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Three-photon ionization cross-section of potassium for single-mode ruby laser radiation
V o l u m e 10, n u m b e r 3
OPTICS COMMUNICATIONS
M a r c h 1974
THREE-PHOTON IONIZATION CROSS-SECTION OF POTASSIUM FOR SINGLE-MODE RUBY LASER RADIATION M.R. CERVENAN and N.R. ISENOR Department of f'hysics, University of Waterloo Waterloo, Ontario, Canada R e c e i v e d 14 J a n u a r y 1974
. . . . . . . . 91 Fhe t h r e e - p h o t o n l O n l z a n o n cross-section l o t K at 6 9 4 3 A has b e e n d e t e r m i n e d to be (8.0 -+ 1.8) X 10 6 2 -3 . . + -91 6 2 -3 . . . . . m s photon for c i r c u l a r l y p o l a r i z e d a n d (3.4 ~ 0 . 8 ) X 10 m s photon lor h n e a r l y p o l a r i z e d r a d m t l o n . T h e ratio o f t h e s e q u a n t i t i e s ( 2 . 3 4 ± 0 . 2 2 ) has b e e n f o u n d to agree s u b s t a n t i a l l y w i t h r e c e n t t h e o r e t i c a l p r e d i c t i o n s .
Recently several papers dealing with the calculation of multiphoton ionization cross-sections for the alkali metal atoms l l, 21 and others dealing with their experimental determination have appeared. The calculations have dealt mainly with single-mode linearly polarized laser fields. The effects of mode structure (photon statistics) [3] in multiphoton absorption processes and of circular polarization on multiphoton ionization processes [4 7] have been considered. The experinrents have been done using multi-mode laser fields [8 12]. In one case [8], a single transverse nmlti-longituciinat mode laser was used. In these experiments, uncertainties introduced by the spatial and temporal fluctuations (mode beating) of the laser beam, along with experimental uncertainties of the ion current, laser power and atom density values contributed to a large assigned error in the cross-section value obtained. Typically, errors o f ~ 10 + 1.3 have been reported. In this paper we report the determination of the 3-photon ionization cross-section of potassium using a single transverse and longitudinal mode ruby laser. For the output of such a laser, the photon correlation ft, nction [3] is unambiguously unity. It is therefore possible to express the photon flux density as a product of a function of time, G(t) and position, D(r). Assuming that the ionization rate is proportional to the kth power of the flux density, the number of ions created in the focal region can be expressed as [ 13]
280
'~,~=p
T-, (_1) n+l 2.5 , , : - - % w ~ , ) n v,, k,
(1)
n = I
where p is the initial atom density, k the order of the process, Wk the cross-section, and F m the peak photon flux at the focal point, rk(= dGk(t)dt) and f~k (= fffly~k(r) d V) are respectively a characteristic time for a kth order process and volume for an (nk)th order process. To a high degree of accuracy a ganssian shape can be assumed for both G(t) and the transverse intensity distribution for the unfocussed beam. When the photon flux distribution in the focus of a spherical lens is calculate d (assuming large )':number), we obtain [ 14 I from eq. ( 1)
_(2k 4)!! ~ ( 1 ) n+l (2nk 5)!t n uk(Pf) (2k 5)!! t7 l - n n [ (2nk 4)!! p f '
(2)
lVi/PV~ This reJationship applies for k ~> 3. For k = 2, the coefficient be-
where Pr = Tk WkFkm and vk =
fore the sum is replaced by unity, as is the coefficient o f p f in first term of the sum. Hence, the ratio of the number o f ions created to the number of neutral atoms in the volume Vk is a unique function, independent of the local length, of a single parameter Pr- For low fluxes, pf represents the probability of an atom at the focal point to be ionized by the laser pulse. For pf 1, the oscillating series is difficult to sum [ 13], how-
Volume 10, number 3
OPTICS COMMUNICATIONS
March 1974
limited only by the uncertainty o f F m and r k and o f tile fit. In our experiments, a 0.7 MW single mode ruby laser beam was focussed by a 41 mm focal length spherical lens into the atomic beam. The potassium atomic beam was produced by a double chamber oven which was dill ferentially heated to eliminate K 2 molecules. Beam densities of ~ 1016 m -3 were used. Identification and detection o f the ions were accomplished by time-offlight analysis and an electron-multiplier. The laser output initially was checked with a Fabry Perot etaIon to ensure a single axial mode and subsequently each shot was monitored by a high speed oscilloscope. Frequent checks were made on the far field pattern to ensure an undistorted TEM00 output. Tile pulse duration (fwhm) was typically 12 nsec and the beam spot size ~ 0.38 mm. Both linearly and circularly polarized beams were used. Experimental data for linear and circular polarization, one run o f each, are fitted to the curve v3(pf) in fig. 1. A gradient-search method was used in the fitting. The averaged results o f several runs for the independently determined values are
go
W3Q= (3.4 -+ 0.8) X 10 -91 m6s2photon - 3 and -2
-2
0
2
4
=(8.0-+ 1.8) X10 -91
m6s2photon - 3 .
