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The linewidth dependence of resonant, small signal sum frequency mixing
Volume 36, number 3
OPTICS COMMUNICATIONS
1 February 1981
THE LINEWIDTH DEPENDENCE OF RESONANT, SMALL SIGNAL SUM FREQUENCY MIXING C. LEUBNER *, H. SCHEINGRABER and C.R. VIDAL
Max-Planck-Institut fffr Extraterrestrische Physik, D-8046 Garehing,FederalRepublic of Germany Received 10 October 1980
For small input intensities, asymptotic approximations are derived for the dependence of sum frequency mixing on the pump laser linewidth, Doppler broadening, pressure broadening, and the pump laser detuning. Compared to the relations given by Stappaerts and coworkers, the results presented have a greatly extended range of validity and cover most situations of practical importance.
For sum frequency mixing in the small signal limit and for plane waves, the intensity qbs of the generated wave at the frequency cos = 2w 1 -+ 662 is given by (cf., e.g., [1]), [4rr266sL ] 2 qb2
*s= ~
" n2n~s(N~l
tibility X~ ) can be factored as [2] X(T3)(66s'661' 661' 602) "" (66r -- 260 1)-1X(F3)(66s), where 2
(3),2Fsin(AkL/2)l 2 XT )
L Ak~/2 j
'
X(3)(66s) "" ~ #
gi
(1) where q51, q52, q5s and n l , n2, n s are the intensities and the refractive indices at the frequencies ca 1, 6°2 and cos, respectively. N is the particle density and L the length of the nonlinear medium; ×~)(66s, 661,661,662) is the third order nonlinear susceptibility at the sum frequency COs,which contains pressure broadening effects, and Ak is the wave vector mismatch. In the vicinity of a two-photon resonance, however, this expression is only valid for a monochromator laser field of frequency 661 and for a negligible Doppler width "YD. In a more realistic situation, the spectral distribution and detuning on the resonant pump laser has to be considered. In addition, the spectral width of the resonance due to the velocity distribution of the interacting particles of density N has to be taken into account. Under two-photon resonant conditions, the suscep* On leave from Theoretical Physics Institute, University of Innsbruck, A-6020 Innsbruck, Austrai.
X ~1
Id~g-ald~ab--F ~
a (%g 66i) -
L T -
l~gal~ab 1
(%g 66s)J -
is real and includes all nonresonant contributions to the nonlinear susceptibility with weight factors gl = 2/3 and g2 = 1/3, while the two photon resonant denominator determines the real and imaginary parts according to (66r -- 2661)-1 = (~2r -- 26°1) + i3'/2
.
(2)
(g2r -- 2661)2 + 3,2/4 7 is the damping constant of the transition, which includes pressure broadening. For a maxwellian velocity distribution, the number of particles with a shifted resonance frequency ~'2D is given by a gaussian,
N(aD) = , / ~ vN~ 3`0 exp[-(~ D - ~r)2/272],
0 0 3 0 - 4 0 1 8 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company
(3)
205
with a width 7D = (kTg22/lac2) 1/2. The laser spectrum is assumed to consist of independent modes with a mode spacing much smaller than any one of the linewidths involved. In most cases the envelope of the intensity distribution of these modes is given by a gaussian, 1
2{I}1 qbl(~ ) = ~ exp [--4(~ -- CO1)2/82] ,
(4)
where 8 denotes the laser linewidth. Since all combinations of laser modes coa and cob with a sum frequency coa + cob = co within the linewidth of the resonant transition contribute to the generated harmonic intensity cos, the square of the pump laser intensity in (1), ~2, must be replaced by the autoconvolution of (4), ~ q~12G(co) = ~
