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Relation between optical breakdown field and stokes spectral broadening
Volume 14, number 1
RELATION
May 1975
OPTICS COMMUNICATIONS
BETWEEN OPTICAL BREAKDOWN FIELD AND STOKES SPECTRAL BROADENING
John MARBURGER Department
of Physics, University of Southern
Californti, Los Angeles, Californin 90007,
USA
Received 21 October 1974, revised version received 3 February 1975 The process which limits the diameter of self focal spots must also limit the Stokes self phase modulated spectral width. A simple analysis is consistent with data for nsec pulses in organic liquids.
That self focusing may lead to optically induced bulk damage tracks in a variety of solid materials is now well established [ 11. There is also good reason to suppose that the damage itself is preceded by avalanche production of electrons whose negative contribution to the non-linear index change limits the evolution of self focusing toward a “singularity” in the local intensity [2]. Many features of the self phase-modulated spectrum of intense pulses are also well understood, and it appears highly probable that in some cases pulse shaping effects due to self focusing play an important role in determining the extent of broadening of the transmitted spectrum [3]. We wish to point out that the process which limits self focusing also limits the extent of the downshifted (Stokes) self phase-modulated spectrum. This circumstance suggests the possibility of inferring the threshold intensity for the limiting process from the transmitted Stokes spectral width. If avalanche electron production is the limiting mechanism, then the spectral width must be related to the breakdown field strength. This relation is most conveniently investigated when the response is not transient, and the stationary theory can be used to find the time dependence of the local optical intensity 1(x, y, z, t). A gaussian equiphase beam of power P(t) entering the medium at z = 0 possesses the axial intensity
Z(O,O,O,t-z/u)
Z(O,O, z, t) = ___
o- Wf(~4J>l
2 (Y/2
1
.
(1)
where (Yapproaches unity for large powers, and 0.13~[$~0.0219+([~~2-Og~2)2~
92
(2)
The limiting forms of these equations for P S P, may be obtained by arguments due to Kelley [4], but the general forms are fits to extensive numerical solutions [5] for P 2 2P2. Here P2 is the least power for which an axial singularity forms at z = 00. Numerical computation shows P, = 0.471 c3/n2wo,
(3)
where the index change is n-n,, = n2 (E2 >, w. is the incident frequency, a the 1/e intensity radius, k = nowO/c, and u the group velocity at wo. Eqs. (1) and (2) may be employed in the WKB expression for the axial optical phase $(z, t’)= kz +-
n2ao
c
z s 0
(E2(z’,
t’)>
dz’,
to find a formula for the “instantaneous w(t) = a0 -
hplat.
(4) frequency”
(5)
In eq. (4) t’ = I -z/u, which is fixed in the integration over z’. Using (l), (2) and (4) for large P/P2 in (5) one finds w(t)
X
= w.
-0.942
[sin-’
(k)
“f ?ka2p2
t$j$],
q dt
(6)
where zf, P and Z(z) are all functions of t’, and Z(z) G Z(O,O, z, v’). This equation shows clearly that wo-w(t) grows without bound as zf (t’) approaches z. If z denotes the exit plane of the medium, where the spectrum is determined, then we expect to see an infinite frequency shift as the first forward moving self focal
Volume
point passes out of the medium. (The WKB approximation breaks down when the local intensity varies too rapidly.) If the self-focusing process is limited, however, by a mechanism whose influence becomes significant at the intensity In, then the spectral shift does not grow much beyond its value at the time when Z(z) = ZB. In the cases of interest here, ZB may be assumed large compared to Z(O), and therefore Z(z) may reach this value only near a self focus. That is, I a ZB implies z w zf, and eq. (6) becomes approximately Wmax - *o = -0.942
Z
ka2K
ldpzB dtIo
= - Aw ZB/2Z(0).