LOG pf
Fig. 1 The data of one run with linear polarization (+) and circular polarization (Xl fitted to the log vkversus log pf (k = 3 in this experiment). The curve and data for circular polarization have been shifted one unit to the left, for clarity.
ever an approximate form, which can be made to approach the exact form to any desired degree of accuracy, has been found [14] for use in this region. The function vk, shown in fig. 1, exhibits typical saturation effects. It can be conveniently used to present data for such experirnents in normalized form. Fitting the experimental data to this curve eliminates the need for absolute knowledge o f the number of ions formed provided that the data extends into both regions. The overall gain and the cross-Nction Wk, can be adjusted independently to achieve this fit. The precision o f the resulting value of Wk is thus
The main contribution to the uncertainty comes from calibration factors used to determine F m. These, o f course, did not contribute to the error in the ratio I~3/14~3, which was determinated to be 2.34 -+ 0.22. Repeat runs gave values well within the quoted uncertainties. The value of W~ reported here can be compared with the results of the perturbation calculations o f Bebb (4.9 X 10 -92 m6s2photon - 3 ) [ 1] and Morton (8.9 X 10 -91 m6s2photon -3) [2]. To our knowledge, no other experimental cross-section value has been published. Klarsfeld and Maquet [4] have predicted a value less than ( 2 k - l)!!/k! for 14~k/W~, ( < 2.5 for k = 3) and that the ratio should be close to that limit for most values o f wavelength. A value o f 2,15 -+ 0.4 has been obtained for this ratio for the 3-photon ionization o f Cs by ruby laser radiation [ 1 1]. Reiss predicts
281
Volume 10, number 3
OPTICS COMMUNICATIONS
[6] that while this ratio may hold for small k, it should get smaller, even less than unity, for large k, and t h a t this reversal may take place for k ~ 4 or 5. This was further c o n f i r m e d by Parzynski [7] who calculated this ratio for the case of a threshold p h o t o n energy hu = 1/k and obtained a value 2.18 for k = 3. While our result supports the prediction of Klarsfeld and Maquet, it does n o t involve a large e n o u g h vNue o f k to be usefully related to the prediction o f Reiss. More e x p e r i m e n t a l work, especially for large k values, is required to clear up this point. Tile financial assistance of tile National Research Council o f Canada is gratefully acknowledged.
References I1] ti.B. Bebb, Phys. Rev. 153 (1967) 23.
282
March 1974
[2] V.M. Morton, Proc. Phys. Soc. 92 11967) 301. [31 J.L. Debethune, 11 Nuovo Cimento 12B (1972) 101. [4] S. Klarsfeld and A Maquet, Phys. Rev. Lett. 29 (1972) 79. [61 P. Lambropoulos, Phys. Rev. Lett. 29 '1972) 453. [61 tI.R. Reiss, Phys. Rev. Lett.29 (1972) 1129. [71 R. Parzynski, Optics Comm. 8 (1973) 79. 181 G.A. Delone, N.B. Delone, V.K. Zolotarev, N.L. Manakov, G.K. Piskova and M.A. Tursunov, P.N. Lebedev Physics Institute USSR Acad. of Sci., preprmt No. 36 ( 1973), (in Russian). [91 (',.A. Delone, N.B. Delone and G.K. Piskova, Proc lt)th Int. Conf. on Ionization phenomena in gases, Oxford, ( 197 t ), (Donald Parsons and Co., Ltd., Oxford), p. 40. 1101 B. lteld, G. Mainfray and .I. Morellec, Phys. Lett. 39A (1972) 57 j i l l R.A. t:ox, R.M. Kogan and E.J. Robinson, Phys. Rev Lett. 26 (197l) 1416. 112i R.G. Evans and P.C. Thonemann, PhiL. Mag. 27 ( 1973) 1387 and Phys. Lett. 39A (1972) 133. 1131 S.L. Chin and N.R. Isenor, (?an. J. Phys. 48 (1970) 1445. 114i M.R. Cervenan and N.R. lsenor, to be published.