[ 2 ]1/2 )21, t~-2 ] e x p [ - 6 - T ( c o - 2001
(5)
with a subsequent integration over co. Upon inserting eqs. (2)-(5) into (1) this yields, +m
q~s
_
C
2n7 2
X
f
l"
dco G(co)
exp[--(~2D -- ~2r)2/2'y2] 2
-~J daD
g2D
coL i@2- - -
'
(6)
where C is given by C
47r2cosL 2 dp2
n2n s
[sin(akL/2)q 2 L ~/2-]
"
As noted by Stappaerts et al. [3], the integrals appearing in the generated intensity (6) can of course be evaluated numerically for a given resonant linewidth 7, Doppler width TD, pump laser line width 6 and pump laser detuning Aoo = 2co 1 -- ~ r ' However, for practical applications, a simple closed form approximation for certain parameter ranges is highly desirable. Indeed, these authors give such approximations for the limiting case where Ace dominates all the other widths, and for the case of 6 being very much larger than either 7D or 7, which, in turn, must be very much larger than the remaining parameters. This assumption of a very large laser linewidth, however, does not hold for many systems which generate 206
1 February 1981
OPTICS COMMUNICATIONS
Volume 36, number 3
tunable, narrowband VUV radiation by means of frequency conversion. In contrast to eximer lasers, for example, two-photon resonant frequency converters allow much smaller output linewidths if a pump laser of sufficiently small width 8 is used. Furthermore, at small input intensities, where these systems are most advantageous, the generated intensity q~s is proportional to the square of the particle density N, according to eqs. (6) (7). Additional pressure broadening is caused by the phase matching gas, which is indispensable for highly efficient systems [2]. In most cases of practical interest therefore, 3' is larger than the pump laser linewidth 8. Frequently, the active medium is a metallic vapour, requiring a high operational temperature in order to achieve the appropriate pressure. This may give rise to an appreciable Doppler width 7D, comparable to the laser linewidth 8. Hence, there is considerable interest in a simple, closed form approximation to (6), which describes the variation of cI,s as a function of the four parameters "Y,"YD,8 and Aco, when 6 and "YD are considerably smaller than 7, and for an arbitrary detuning Aco. In this paper, we will provide a systematic procedure for deriving such an approxnnation from (6). In addition, our approach is capable of producing simple, closed form expressions for several other cases of practical interest. In particular, we obtain asymptotic series for the cases which are dominated either by Doppler broadening or by pressure broadening. The series thus derived yield the variation of q~s as a function of all its variables over an appropriate range and reduce under the extreme conditions assumed by Stappaerts et al. [3] to tire simple relationships given by these authors. As a key step we cast eq. (6) into a form which is amenable to standard asymptotic techniques. To this end, we represent exp [-(g2 D - ~2r)2/2T~ ] by its Fourier integral and interchange the order of integration, upon which the gZD integral can be done in closed form. After taking the squared modulus and interchanging the order of integrations again, also the co-integration can be carrier out. With appropriate new variables, (6) now reads
% = 4CRel/d~/exp
[-('y2/4 + 82/8)r/2 + iAcor/]
tO X nf
d~exp[ T 2 ~ 2 / 4 - 7 ~ / 2 ] } .