(7)
Here Z(0) and dP/dt are evaluated at that power which yields a self focus at .z on the rising portion of the pulse. Since dP/dt is positive there, eq. (7) is a formula for the maximum Stokes frequency shift. Of course light from later more powerful portions of the pulse must also pass through a self focus, but that focus lies within the medium, where electron production may occur. Light from such a focus must always pass through a subsequent region in which avalanche electron production is taking place. That is, it is exposed to a rapidly decreasing refractive index which diminishes and may even reverse the net frequency sweep. The only light on the rising portion of the pulse which passes through a self focus, but does not pass through an avalanche region, is that in the portion which self focuses at the exit face. If the limiting mechanism is not avalanche electron production, then this reasoning is not directly applicable. Eq. (7) is nevertheless rather insensitive to the precise time at which Z(0) and dP/dt are evaluated, and for numerical estimates, we shall use the approximation dP/dt = 2ra2Z(O)/T,
(8)
which is appropriate at the leading edge of an incident parabolic pulse of duration 2 T. Eq. (7) should be compared with the corresponding expression without self focusing: o-w0 = - A.w. The effect of self focusing is evidently to increase the spectral width by the factor ZB/2Z(0), where Z(0) is the incident axial intensity required to form a self focus at the exit plane of the non-linear medium. Using (8), one might also say that self focusing, limited 2
May 1975
OPTICS COMMUNICATIONS
14, number 1
by avalanche breakdown, leads to an effective pulse length for estimating self broadening in a model which ignores self focusing: Teff = 21(O) T/ZB.
(9)
Gustafson et al. [3] invoked an effective pulse length shortened by self focusing to interpret spectral broadening of Q-switched ruby laser pulses in CS,, but at that time there was no theory available for estimating its magnitude a priori. Shen and Ioy [6] also employed an effective pulse whose duration was estimated to be comparable to the transient response time of the non-linear index. In both cases a reduction in pulse length of up to two orders of magnitude was required to obtain agreement with experimental data. With an incident power of 5 MW in a f mm radius beam, this implies a maximum breakdown intensity ofZB = 1.3 X 1011 W/cm2 which is not an unreasonable upper limit [2]. Better comparison with data is difficult because complete details of beam parameters are not reported in the experimental papers, and indeed are difficult to measure. Recent advances in beam diagnostics, however, should allow more precise studies of the quantitative relation between spectral broadening and breakdown threshold. The theory presented here indicates that the avalanche electron production mechanism alone is adequate to explain observed Stokes spectral widths, just as it accounts well for the limiting diameters of self-focussed beams [2]. This theory can be refined in many details to permit more credible comparison with real experiments. For example, the effect of transiency may be estimated by ignoring self focusing, and by assuming that the index change 6n is driven by a parabolic pulse of duration 2 Teff. If 6 n satisfies rat
a
6n = n2(E2)
then the maximum found to be w(t’)-o,=
-Ao-
- 6n, Stokes frequency
shift is readily
(10)
The accuracy of the last factor can be checked by comparison with numerical calculations of self phasemodulated spectra reported by Gustafson et al. [3]. In their fig. 6, for example, the product Awn,& T is 119, while eq. (10) yields 115 for their parameters. 93
Volume 14, number 1
OPTICS COMMUNICATIONS
One may also verify that the general form of eq. (7) is not sensitive to the detailed form of eqs. (1) and (2). It is only necessary that the axial intensity increase precipitously at z = zf, and that 1/zf” is proportional to P/Pz. These conditions are satisfied by all forms of self focusing, including that arising from enhancement of hot spots on an otherwise smooth beam [7]. Finally, one should realize that although extensive spectral broadening of picosecond pulses in solids occurs in small hot spots, it is not always necessary to invoke self focusing induced pulse shortening to understand the observed spectral width. Alfano and Shapiro have shown that the broadening in their experiments is consistent with the known incident pulse duration
181.
94
May 1975
References [l] C.R. Giuliano and J.H. Marburger, Phys. Rev. Lett. 27 (1971) 905. [2] E. Yablonovitch and N. Bloembergen, Phys. Rev. Lett. 29 (1972) 907. [3] T.K. Gustafson, J.P. Taran, H.A. Haus, J.R. Lifsitz and P.L. Kelley, Phys. Rev. 177 (1969) 306; [4] P.L. Kelley, Phys. Rev. Lett. 15 (1965) 1005. [5] E.L. Dawes and J.H. Marburger, Phys. Rev. 179 (1969) 862. [6] Y.R. Shen and M.M.T. Loy, Phys. Rev. A3 (1971) 2099. [7] V.I. Bespalov and V.I. Talanov, JETP Lett. 3 (1966) 307. [8] R.R. Alfano and S.L. Shapiro, Phys. Rev. Lett. 24 (1970) 592.
