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Nonlinear optics in radiatively cooled vapors
Volume 30, number 1
OPTICS COMMUNICATIONS
July 1979
NONLINEAR OPTICS IN R A D I A T W E L Y COOLED VAPORS A.J. PALMER Hughes Research Laboratories, Malibu, CA 90265, USA
Received 26 February 1979 Revised manuscript received 9 April 1979
We consider the use of a radiatively cooled resonant vapor as a media for 3rd order nonlinear optics processes. The 3rd order nonlinear refractive index coefficient due to both steady state saturation of the atomic polarizability and electrostrictive induced density changes are computed for a radiatively cooled vapor with sodium as an example. For pump detuning on the order of the natural linewidth the electrostrictive and saturation induced nonlinear refractive index coefficients are comparable and several orders of magnitude greater than the highest values yet reported. At pump detunings larger than the natural linewidth the electrostrictive mechanism is shown to dominate in the steady state.
There has been a resurgence of interest in third order nonlinear optics processes recently due in large measure to newly discovered adaptive optics applications which exploit the phase conjugation properties of these processes [ 1 - 1 0 ] . One problem in the practical implementation of these processes is the high required pump power. In this note we identify a radiatively cooled resonant vapor as the ultimate low power media for these processes and compute the nonlinear refractive index coefficient for two separate mechanisms operating within such a vapor. Nonlinear optics processes are governed b y a field dependent susceptibility. The susceptibility is equal to the product o f the atomic or molecular polarizability times the number density. Third order nonlinear optics effects can occur via either an intensity dependent polarizability or an intensity dependent density. The mechanism operating in the recent four wave mixing experiments in atomic vapors belong to the former category and is brought about b y saturation effects [ 6 - 8 ] . An intensity dependent density contribution to the nonlinear susceptibility will also be present in resonant atomic vapors due to influence of electrostrictive pressure gradients on the vapor density. While the contribution of this latter mechanism to the susceptibility in resonant vapors is small compared with the nonlinear polarizability contribution under normal equilibrium temperatures it can dominate under steady state conditions in a radiatively cooled resonant vapor as 104
will be seen below. The nonlinear susceptibility due to both mechanisms can be increased dramatically by reducing the translational temperature of the vapor. For the former mechanism this temperature dependence is due to the reduced doppler bandwidth, yielding a decreased saturation flux for the transition. In the latter case it results from a higher effective compressibility of the vapor at lower temperatures. The technique o f radiative cooling of resonant vapors has been discussed theoretically b y several authors [ 1 1 - 1 4 ] and has recently been demonstrated experimentally [ 15,16]. It has been pointed out that cooling of translational temperatures can be achieved to [14]:
r ~ h'rN/k,
(1)
where h is Plank's constant, k is Bolzmann's constant and 7N is the natural linewidth of the resonance line. This is achieved through damping of the atom velocities b y radiation pressure forces. Such forces will be exerted b y counter propagating laser beams tuned into the lower half o f the doppler contour o f the resonance line of an atomic vapor. For the sodium 5890 A resonance line the temperature defined by eq. (1) is 10 - 4 K. The minimum cooling time occurs at saturation power density for the resonance line and is estimated to be ~ 10 - 5 sec for sodium [14]. A vapor which has been cooled to the limit defined by eq. (1) is left with a naturally broadened linewidth. Near this resonance line the nonlinear refractive index
Volume 30, number 1
OPTICS COMMUNICATIONS
due to the steady state intensity dependent polarizability is given by [8,9] n2sat = 2rrN Oo~/aE2 = 21rN a2 /h (V-Vo) ,
(2)
where the polarizability is given by a -
)~3
('YN/2) (V--VO)
(3)
(2rr) 3 (v--v0)2 + ("/N/2) 2 Here X is the radiation wavelength, "YN is the natural linewidth of the resonance line and (V-Vo) is the detuning frequency from the resonance line center frequency, v0 . Since collisions are absent we have equated the transverse relaxation time to (rr),N)-I [7]. Consider now the contribution of the electrostrictively induced density changes to the nonlinear susceptibility of a resonant vapor. The electrostrictive pressure acting on the vapor is [13] :
Ps _- 1-¢ N°tE2,
(4)
where a is again the atomic polarizability and E is the peak electric field amplitude. For third order nonlinear processes such as degenerate four-wave mixing and self-focusing which transfer no momentum to the medium, the steady state density change caused by the electrostrictive pressure is determined simply by the effective compressibility of the vapor. If we restrict the induced density change to be less than the average density of the vapor, and assume that the vapor is collisionless and has a thermal distribution of velocities, then it is elementary to show from the collisionless linearized Boltzmann equation that the effective compressibility of the vapor is given by [18]:
aN/aPs IT = 1/kT.