(8)
Volume 36, number 3
OPTICS COMMUNICATIONS
Compared with (6), the representation (8) has the virtue that all parameters appear in the exponent of exponentials. Furthermore, the integral can be written as a product of two factors depending on a single variable each, thus permitting the successive application of standard asymptotic techniques appropriate for one dimensional integrals that contain one or more large parameters (cf., e.g., [4] ). If, for example, 3' >~ 7 D ,
(9a)
then the integrand of the G-integral is a product of a fast varying factor, exp(-7~/2), and of a slowly varying one, exp(-3'2~2/4). Since there are no stationary points of the rapidly varying portion of the integrand, the only critical point is the endpoint ~7 of the interval of integration, and the asymptotic expansion of the integral may be obtained by repeated integration by parts. To lowest order in (3'D/7) we thus find Cs ~- (8C/3') × Re
drl exp [-(3'/2 + iA¢o)~ -- (3'20/2 + 52/2)7 2] t0 /
If now (9b)
the same reasoning applies once more with the result ~ s
4C 1 Aw 2 + 3'2/4
Aw 2 + 3'2/4
+ (3A6°2 : 72/4) (_3'2D+ 62/4)]. (Aco 2 + 3'2/4)2
comparison using values as obtained experimentally from a Mg-Kr system [5], namely, ")'D = 0.14 cm -1 , 6 = 0.1 cm -1 , 3' = 1.0 cm -1 and arbitrary detuning Aw, shows that the maximum relative error of (10) is less than 4 X 10 -2 for Aco = 0 and decreases rapidly for increasing values of Aco. With the same ease, accurate asymptotic series for ~s can be derived from (8) for the remaining two limiting cases considered by Stappaerts et al. [3]. If Doppler broadening dominates pressure broadening, 3'D >~ 7,
(1 la)
we note that the rapidly varying factor e x p ( - 7 2 ~ 2 / 4 ) gives rise to two critical points in the G-integral, which coalesce for ~ -+ 0. One is the left end point ~ = ~7, the other a saddle point at ~ = 0. A power series expansion of the slowly varying factor exp(-3,~/2) about ~ = r/ and a subsequent term by term integration leaves us with the rbintegral, whose integrand again contains a rapidly varying factor with a sharp maximum at 7) = 0. We therefore expand the slowly varying portion of the integrand for 72+62/2>>A6o
(llb)
in a series about this point and integrate term by term, where the necessary integrals can be found in standard tables [6]. The result is
X [1 -- 2 ( % / 3 ' ) 2 - (3'D/3')23'~1 }.
3'2/4 + Aw 2 >> 3'2/2 + 62/8,
I February 198l
(10)
Note that if the atomic linewidth satisfies 3"2/4 >>3"2Dl2 + 62/8, the requirements (9a) and (9b) are simultaneously fulfilled and (10) holds for arbitrary detuning, 0 ~< Aw < oo. Upon setting 3 ' D = 6 = 0 and neglecting 3', (10) reduces to the limiung case given by Stappaerts et al. in eq. (5) of ref. [3], which can of course also be obtained directly from (8). For most practical situations, (10) is sufficiently accurate to replace the exact expression (8) throughout a wide range of parameters. For example, a numerical
q5s ~-- (4C/3 ,2) [(Z/r) tan -1 (r) (1 + v2/4 - w 2) + (1 + r2)-1(02/2 + 2w 2) - x/t~v(1 + r2)-1/2], (12) where the abbreviations r = (1 + 62/2"),2) 1/2, v = 3"/3"D and w = Aco/3,D r have been introduced. The expansion (12) is remarkably accurate. If we choose, for example, similar values as above, i.e., 3 ' D = 0.14 cm -1 , 7 = 0.01 cm -1 , Aco = 0 and 6 arbitrary, 0 ~< 6 < o% the maximum relative error is less than 2 × 10 -4 . Note that the leading term of the expansion (12) can again be found directly from (8) for Aco = 3' = 0. The simple relationship given for the present case by Stappaerts et al. [3] in their eq. (6), in which ~s depends on the linewidths solely through a factor (3'D6) -1 , represents -- apart from a discrepancy in the numerical factor - the limiting case of this leading term for 7D/6 ~ 1. Since these authors failed to mention this fact, their relationship as it stands is misleading for many practical situations, where the laser linewidth is not much larger than the Doppler width and where ~s is 207
Volume 36, number 3
OPTICS COMMUNICATIONS
therefore a more complicated function of its parameters, as reflected by (12). When pressure broadening dominates Doppler broadening, 7 >;>7 D ,
(13a)