RELATION
May 1975
OPTICS COMMUNICATIONS
BETWEEN OPTICAL BREAKDOWN FIELD AND STOKES SPECTRAL BROADENING
John MARBURGER Department
of Physics, University of Southern
Californti, Los Angeles, Californin 90007,
USA
Received 21 October 1974, revised version received 3 February 1975 The process which limits the diameter of self focal spots must also limit the Stokes self phase modulated spectral width. A simple analysis is consistent with data for nsec pulses in organic liquids.
That self focusing may lead to optically induced bulk damage tracks in a variety of solid materials is now well established [ 11. There is also good reason to suppose that the damage itself is preceded by avalanche production of electrons whose negative contribution to the non-linear index change limits the evolution of self focusing toward a “singularity” in the local intensity [2]. Many features of the self phase-modulated spectrum of intense pulses are also well understood, and it appears highly probable that in some cases pulse shaping effects due to self focusing play an important role in determining the extent of broadening of the transmitted spectrum [3]. We wish to point out that the process which limits self focusing also limits the extent of the downshifted (Stokes) self phase-modulated spectrum. This circumstance suggests the possibility of inferring the threshold intensity for the limiting process from the transmitted Stokes spectral width. If avalanche electron production is the limiting mechanism, then the spectral width must be related to the breakdown field strength. This relation is most conveniently investigated when the response is not transient, and the stationary theory can be used to find the time dependence of the local optical intensity 1(x, y, z, t). A gaussian equiphase beam of power P(t) entering the medium at z = 0 possesses the axial intensity
Z(O,O,O,t-z/u)
Z(O,O, z, t) = ___
o- Wf(~4J>l
2 (Y/2
1
.
(1)
where (Yapproaches unity for large powers, and 0.13~[$~0.0219+([~~2-Og~2)2~
92
(2)
The limiting forms of these equations for P S P, may be obtained by arguments due to Kelley [4], but the general forms are fits to extensive numerical solutions [5] for P 2 2P2. Here P2 is the least power for which an axial singularity forms at z = 00. Numerical computation shows P, = 0.471 c3/n2wo,
(3)
where the index change is n-n,, = n2 (E2 >, w. is the incident frequency, a the 1/e intensity radius, k = nowO/c, and u the group velocity at wo. Eqs. (1) and (2) may be employed in the WKB expression for the axial optical phase $(z, t’)= kz +-
n2ao
c
z s 0
(E2(z’,
t’)>
dz’,
to find a formula for the “instantaneous w(t) = a0 -
hplat.
(4) frequency”
(5)
In eq. (4) t’ = I -z/u, which is fixed in the integration over z’. Using (l), (2) and (4) for large P/P2 in (5) one finds w(t)
X
= w.
-0.942
[sin-’
(k)
“f ?ka2p2
t$j$],
q dt
(6)
where zf, P and Z(z) are all functions of t’, and Z(z) G Z(O,O, z, v’). This equation shows clearly that wo-w(t) grows without bound as zf (t’) approaches z. If z denotes the exit plane of the medium, where the spectrum is determined, then we expect to see an infinite frequency shift as the first forward moving self focal
Volume
point passes out of the medium. (The WKB approximation breaks down when the local intensity varies too rapidly.) If the self-focusing process is limited, however, by a mechanism whose influence becomes significant at the intensity In, then the spectral shift does not grow much beyond its value at the time when Z(z) = ZB. In the cases of interest here, ZB may be assumed large compared to Z(O), and therefore Z(z) may reach this value only near a self focus. That is, I a ZB implies z w zf, and eq. (6) becomes approximately Wmax - *o = -0.942
Z
ka2K
ldpzB dtIo
= - Aw ZB/2Z(0).
