E.E. Hagenlocker, R.W. Minck and W.G. Rado, Phys. Rev. 154 (1967) 226. [ 161 A. Corvo and A. Gavrieldes, J. Appl. Phys. 63 (1988) 5220. [ 171 M.G. Littman and H.J. Metcalf, Appl. Optics 17 ( 1978) 2224. [ 181 D.C. Smith and R.G. Meyerand, in: Principles of laser plasmas, ed. G. Bekefi (Wiley, New York, 1976). [ 191 Y.R. Shen, The principles of nonlinear optics (Wiley, New York, 1984). [ 201 D.R. Cohn, C.E. Chase, W. Halverson and B. Lay, Appl. Phys. Lett. 20 (1972) 225. [21] R.A. Mullen and J.N. Matossian, Optics Lett. 15 (1990) 601. [ 22 ] H.T. Buscher, R.G. Tomlinson and E.K. Damon, Phys. Rev. Lett. 15 (1965) 847. [23] R.H. Moyer, in: Laser wavefront control, ed. J.F. Reintjes, SPIE Vol. 1000 (1989).
94 (1992)
346-352
OPTICS COMMUNICATIONS
Temporal response of stimulated Brillouin scattering phase conjugation M.R. Osborne and M.A. O’Key Laser Technology Centre, AEA Technology, Culham Laboratory, Abingdon, Oxon, OX14 3DB. UK Received
I April 1992
The response of stimulated Brillouin scattering phase conjugate reflectors to rapidly changing input beam profiles is investigated experimentally. Efficient, high fidelity response on timescales several times shorter than the acoustic decay time, ra, is demonstrated. The data obtained agree with a simple, semi-analytical theory which can be used to scale the results to other wavelengths, intensities and media.
1. Introduction
2. Theoretical background
Stimulated Brillouin scattering (SBS) has been extensively investigated as a technique for phase conjugating laser radiation [l-4]. Correction for a wide range of aberrations has been demonstrated in double pass configurations. Although, in principle, capable of compensating highly dynamic aberrations, most work has been carried out using static or slowly varying aberrators. This is because the majority of effort has been directed towards thermally generated or atmospheric aberrations, which vary only relatively slowly. However, the recent increased interest in using SBS both to correct for very rapidly varying aberrations (such as those that may be produced in plasmas or discharge excited lasers [ 5,6] ) and at longer wavelengths (e.g. for 3-5 pm HL laser emission [ 71) where the response is inherently slower, have made an investigation of the speed of response of the SBS process to temporally varying optical inputs of great importance. After a brief description of the theoretical background, the present paper presents the results of an experimental investigation of the speed and fidelity of response of several SBS media, and shows how these results may be interpreted through a semi analytical treatment.
In SBS, the input wave can be considered as being backscattered from a moving acoustic wave within the nonlinear medium. The Doppler shifted backscatter then beats with the incoming wave, and this beat frequency enhances the acoustic wave through electrostriction, establishing a positive feedback loop. The hypersonic acoustic wave assumes a phase structure which mimics that of the input beam, resulting in the backscatter approximating to the phase conjugate of the input. However, if the phasefront of the input wave were to change instantaneously, the existing acoustic wave would have the wrong phase structure, and non-conjugate backscatter would occur until the SBS medium has had time to respond to the change in input, and adjust the acoustic wavefront accordingly. It has been widely qualitatively accepted that this “response time” is closely related to the time taken for the pre-existing acoustic wave to decay, although this argument can only readily be made quantitative for the most simple of cases as described below. The acoustic decay time, fg, is a well documented figure for several materials [8,9] and has been determined both spectroscopically (as the inverse of the spontaneous Brillouin scattering linewidth) [lo] and directly by a probing technique [ 111. The decay time is a function of the wavelength of the acoustic grating, Izg, which, in backward scat-
346
0030-4018/92/$
05.00 0 1992 Elsevier Science Publishers
B.V. All rights reserved.
Volume 94, number
5
OPTICS
COMMUNICATIONS
tering, closely approximates to half of the optical wavelength, jlL. Considering the classical acoustic damping mechanisms of bulk and shear viscosity, and thermal conductivity, one obtains the relationship [1,9,111 z,aA;ul~.