the analysis proceeds essentially along the same lines, with e x p ( 7 ~ / 2 ) now playing the part of the rapidly varying portion of the ~-integrand. Since the only critical point is the end point ~ = 7? of the interval of integration, the appropriate procedure to arrive at an asymptotic expansion is again repeated integration by parts. In the rT-integral, the critical points are then the left end point r/= 0 and a saddle point of the rapidly varying portion of the integrand, exp(-82r?2/8 7rff2), at r~S = - 2 7 / 8 2, which approaches the endpoint 77 = 0 for 8/7 ~ 1. If we want the asymptotic approximation to hold for arbitrary 8, the contribution of these two coalescing critical points must be incorporated uniformly (cf., e.g., [7] ). To this end, we expand the slowly varying portion of the integrand in a series about r/= 0, assuming 7 > Aoo,
(1 3b)
and find after integrating this series term b y term
[6],
1 February 1981
In a similar fashion, asymptotic expansions of q)s in terms of elementary functions could be derived from (8) for a few other suitably restricted regions of the space of the four parameters 7, 7D, 8, and Aco. In general, however, it must be kept in mind that for a range of parameters which is less restricted, the corresponding asymptotic series becomes less elementary [4,7]. For example, an expansion of q~s holding uniformly for large 7, 8 and Aco, with TD being the only small parameter, is necessarily of a similar type as (14), but with the error function now depending on a complex argument of the form (x/5~/8) (3,/2 - iAoo). If, ultimately, we desire a description of 4}s holding uniformly throughout the space of parameters 7, 7D, (5, and Aco, then there is no simpler adequate representation than the integral (8), or, equivalently, (6). This work was performed while one of the authors (C.L.) held an Alexander von Humboldt fellowship and he wishes to thank the trustees for their support. The financial assistance of the Osterreichische Fonds zur F6rderung der wissenschaftlichen Forschung under contract 3852 is also gratefully acknowledged. H.S. greatly appreciates the support given to him by the Deutsche Forschungsgemeinschaft.
% ~-- 8x/2C(~8) -1 X {x/-r7exp(r 2) erfc(r)[l - 202(l + 4r 2) _ 4w2r2(1 + 2r2)] - 4o2r(1
(14)
2r 2) + 8w2r3},
where the abbreviations r = 7/(x/28), v = 7D/7 and w = 2xco/7 have been introduced. The appearance of the complementary error function erfc(r) is a necessary consequence of the uniform incorporation of the contributions of a coalescing end point and a saddle point [7]. It may be conveniently and sufficiently accurately expressed through elementary functions for all values of the argument r by standard approximations [8]. If the expansion (14) is tested against a numerical evaluation of (8) for 7 = 0.14 cm -1 , 71) = 0.01 cm -1 , /',co = 0 and arbitrary 6 , 0 < 8 < 0% the maximum relative error is found to be less than 2 X 10 -3. As in (12), the leading term of the expansion (14) may be obtained directly from (8) and again reduces to the simple relationship given in eq. (7) of ref. [3] for (5/7 >> 1, i.e., when TD = Aco = 0 and exp(r2)erfc(r) m a y be replaced by unity. 208
References [ 1 ] J.A. Armstrong, N. Bloeinbergen, J. Ducuing and P.S. l}ershan, Phys. Rev. 127 (1962) 1918. I I. Puell, K. Spanner, W. Falkenstein, W. Kaiser and C.R. Vidal, Phys. Rev. A14 (1976) 2240. [21 tI. Puelt, II. Scheingraber and C.R. Vidal, Phys. Rev. A22 (1980) 1165; C.R. Vidal, Applied Optics, to be published. [3 ] I£.A. Stappaerts, G.W. Bekkers, J.F. Young and S.E. lfarris, IEEE J. Quantum Electr. QE-12 (1976) 330. I4] N. Bleistein and R.A. ttandelsman, Asymptotic expansions of integrals (tlolt, Rinehart and Winston, New York, 1975), [5] H. Junginger, H. Puell, tl. Scheingraber and C.R. Vidal, IEEE J. Quantum Electr. QE-16, Oct. (1980), to be published. [6] I.S. Gradshteyn and I.M. Ryshik, Table of integrals, series and products (Academic, New York, 1965) pp. 648- 650. [7] A. Erdelyi, in: Analytic methods in mathematical physics, eds. R. Gilbert and R. Newton (Gordon and Breach, New York, 1970) pp. 149-168. [8] M. Abramowitz and I.A. Stegun, eds., tlandbook of mathematical functions (National Bureau of Standards, Washington, 1964) p. 298, 7.1.23 and p. 299, 7.1,26.