(7)
Here Z(0) and dP/dt are evaluated at that power which yields a self focus at .z on the rising portion of the pulse. Since dP/dt is positive there, eq. (7) is a formula for the maximum Stokes frequency shift. Of course light from later more powerful portions of the pulse must also pass through a self focus, but that focus lies within the medium, where electron production may occur. Light from such a focus must always pass through a subsequent region in which avalanche electron production is taking place. That is, it is exposed to a rapidly decreasing refractive index which diminishes and may even reverse the net frequency sweep. The only light on the rising portion of the pulse which passes through a self focus, but does not pass through an avalanche region, is that in the portion which self focuses at the exit face. If the limiting mechanism is not avalanche electron production, then this reasoning is not directly applicable. Eq. (7) is nevertheless rather insensitive to the precise time at which Z(0) and dP/dt are evaluated, and for numerical estimates, we shall use the approximation dP/dt = 2ra2Z(O)/T,
(8)
which is appropriate at the leading edge of an incident parabolic pulse of duration 2 T. Eq. (7) should be compared with the corresponding expression without self focusing: o-w0 = - A.w. The effect of self focusing is evidently to increase the spectral width by the factor ZB/2Z(0), where Z(0) is the incident axial intensity required to form a self focus at the exit plane of the non-linear medium. Using (8), one might also say that self focusing, limited 2
May 1975
OPTICS COMMUNICATIONS
14, number 1
by avalanche breakdown, leads to an effective pulse length for estimating self broadening in a model which ignores self focusing: Teff = 21(O) T/ZB.
(9)
Gustafson et al. [3] invoked an effective pulse length shortened by self focusing to interpret spectral broadening of Q-switched ruby laser pulses in CS,, but at that time there was no theory available for estimating its magnitude a priori. Shen and Ioy [6] also employed an effective pulse whose duration was estimated to be comparable to the transient response time of the non-linear index. In both cases a reduction in pulse length of up to two orders of magnitude was required to obtain agreement with experimental data. With an incident power of 5 MW in a f mm radius beam, this implies a maximum breakdown intensity ofZB = 1.3 X 1011 W/cm2 which is not an unreasonable upper limit [2]. Better comparison with data is difficult because complete details of beam parameters are not reported in the experimental papers, and indeed are difficult to measure. Recent advances in beam diagnostics, however, should allow more precise studies of the quantitative relation between spectral broadening and breakdown threshold. The theory presented here indicates that the avalanche electron production mechanism alone is adequate to explain observed Stokes spectral widths, just as it accounts well for the limiting diameters of self-focussed beams [2]. This theory can be refined in many details to permit more credible comparison with real experiments. For example, the effect of transiency may be estimated by ignoring self focusing, and by assuming that the index change 6n is driven by a parabolic pulse of duration 2 Teff. If 6 n satisfies rat
a
6n = n2(E2)
then the maximum found to be w(t’)-o,=
-Ao-
- 6n, Stokes frequency
shift is readily
(10)
The accuracy of the last factor can be checked by comparison with numerical calculations of self phasemodulated spectra reported by Gustafson et al. [3]. In their fig. 6, for example, the product Awn,& T is 119, while eq. (10) yields 115 for their parameters. 93
Volume 14, number 1
OPTICS COMMUNICATIONS
One may also verify that the general form of eq. (7) is not sensitive to the detailed form of eqs. (1) and (2). It is only necessary that the axial intensity increase precipitously at z = zf, and that 1/zf” is proportional to P/Pz. These conditions are satisfied by all forms of self focusing, including that arising from enhancement of hot spots on an otherwise smooth beam [7]. Finally, one should realize that although extensive spectral broadening of picosecond pulses in solids occurs in small hot spots, it is not always necessary to invoke self focusing induced pulse shortening to understand the observed spectral width. Alfano and Shapiro have shown that the broadening in their experiments is consistent with the known incident pulse duration
181.
94
May 1975
References [l] C.R. Giuliano and J.H. Marburger, Phys. Rev. Lett. 27 (1971) 905. [2] E. Yablonovitch and N. Bloembergen, Phys. Rev. Lett. 29 (1972) 907. [3] T.K. Gustafson, J.P. Taran, H.A. Haus, J.R. Lifsitz and P.L. Kelley, Phys. Rev. 177 (1969) 306; [4] P.L. Kelley, Phys. Rev. Lett. 15 (1965) 1005. [5] E.L. Dawes and J.H. Marburger, Phys. Rev. 179 (1969) 862. [6] Y.R. Shen and M.M.T. Loy, Phys. Rev. A3 (1971) 2099. [7] V.I. Bespalov and V.I. Talanov, JETP Lett. 3 (1966) 307. [8] R.R. Alfano and S.L. Shapiro, Phys. Rev. Lett. 24 (1970) 592.