OPTICS COMMUNICATIONS
M a r c h 1974
THREE-PHOTON IONIZATION CROSS-SECTION OF POTASSIUM FOR SINGLE-MODE RUBY LASER RADIATION M.R. CERVENAN and N.R. ISENOR Department of f'hysics, University of Waterloo Waterloo, Ontario, Canada R e c e i v e d 14 J a n u a r y 1974
. . . . . . . . 91 Fhe t h r e e - p h o t o n l O n l z a n o n cross-section l o t K at 6 9 4 3 A has b e e n d e t e r m i n e d to be (8.0 -+ 1.8) X 10 6 2 -3 . . + -91 6 2 -3 . . . . . m s photon for c i r c u l a r l y p o l a r i z e d a n d (3.4 ~ 0 . 8 ) X 10 m s photon lor h n e a r l y p o l a r i z e d r a d m t l o n . T h e ratio o f t h e s e q u a n t i t i e s ( 2 . 3 4 ± 0 . 2 2 ) has b e e n f o u n d to agree s u b s t a n t i a l l y w i t h r e c e n t t h e o r e t i c a l p r e d i c t i o n s .
Recently several papers dealing with the calculation of multiphoton ionization cross-sections for the alkali metal atoms l l, 21 and others dealing with their experimental determination have appeared. The calculations have dealt mainly with single-mode linearly polarized laser fields. The effects of mode structure (photon statistics) [3] in multiphoton absorption processes and of circular polarization on multiphoton ionization processes [4 7] have been considered. The experinrents have been done using multi-mode laser fields [8 12]. In one case [8], a single transverse nmlti-longituciinat mode laser was used. In these experiments, uncertainties introduced by the spatial and temporal fluctuations (mode beating) of the laser beam, along with experimental uncertainties of the ion current, laser power and atom density values contributed to a large assigned error in the cross-section value obtained. Typically, errors o f ~ 10 + 1.3 have been reported. In this paper we report the determination of the 3-photon ionization cross-section of potassium using a single transverse and longitudinal mode ruby laser. For the output of such a laser, the photon correlation ft, nction [3] is unambiguously unity. It is therefore possible to express the photon flux density as a product of a function of time, G(t) and position, D(r). Assuming that the ionization rate is proportional to the kth power of the flux density, the number of ions created in the focal region can be expressed as [ 13]
280
'~,~=p
T-, (_1) n+l 2.5 , , : - - % w ~ , ) n v,, k,
(1)
n = I
where p is the initial atom density, k the order of the process, Wk the cross-section, and F m the peak photon flux at the focal point, rk(= dGk(t)dt) and f~k (= fffly~k(r) d V) are respectively a characteristic time for a kth order process and volume for an (nk)th order process. To a high degree of accuracy a ganssian shape can be assumed for both G(t) and the transverse intensity distribution for the unfocussed beam. When the photon flux distribution in the focus of a spherical lens is calculate d (assuming large )':number), we obtain [ 14 I from eq. ( 1)
_(2k 4)!! ~ ( 1 ) n+l (2nk 5)!t n uk(Pf) (2k 5)!! t7 l - n n [ (2nk 4)!! p f '
(2)
lVi/PV~ This reJationship applies for k ~> 3. For k = 2, the coefficient be-
where Pr = Tk WkFkm and vk =
fore the sum is replaced by unity, as is the coefficient o f p f in first term of the sum. Hence, the ratio of the number o f ions created to the number of neutral atoms in the volume Vk is a unique function, independent of the local length, of a single parameter Pr- For low fluxes, pf represents the probability of an atom at the focal point to be ionized by the laser pulse. For pf 1, the oscillating series is difficult to sum [ 13], how-
Volume 10, number 3
OPTICS COMMUNICATIONS
March 1974
limited only by the uncertainty o f F m and r k and o f tile fit. In our experiments, a 0.7 MW single mode ruby laser beam was focussed by a 41 mm focal length spherical lens into the atomic beam. The potassium atomic beam was produced by a double chamber oven which was dill ferentially heated to eliminate K 2 molecules. Beam densities of ~ 1016 m -3 were used. Identification and detection o f the ions were accomplished by time-offlight analysis and an electron-multiplier. The laser output initially was checked with a Fabry Perot etaIon to ensure a single axial mode and subsequently each shot was monitored by a high speed oscilloscope. Frequent checks were made on the far field pattern to ensure an undistorted TEM00 output. Tile pulse duration (fwhm) was typically 12 nsec and the beam spot size ~ 0.38 mm. Both linearly and circularly polarized beams were used. Experimental data for linear and circular polarization, one run o f each, are fitted to the curve v3(pf) in fig. 1. A gradient-search method was used in the fitting. The averaged results o f several runs for the independently determined values are
go
W3Q= (3.4 -+ 0.8) X 10 -91 m6s2photon - 3 and -2
-2
0
2
4
=(8.0-+ 1.8) X10 -91
m6s2photon - 3 .