OPTICS COMMUNICATIONS
1 February 1981
THE LINEWIDTH DEPENDENCE OF RESONANT, SMALL SIGNAL SUM FREQUENCY MIXING C. LEUBNER *, H. SCHEINGRABER and C.R. VIDAL
Max-Planck-Institut fffr Extraterrestrische Physik, D-8046 Garehing,FederalRepublic of Germany Received 10 October 1980
For small input intensities, asymptotic approximations are derived for the dependence of sum frequency mixing on the pump laser linewidth, Doppler broadening, pressure broadening, and the pump laser detuning. Compared to the relations given by Stappaerts and coworkers, the results presented have a greatly extended range of validity and cover most situations of practical importance.
For sum frequency mixing in the small signal limit and for plane waves, the intensity qbs of the generated wave at the frequency cos = 2w 1 -+ 662 is given by (cf., e.g., [1]), [4rr266sL ] 2 qb2
*s= ~
" n2n~s(N~l
tibility X~ ) can be factored as [2] X(T3)(66s'661' 661' 602) "" (66r -- 260 1)-1X(F3)(66s), where 2
(3),2Fsin(AkL/2)l 2 XT )
L Ak~/2 j
'
X(3)(66s) "" ~ #
gi
(1) where q51, q52, q5s and n l , n2, n s are the intensities and the refractive indices at the frequencies ca 1, 6°2 and cos, respectively. N is the particle density and L the length of the nonlinear medium; ×~)(66s, 661,661,662) is the third order nonlinear susceptibility at the sum frequency COs,which contains pressure broadening effects, and Ak is the wave vector mismatch. In the vicinity of a two-photon resonance, however, this expression is only valid for a monochromator laser field of frequency 661 and for a negligible Doppler width "YD. In a more realistic situation, the spectral distribution and detuning on the resonant pump laser has to be considered. In addition, the spectral width of the resonance due to the velocity distribution of the interacting particles of density N has to be taken into account. Under two-photon resonant conditions, the suscep* On leave from Theoretical Physics Institute, University of Innsbruck, A-6020 Innsbruck, Austrai.
X ~1
Id~g-ald~ab--F ~
a (%g 66i) -
L T -
l~gal~ab 1
(%g 66s)J -
is real and includes all nonresonant contributions to the nonlinear susceptibility with weight factors gl = 2/3 and g2 = 1/3, while the two photon resonant denominator determines the real and imaginary parts according to (66r -- 2661)-1 = (~2r -- 26°1) + i3'/2
.
(2)
(g2r -- 2661)2 + 3,2/4 7 is the damping constant of the transition, which includes pressure broadening. For a maxwellian velocity distribution, the number of particles with a shifted resonance frequency ~'2D is given by a gaussian,
N(aD) = , / ~ vN~ 3`0 exp[-(~ D - ~r)2/272],
0 0 3 0 - 4 0 1 8 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company
(3)
205
with a width 7D = (kTg22/lac2) 1/2. The laser spectrum is assumed to consist of independent modes with a mode spacing much smaller than any one of the linewidths involved. In most cases the envelope of the intensity distribution of these modes is given by a gaussian, 1
2{I}1 qbl(~ ) = ~ exp [--4(~ -- CO1)2/82] ,
(4)
where 8 denotes the laser linewidth. Since all combinations of laser modes coa and cob with a sum frequency coa + cob = co within the linewidth of the resonant transition contribute to the generated harmonic intensity cos, the square of the pump laser intensity in (1), ~2, must be replaced by the autoconvolution of (4), ~ q~12G(co) = ~