(1)
This has been shown to apply accurately for nominally monatomic gasses e.g. xenon, but requires some modification in polyatomic gases due to the vibrational and rotational degrees of freedom [ 111. Although there is little experimental evidence, eq. ( 1) is presumed to be reasonably accurate in liquids [ 11 where the relatively high viscosity dominates. With no driving term and taking the plane wave approximation, an acoustic wave, Q (where Q is proportional to the density change, Ap, in the medium) will decay with time as
Q(O=Q(o> exp(--U&J
,
(2)
where rr, is the intensity acoustic damping time. The reflectivity, R, of the acoustic grating will vary in proportion to the square of the modulation of the refractive index [ 12 1, and hence as &* and Q*. It will therefore decay according to R(t)=R(O)
exp( -t/q.,)
.
(3)
If a low power (i.e. below SBS threshold) probe beam with the same spatial profile and colinear with the input is then applied to this acoustic wave, the reflectivity may be probed directly. This has been reported in ref. [ 111. If the below threshold probe has a different spatial profile, or is not colinear with the original beam, a more complex situation applies. The simplest case is that of a plane probe, displaced at an angle A6 from the original wave. In this case, it can be shown [ 12,13 ] that relatively efficient (i.e. 50% of the value for colinear incidence) specular reflection will occur for angles A0 in the range 805
-tJm,
(4)
where b is the effective depth of the grating, which will be between one and five [ 14 ] times the Rayleigh range in the focused geometry, assuming this to be shorter than both the cell length and three times the coherence length. The Rayleigh range for a beam with a divergence which is m * times diffraction limited is given by
b=m*Af */nd2,
1 December
1992
(5)
wherefis the focal length of the lens and d the beam diameter. Substituting eq. (5) in eq. (4) and neglecting the factor of Jn!2 which is of order unity, gives AOsd/m*f.
(6)
Inequality ( 6 ) implies that efficient reflection should occur for angles within the focussing cone of the lens. Hence, significant non-conjugate (near) backscatter may be expected to occur for rapid variations in spatial profile which may be considered as a perturbation of the profile within a fixed beam diameter. The reflectivity will be slightly reduced from, but have the same temporal form as eq. (3 ). If the probe beam is of high power (i.e. above SBS threshold) it will eventually generate its own acoustic wave, producing a phase conjugate return. However, this will not occur instantaneously, being determined by the transient SBS threshold. In the case of a single input beam with backscatter starting from thermal noise phonons, the transient threshold may be expressed as
[151, (g,l j IL dt/rie)ii2=25,
(7)
-cc where gB is the Brillouin gain coefficient, Zis the interaction length and IL the intensity of the incident laser. It is difficult to quantify how relevant eq. (7) is to the present situation of rapidly changing small variations of the input spatial profile. Several factors may significantly alter eq. ( 7 ) . Firstly, for times prior to the attainment of the transient threshold, a fraction of the incident radiation may be Bragg reflected from the existing (decaying) grating. This will give rise to non-conjugate backscatter as described previously. It will also modify the expected value and distribution of I,_ in the focal region. Perhaps more significantly, the input and Bragg reflected waves will be perfectly matched in angle and frequency to enhance the existing grating by non 180’ SBS [ 161. In conventional, single beam SBS, non 180’ scattering is suppressed by a combination of increased transient threshold, reduced interaction length due to beam walk-off, and, in some cases, increased laser bandwidth restrictions. In the present 347
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case, however, the transient threshold does not apply as the required grating is already in existence. Also, the greater bandwidth restrictions are of no significance at the small angular deviation from 180’ considered. Only beam walk-off remains as a detriment to producing SBS at the Bragg diffraction angle from the original acoustic grating. Secondly, even assuming that the true phase conjugate return of the probe beam eventually dominates, the question arises as to whether the relevant acoustic wave is generated from noise, as eq. (5 ) presumes, or is produced as some form of mutation of the old acoustic wave which will be many orders of magnitude (perhaps 1O-l 3) above the thermal noise background. Finally, the existence of more than two beams in the interaction region, as could be produced by the existence of both a Bragg reflected and a phase conjugate reflected version of the new input pulse could allow four-wave mixing processes producing a beam counter-propagating to the specular reflection. Any temporal overlap between the original and the new input (or Stokes) beams, will further complicate the interaction. The precise effects of all of the above processes will depend critically on the exact form of the spatiotemporal variation of the input radiation - a single general case cannot be identified. Therefore, an experimental investigation was performed to determine the ability of SBS conjugators to respond to rapid variations, and which factors control this ability.