LOG pf
Fig. 1 The data of one run with linear polarization (+) and circular polarization (Xl fitted to the log vkversus log pf (k = 3 in this experiment). The curve and data for circular polarization have been shifted one unit to the left, for clarity.
ever an approximate form, which can be made to approach the exact form to any desired degree of accuracy, has been found [14] for use in this region. The function vk, shown in fig. 1, exhibits typical saturation effects. It can be conveniently used to present data for such experirnents in normalized form. Fitting the experimental data to this curve eliminates the need for absolute knowledge o f the number of ions formed provided that the data extends into both regions. The overall gain and the cross-Nction Wk, can be adjusted independently to achieve this fit. The precision o f the resulting value of Wk is thus
The main contribution to the uncertainty comes from calibration factors used to determine F m. These, o f course, did not contribute to the error in the ratio I~3/14~3, which was determinated to be 2.34 -+ 0.22. Repeat runs gave values well within the quoted uncertainties. The value of W~ reported here can be compared with the results of the perturbation calculations o f Bebb (4.9 X 10 -92 m6s2photon - 3 ) [ 1] and Morton (8.9 X 10 -91 m6s2photon -3) [2]. To our knowledge, no other experimental cross-section value has been published. Klarsfeld and Maquet [4] have predicted a value less than ( 2 k - l)!!/k! for 14~k/W~, ( < 2.5 for k = 3) and that the ratio should be close to that limit for most values o f wavelength. A value o f 2,15 -+ 0.4 has been obtained for this ratio for the 3-photon ionization o f Cs by ruby laser radiation [ 1 1]. Reiss predicts
281
Volume 10, number 3
OPTICS COMMUNICATIONS
[6] that while this ratio may hold for small k, it should get smaller, even less than unity, for large k, and t h a t this reversal may take place for k ~ 4 or 5. This was further c o n f i r m e d by Parzynski [7] who calculated this ratio for the case of a threshold p h o t o n energy hu = 1/k and obtained a value 2.18 for k = 3. While our result supports the prediction of Klarsfeld and Maquet, it does n o t involve a large e n o u g h vNue o f k to be usefully related to the prediction o f Reiss. More e x p e r i m e n t a l work, especially for large k values, is required to clear up this point. Tile financial assistance of tile National Research Council o f Canada is gratefully acknowledged.
References I1] ti.B. Bebb, Phys. Rev. 153 (1967) 23.
282
March 1974
[2] V.M. Morton, Proc. Phys. Soc. 92 11967) 301. [31 J.L. Debethune, 11 Nuovo Cimento 12B (1972) 101. [4] S. Klarsfeld and A Maquet, Phys. Rev. Lett. 29 (1972) 79. [61 P. Lambropoulos, Phys. Rev. Lett. 29 '1972) 453. [61 tI.R. Reiss, Phys. Rev. Lett.29 (1972) 1129. [71 R. Parzynski, Optics Comm. 8 (1973) 79. 181 G.A. Delone, N.B. Delone, V.K. Zolotarev, N.L. Manakov, G.K. Piskova and M.A. Tursunov, P.N. Lebedev Physics Institute USSR Acad. of Sci., preprmt No. 36 ( 1973), (in Russian). [91 (',.A. Delone, N.B. Delone and G.K. Piskova, Proc lt)th Int. Conf. on Ionization phenomena in gases, Oxford, ( 197 t ), (Donald Parsons and Co., Ltd., Oxford), p. 40. 1101 B. lteld, G. Mainfray and .I. Morellec, Phys. Lett. 39A (1972) 57 j i l l R.A. t:ox, R.M. Kogan and E.J. Robinson, Phys. Rev Lett. 26 (197l) 1416. 112i R.G. Evans and P.C. Thonemann, PhiL. Mag. 27 ( 1973) 1387 and Phys. Lett. 39A (1972) 133. 1131 S.L. Chin and N.R. Isenor, (?an. J. Phys. 48 (1970) 1445. 114i M.R. Cervenan and N.R. lsenor, to be published.