[ 2 ]1/2 )21, t~-2 ] e x p [ - 6 - T ( c o - 2001
(5)
with a subsequent integration over co. Upon inserting eqs. (2)-(5) into (1) this yields, +m
q~s
_
C
2n7 2
X
f
l"
dco G(co)
exp[--(~2D -- ~2r)2/2'y2] 2
-~J daD
g2D
coL i@2- - -
'
(6)
where C is given by C
47r2cosL 2 dp2
n2n s
[sin(akL/2)q 2 L ~/2-]
"
As noted by Stappaerts et al. [3], the integrals appearing in the generated intensity (6) can of course be evaluated numerically for a given resonant linewidth 7, Doppler width TD, pump laser line width 6 and pump laser detuning Aoo = 2co 1 -- ~ r ' However, for practical applications, a simple closed form approximation for certain parameter ranges is highly desirable. Indeed, these authors give such approximations for the limiting case where Ace dominates all the other widths, and for the case of 6 being very much larger than either 7D or 7, which, in turn, must be very much larger than the remaining parameters. This assumption of a very large laser linewidth, however, does not hold for many systems which generate 206
1 February 1981
OPTICS COMMUNICATIONS
Volume 36, number 3
tunable, narrowband VUV radiation by means of frequency conversion. In contrast to eximer lasers, for example, two-photon resonant frequency converters allow much smaller output linewidths if a pump laser of sufficiently small width 8 is used. Furthermore, at small input intensities, where these systems are most advantageous, the generated intensity q~s is proportional to the square of the particle density N, according to eqs. (6) (7). Additional pressure broadening is caused by the phase matching gas, which is indispensable for highly efficient systems [2]. In most cases of practical interest therefore, 3' is larger than the pump laser linewidth 8. Frequently, the active medium is a metallic vapour, requiring a high operational temperature in order to achieve the appropriate pressure. This may give rise to an appreciable Doppler width 7D, comparable to the laser linewidth 8. Hence, there is considerable interest in a simple, closed form approximation to (6), which describes the variation of cI,s as a function of the four parameters "Y,"YD,8 and Aco, when 6 and "YD are considerably smaller than 7, and for an arbitrary detuning Aco. In this paper, we will provide a systematic procedure for deriving such an approxnnation from (6). In addition, our approach is capable of producing simple, closed form expressions for several other cases of practical interest. In particular, we obtain asymptotic series for the cases which are dominated either by Doppler broadening or by pressure broadening. The series thus derived yield the variation of q~s as a function of all its variables over an appropriate range and reduce under the extreme conditions assumed by Stappaerts et al. [3] to tire simple relationships given by these authors. As a key step we cast eq. (6) into a form which is amenable to standard asymptotic techniques. To this end, we represent exp [-(g2 D - ~2r)2/2T~ ] by its Fourier integral and interchange the order of integration, upon which the gZD integral can be done in closed form. After taking the squared modulus and interchanging the order of integrations again, also the co-integration can be carrier out. With appropriate new variables, (6) now reads
% = 4CRel/d~/exp
[-('y2/4 + 82/8)r/2 + iAcor/]
tO X nf
d~exp[ T 2 ~ 2 / 4 - 7 ~ / 2 ] } .