3. Experimental Given that the effects to be investigated were expected to scale as the square of the laser wavelength, eq. ( I), a tunable source of radiation was favoured. For ease of diagnostics, the photographic region of the spectrum was preferred. The relatively narrow bandwidth and high peak powers necessary to reach SBS threshold, together with the above factors, lead to the choice of an excimer-pumped tunable dye laser as the source of radiation. Moreover, previous experience had indicated that it was relatively easy to produce severely aberrated beams directly from such systems. The kind of structure which can be produced is shown graphically in the upper part of fig. 348
I December
1992
time
2mm
Input
I
Phase conjugate return
’
Fig.1
1Ons
’ 710nm
n-hexane
Fig. 1. Streak of input and backscatter of 7 10 nm radiation using n-hexane (~~-0.76 ns) as the SBS medium. Good spatial and temporal fidelity is observed.
1 which is a streak record of the dye laser output. The resolution in this figure is approximately 150 km in the vertical dimension (cf beam diameter of - 1.5 mm) and approximately 200 ps in the horizontal dimension (cf pulse length of - 15 ns). The modulation observed is caused by a number of factors. The excimer laser pump beam naturally shows two components of different divergence, which have different temporal evolutions and can be angularly separated by adjustment of the excimer cavity, so that they are focussed in different places by the cylindrical lens which concentrates the radiation onto the dye oscillator cell. The focal regions are also separated in the direction of beam propagation. This complex pump beam behaviour added to the longitudinal mode beating of the dye oscillator produces the highly modulated beam which, after one stage of unsaturated amplification is shown in fig. 1. Operated in this mode, the output from the Littman type [ 17 ] dye laser has a temporally averaged bandwidth of typically 4-6 GHz, as measured with a Fabry-Ptrot etalon (corresponding to 4-6 longitudinal modes) and a beam divergence of typically 2 mrad (full angle), being approximately three times the diffraction limit. The phase of this beam is clearly complex and difficult to define precisely, but will certainly vary on a timescale not slower than the observed intensity fluctuations. The input (reference) and phase conjugate beams were recorded with
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the experimental arrangement shown in fig. 2. Mirror M is slightly misaligned vertically to separate the two beams, and the distances x and z were made equal to produce temporally synchronised beams on the diffusing screen thereby greatly simplifying the recording of the streaks. However, in this arrangement the input and backscattered waves are not recorded in conjugate planes, the input beam effectively propagating a distance 2x+ 2y further than the conjugated beam before reaching the screen. The effects of this mismatch were minimised by a combination of two procedures. Firstly, the distances x and y were minimised, giving 2x+2y=40 cm, and secondly the input beam was propagated a considerable distance ( > 6 m) from any source of aberration before reaching the beamsplitter in fig. 2. At such distances, the change in spatial profile with propagation is slow, and is indeed found to be negligible over the 40 cm path difference. Several gaseous and liquid SBS media of differing acoustic decay times were investigated (see table 1) . The acoustic decay time was varied in two ways. Firstly by tuning the dye laser, and using eq. ( 1)) or one of its modified forms [ 111. This method keeps several SBS material parameters constant (steady state gain coefficient, pressure, density, viscosity,
temperature) but does vary the transient threshold, as described by eq. (7 ). To obtain some other decay times, the pressure of the gaseous media was varied, although this has the disadvantage of altering the parameters kept constant above. In all, a range of r, of approximately two orders of magnitude from 0.14 to 16 ns was covered.
4. Results and discussion Under some conditions both the spatial and temporal fidelity of the conjugation were good, as can be seen from the comparison of input and SBS return beams shown in fig. 1. This result was obtained using n-hexane as SBS medium with an input wavelength of 7 10 nm. As shown in table 1, the acoustic decay time in this case of -0.76 ns is comparable to the timescale on which gross changes of the input beam occur. Also, in this case, it can be seen that there is very little delay to the initiation of the backscatter. In fig. 3, the same wavelength (7 10 nm) is used with 30 bar of CClF3 as the SBS medium. In this case the acoustic decay time is -8.8 ns. However, after an onset delay of approximately 3 ns, high fidelity response on a nanosecond timescale is still demon-
From Dye Laser
SBS Cell
t-z-
\Diffusing Screen
Fig. 2. Experimental arrangement for the recording of input and backscatter streaks. The distances x, y and z are discussed in the text.
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Table 1 Material
gn (cmMW-I)
rB (ns) 710nm
490 nm
455 nm
390 nm
308 nm
Xe a) (Sob) (25b) CClF, b, (30b) CZFeC) (23b) CS2 d, (liq)
0.09 0.015 0.056 0.016 0.15
16.1 8.1 8.8 4.9 2.83
11.1 5.6 7.5 3.4 1.95
6.6 3.3 5.6 2.0 1.16
4.9 2.5 4.6 1.5 0.85 ”
3.0 1.5 3.3 0.92 _ e)
n-hex d, (liq)
0.023
0.76
0.52
0.31
0.23
0.14
a) Assuming d, Assuming
b, ri’ =4.78X 109/& ideal L* variation. ‘) Absorbs. viscosity high, :.1’ variation.