(8)
Volume 36, number 3
OPTICS COMMUNICATIONS
Compared with (6), the representation (8) has the virtue that all parameters appear in the exponent of exponentials. Furthermore, the integral can be written as a product of two factors depending on a single variable each, thus permitting the successive application of standard asymptotic techniques appropriate for one dimensional integrals that contain one or more large parameters (cf., e.g., [4] ). If, for example, 3' >~ 7 D ,
(9a)
then the integrand of the G-integral is a product of a fast varying factor, exp(-7~/2), and of a slowly varying one, exp(-3'2~2/4). Since there are no stationary points of the rapidly varying portion of the integrand, the only critical point is the endpoint ~7 of the interval of integration, and the asymptotic expansion of the integral may be obtained by repeated integration by parts. To lowest order in (3'D/7) we thus find Cs ~- (8C/3') × Re
drl exp [-(3'/2 + iA¢o)~ -- (3'20/2 + 52/2)7 2] t0 /
If now (9b)
the same reasoning applies once more with the result ~ s
4C 1 Aw 2 + 3'2/4
Aw 2 + 3'2/4
+ (3A6°2 : 72/4) (_3'2D+ 62/4)]. (Aco 2 + 3'2/4)2
comparison using values as obtained experimentally from a Mg-Kr system [5], namely, ")'D = 0.14 cm -1 , 6 = 0.1 cm -1 , 3' = 1.0 cm -1 and arbitrary detuning Aw, shows that the maximum relative error of (10) is less than 4 X 10 -2 for Aco = 0 and decreases rapidly for increasing values of Aco. With the same ease, accurate asymptotic series for ~s can be derived from (8) for the remaining two limiting cases considered by Stappaerts et al. [3]. If Doppler broadening dominates pressure broadening, 3'D >~ 7,
(1 la)
we note that the rapidly varying factor e x p ( - 7 2 ~ 2 / 4 ) gives rise to two critical points in the G-integral, which coalesce for ~ -+ 0. One is the left end point ~ = ~7, the other a saddle point at ~ = 0. A power series expansion of the slowly varying factor exp(-3,~/2) about ~ = r/ and a subsequent term by term integration leaves us with the rbintegral, whose integrand again contains a rapidly varying factor with a sharp maximum at 7) = 0. We therefore expand the slowly varying portion of the integrand for 72+62/2>>A6o
(llb)
in a series about this point and integrate term by term, where the necessary integrals can be found in standard tables [6]. The result is
X [1 -- 2 ( % / 3 ' ) 2 - (3'D/3')23'~1 }.
3'2/4 + Aw 2 >> 3'2/2 + 62/8,
I February 198l
(10)
Note that if the atomic linewidth satisfies 3"2/4 >>3"2Dl2 + 62/8, the requirements (9a) and (9b) are simultaneously fulfilled and (10) holds for arbitrary detuning, 0 ~< Aw < oo. Upon setting 3 ' D = 6 = 0 and neglecting 3', (10) reduces to the limiung case given by Stappaerts et al. in eq. (5) of ref. [3], which can of course also be obtained directly from (8). For most practical situations, (10) is sufficiently accurate to replace the exact expression (8) throughout a wide range of parameters. For example, a numerical
q5s ~-- (4C/3 ,2) [(Z/r) tan -1 (r) (1 + v2/4 - w 2) + (1 + r2)-1(02/2 + 2w 2) - x/t~v(1 + r2)-1/2], (12) where the abbreviations r = (1 + 62/2"),2) 1/2, v = 3"/3"D and w = Aco/3,D r have been introduced. The expansion (12) is remarkably accurate. If we choose, for example, similar values as above, i.e., 3 ' D = 0.14 cm -1 , 7 = 0.01 cm -1 , Aco = 0 and 6 arbitrary, 0 ~< 6 < o% the maximum relative error is less than 2 × 10 -4 . Note that the leading term of the expansion (12) can again be found directly from (8) for Aco = 3' = 0. The simple relationship given for the present case by Stappaerts et al. [3] in their eq. (6), in which ~s depends on the linewidths solely through a factor (3'D6) -1 , represents -- apart from a discrepancy in the numerical factor - the limiting case of this leading term for 7D/6 ~ 1. Since these authors failed to mention this fact, their relationship as it stands is misleading for many practical situations, where the laser linewidth is not much larger than the Doppler width and where ~s is 207
Volume 36, number 3
OPTICS COMMUNICATIONS
therefore a more complicated function of its parameters, as reflected by (12). When pressure broadening dominates Doppler broadening, 7 >;>7 D ,
(13a)