+ 3.25 x 10’~
Input
Return
Fig. 3. Streak of input and backscatter of 7 10 nm radiation using CCIF, (30 bar) as the SBS medium. Temporal and spatial scales are as fig. 1. After an onset delay the fidelity remains good.
Input
Return
Fig. 4. As fig. 3, except reduced increased, and fidelity reduced.
input intensity.
Onset delay is
strated. At lower input intensities, however, a different behaviour is observed, as shown in fig. 4, which is for the same wavelength and material as fig. 3. The delay to the onset of backscatter is increased, and both the temporal and spatial profiles fail to follow the rapid variations of the input pulse. The above results may be explained by a simple qualitative argument based on the familiar transient threshold condition, eq. (7). This equation describes the growth of the Stokes wave by scattering from thermal noise acoustic phonons, and hence is 350
[ 111.
‘) Assuming
ideal at pressures
$23 bar
[ 111.
applicable to the leading edge of the input. The response of the medium to variations in the spatial or temporal profile of the input intensity which occur during the pulse may be expected to obey a similar formula. In this case, however, the acoustic wave is unlikely to grow from noise, as waves some lo- 13 orders of magnitude greater than this already exist (albeit decaying) in the medium, Accordingly, we may introduce a phenomenological factor A on the RHS of eq. (7) where A varies according to the magnitude of the change in the input profile (being unity for 100% change, i.e. the onset or cesation of the input, reducing towards zero for smaller changes). If the input laser intensity can be taken as the temporally averaged value, 4 over the response time, t’, one obtains the response time from eq. (7 ) as t’=25=A2rB/~gBl.
(8)
Recalling that steady-state Brillouin threshold can be written as g,ZZ= 30 [ 191, eq. (8) can be recast as t’x7JN where N represents the number of times above the steady-state threshold. The apparent linear relationship between response time and acoustic decay time, is tempered by the fact that g, depends on 7, via [9] gB =
yhk.rB 2n2c2VQo
’
where ye is the electrostrictive coefficient, n is the medium refractive index, V, the acoustic velocity and p0 the density of the SBS medium. Substituting eq. (9) in eq. (8) yields
(10)
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Recalling that rBaL2 (eq. ( 1) ), the linear relationship between t’ and 7B remains, and this clearly implies faster response at shorter wavelengths. This dependence has been confirmed experimentally in the present work. Media with large Brillouin nonlinearities, large ye, will also result in faster response. Equation ( 10) also indicates faster response as the interaction length is increased, as a result of the greater Brillouin gain. However, this trend must be tempered by the realisation that both reflectivity and, in particular, fidelity will suffer once the optical (double) transit time of the interaction region becomes comparable to the predicted response time, i.e. t’ cannot reasonably be lower than 2nllc. Perhaps of most interest is the inverse dependence of the response time, t’, on the laser intensity, I. This effect is shown quite graphically in figs. 3 and 4 where both the initial and intra-pulse response times are seen to vary with overall intensity. Of greatest practical use is the demonstration that, at sufficiently high intensities, the effective response time can be much faster than zB. Figure 3 shows intra-pulse response almost an order of magnitude faster than tB of 8.8 ns. Response times better than r,/20 have been achieved using high pressure xenon as the SBS medium. For variations of the type and magnitude of the present work, the fidelity appears to remain high during the majority of the pulse once the backscatter is initiated. The limit to the response time which can be achieved in a given medium by the simple expedient of increasing the focal intensity is governed by the onset of parasitic processes, and in particular breakdown. The effects on fidelity can be quite severe. Figure 5 shows the results of an experiment under the same conditions as gave the good conjugation
IflplJt
Return
Fig. 5. As fig. 1, except increased input intensity. Breakdown related phenomena greatly reduce the spatial fidelity.