the analysis proceeds essentially along the same lines, with e x p ( 7 ~ / 2 ) now playing the part of the rapidly varying portion of the ~-integrand. Since the only critical point is the end point ~ = 7? of the interval of integration, the appropriate procedure to arrive at an asymptotic expansion is again repeated integration by parts. In the rT-integral, the critical points are then the left end point r/= 0 and a saddle point of the rapidly varying portion of the integrand, exp(-82r?2/8 7rff2), at r~S = - 2 7 / 8 2, which approaches the endpoint 77 = 0 for 8/7 ~ 1. If we want the asymptotic approximation to hold for arbitrary 8, the contribution of these two coalescing critical points must be incorporated uniformly (cf., e.g., [7] ). To this end, we expand the slowly varying portion of the integrand in a series about r/= 0, assuming 7 > Aoo,
(1 3b)
and find after integrating this series term b y term
[6],
1 February 1981
In a similar fashion, asymptotic expansions of q)s in terms of elementary functions could be derived from (8) for a few other suitably restricted regions of the space of the four parameters 7, 7D, 8, and Aco. In general, however, it must be kept in mind that for a range of parameters which is less restricted, the corresponding asymptotic series becomes less elementary [4,7]. For example, an expansion of q~s holding uniformly for large 7, 8 and Aco, with TD being the only small parameter, is necessarily of a similar type as (14), but with the error function now depending on a complex argument of the form (x/5~/8) (3,/2 - iAoo). If, ultimately, we desire a description of 4}s holding uniformly throughout the space of parameters 7, 7D, (5, and Aco, then there is no simpler adequate representation than the integral (8), or, equivalently, (6). This work was performed while one of the authors (C.L.) held an Alexander von Humboldt fellowship and he wishes to thank the trustees for their support. The financial assistance of the Osterreichische Fonds zur F6rderung der wissenschaftlichen Forschung under contract 3852 is also gratefully acknowledged. H.S. greatly appreciates the support given to him by the Deutsche Forschungsgemeinschaft.
% ~-- 8x/2C(~8) -1 X {x/-r7exp(r 2) erfc(r)[l - 202(l + 4r 2) _ 4w2r2(1 + 2r2)] - 4o2r(1
(14)
2r 2) + 8w2r3},
where the abbreviations r = 7/(x/28), v = 7D/7 and w = 2xco/7 have been introduced. The appearance of the complementary error function erfc(r) is a necessary consequence of the uniform incorporation of the contributions of a coalescing end point and a saddle point [7]. It may be conveniently and sufficiently accurately expressed through elementary functions for all values of the argument r by standard approximations [8]. If the expansion (14) is tested against a numerical evaluation of (8) for 7 = 0.14 cm -1 , 71) = 0.01 cm -1 , /',co = 0 and arbitrary 6 , 0 < 8 < 0% the maximum relative error is found to be less than 2 X 10 -3. As in (12), the leading term of the expansion (14) may be obtained directly from (8) and again reduces to the simple relationship given in eq. (7) of ref. [3] for (5/7 >> 1, i.e., when TD = Aco = 0 and exp(r2)erfc(r) m a y be replaced by unity. 208
References [ 1 ] J.A. Armstrong, N. Bloeinbergen, J. Ducuing and P.S. l}ershan, Phys. Rev. 127 (1962) 1918. I I. Puell, K. Spanner, W. Falkenstein, W. Kaiser and C.R. Vidal, Phys. Rev. A14 (1976) 2240. [21 tI. Puelt, II. Scheingraber and C.R. Vidal, Phys. Rev. A22 (1980) 1165; C.R. Vidal, Applied Optics, to be published. [3 ] I£.A. Stappaerts, G.W. Bekkers, J.F. Young and S.E. lfarris, IEEE J. Quantum Electr. QE-12 (1976) 330. I4] N. Bleistein and R.A. ttandelsman, Asymptotic expansions of integrals (tlolt, Rinehart and Winston, New York, 1975), [5] H. Junginger, H. Puell, tl. Scheingraber and C.R. Vidal, IEEE J. Quantum Electr. QE-16, Oct. (1980), to be published. [6] I.S. Gradshteyn and I.M. Ryshik, Table of integrals, series and products (Academic, New York, 1965) pp. 648- 650. [7] A. Erdelyi, in: Analytic methods in mathematical physics, eds. R. Gilbert and R. Newton (Gordon and Breach, New York, 1970) pp. 149-168. [8] M. Abramowitz and I.A. Stegun, eds., tlandbook of mathematical functions (National Bureau of Standards, Washington, 1964) p. 298, 7.1.23 and p. 299, 7.1,26.