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shown in fig. 1, but with a higher intensity. The spatial profile is severely degraded, although the overall temporal profile shows fast response and high fidelity. The optical breakdown intensity for a given medium is a complex function of wavelength, pressure, pulse duration, focal spot size and medium purity [ 18,19 1. Both magnetic [ 201 and electric [ 2 1] fields can also be used to change the breakdown threshold. Only the most basic trends are considered in the present work. At laser wavelengths longer than approximately 1 urn, cascade ionisation processes dominate resulting in a breakdown threshold, IaD, which scales as ZBDa 1/A’ [ 18,191. In the limit of I= ZBD,eq. ( 10) then predicts that the response time is given as t&,,aA4 (or 7d2) indicating significant practical difficulties in obtaining rapid response at longer wavelengths. Rather conversely, for wavelength less than approximately 500 nm, multi photon ionisation becomes the dominant breakdown mechanism, and for n photon ionisation, this process has a breakdown threshold which scales roughly as Z,,aA”. This leads to t& aA2-” (or zB/An) which, as n is usually greater than 2, leads to a slower optimimal response time as the wavelength becomes shorter. In the intermediate optical breakdown region which incorporates the important spectral region ( 1 urn 212 500 nm) where both cascade and multiphoton ionisation effects are important, the breakdown intensity has a maximum and varies only relatively slowly with wavelength [ 22 1. It will be in this wavelength range where many media display their optimum, breakdown limited response time, GD.
5. Conclusions Efficient, high fidelity conjugation has been demonstrated on timescales much faster than the acoustic decay time, and the predictions of a simple semianalytical description of the response time have been qualitatively verified. The effects on spatial and temporal fidelity of using both too low an intensity, leading to sluggish response, and too high an intensity, leading to optical breakdown, have been graphically demonstrated. Using the present beam profiles, specular reflections [ 231 were not observed. The semi-analytical description allows the effects ob351
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served in the present work at high intensities and short pulses to be scaled to slower timescales, longer pulses and longer wavelengths. The areas of greatest applicability are likely to include phase conjugate resonators, where optical transit times of the aberrator (discharge, for example) are relatively short, and infra red conjugation, where the response is inherently slow.
Acknowledgements The authors wish to acknowledge the financial support of AEA Technology Corporate Research for this work.
References [ 1 ] B.Ya. Zeldovich, N.F. Pilipetskii and V.V. Shkunov, Principles of phase conjugation (Springer, Berlin 1985 ) [2] N.G. Basov, ed., Phase conjugation of laser emmission, (Nova Science, New York, 1988). [ 31 R.A Fisher, ed., Optical phase conjugation (Academic, New York, 1983). [4] R.W. Hellwarth, Opt. Eng. 21 (1982) 257. [ 5 ] M. Sugii, 0. Sugihara, M. Ando and K. Sasaki, J. Appl. Phys, 62 (1987) 3480. [ 61 M.R. Osborne, W.A. Schroeder, M.J. Damzen and M.H.R. Hutchinson, Apppl. Phys. B 48 ( 1989) 35 1.
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[ 71 M.T. Duignan, B.J. Feldman and W.T. Whitney, Optics Lett. 12 (1987) 111. [8] V.S. Starunov and I.L. Fabelinski, Sov. Phys. Uspk. 12 (1970) 463. [9] W. Kaiser and M. Maier, in: Laser handbook, Vol II, eds. Arecchi and Dubois (North-Holland, Amsterdam, 1972). [lo] R.Y. Chiao and P.A. Fleury, in: Physics of quantum electronics, eds. Kelley, Lax and Tannerwald (McGraw-Hill, New York, 1966). [ 111 M.J. Damzen, M.H.R. Hutchinson and W.A. Schroeder, IEEE J. Quantum Electron. QE23 (1987) 328. [ 121 H. Kogelnik, Bell Syst. Techn. J. 48 (1969) 2909. [ 13lM.R. Osborne, M.A. G’Key and A.J.B. Travis, Culham International Report, APN 547 (unpublished). [ 141 J. Munch, R.F. Wuerker and M.J. LeFebvre, Appl. Optics 28 (1989) 3099. [ 151 E.E. Hagenlocker, R.W. Minck and W.G. Rado, Phys. Rev. 154 (1967) 226. [ 161 A. Corvo and A. Gavrieldes, J. Appl. Phys. 63 (1988) 5220. [ 171 M.G. Littman and H.J. Metcalf, Appl. Optics 17 ( 1978) 2224. [ 181 D.C. Smith and R.G. Meyerand, in: Principles of laser plasmas, ed. G. Bekefi (Wiley, New York, 1976). [ 191 Y.R. Shen, The principles of nonlinear optics (Wiley, New York, 1984). [ 201 D.R. Cohn, C.E. Chase, W. Halverson and B. Lay, Appl. Phys. Lett. 20 (1972) 225. [21] R.A. Mullen and J.N. Matossian, Optics Lett. 15 (1990) 601. [ 22 ] H.T. Buscher, R.G. Tomlinson and E.K. Damon, Phys. Rev. Lett. 15 (1965) 847. [23] R.H. Moyer, in: Laser wavefront control, ed. J.F. Reintjes, SPIE Vol. 1000 (1989).