(5)
The corresponding nonlinear refractive index coefficient is: n2stric t . 21ra . ~N . . 2 n a.S N aPs ~E 2 ~P ~E 2
nN ~-~. a2
(6)
The ratio of the striction induced to saturation induced nonlinear refractive index coefficients is therefore: r/2strict/n2sat _1 - ~- h(v_Vo)/kT"
n2strict/n2sat =-~ (V--V0)')'N,
July 1979 (8)
thus electrostriction induced density changes will dominate the nonlinear refractive index coefficient in a radiatively cooled vapor for detunings larger than twice the natural linewidth. Note that n2sat changes sign as one goes through resonance while n 2 striction does not. This feature can be used as a signature of the mechanism operating in a given third order nonlinear optics experiment in resonant vapors. We also note that relatively long transient times will be associated with the electrostriction mechanism operating within a radiatively cooled vapor due to the slow atomic motions. This will be especially so, for example, for nearly collinear pump and object beams in four wave mixing where the spatial period of the established density grating is long. The spatial gain coefficient of a typical third order nonlinear optics process such as degenerate four wave mixing or self-focusing is characteristically on the order of * : g ~ (27r/X) (n 2E2).
(9)
It is of interest to compute the conditions under which the spatial gain will exceed the attenuation due to linear absorption in a radiatively cooled resonant vapor. The linear absorption coefficient can be written in terms of the polarizability as: 13= (27r/X) a (~/N/2)/(v-v 0).
(1 O)
Upon comparing eqs. (10, (9), (6) and (12) one sees that the threshold power density defined by putting g -- ~ is independent of detuning for the electrostriction mechanism while it increases linearly with detuning for the saturation mechanism. For the sodium D 2 line the electrostrictive threshold power density is ~30/aW/cm 2. This power level is roughly four orders of magnitude below that utilized in the resonant vapors of ref. [7] and over eight orders of magnitude below that utilized in the resonant vapors of ref. [6]. [This power level is a couple of orders of magnitude below saturation power for either nonlinearity for detunings on the order of the natural linewidth. For so-
(7)
At the radiatively cooled temperature limit defined by eq. (1) this ratio becomes
We consider the gain due only to the real part of the nonlinear refractive index. In ref. [8] it is shown that a net gain in degenerate four wave mixing can also exist at line center due to the imaginary part of the nonlinear refractive index.
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dium the latter is ~ 6 × 10 - 2 W/cm 2.] The gain and attenuation themselves depend on b o t h the density and the detuning. The maximum value of the gain occurs for (V-Vo) ~ 7N/2. At this detuning in the radiation cooling limit we can write: a ~ X3/(27r) 2,
~3~ ;k2/47r
g ~ n(k2/(2rr) 3 ) N ( E 2 X 3 / h T N ) .
(1 1 , 1 2 ) (13)
For the sodium D 2 line* net gain coefficients on the order o f 1 cm -1 will occur at threshold power densities for densities on the order o f 1 X 1010 cm -3. A trapped one centimeter long cylindrical volume o f radiatively cooled sodium vapor at this density can be prepared with cw single mode dye laser beams. The cooling can be accomplished on a 10/as time scale through the use of counterpropagating single mode cw beams each at a power level o f ~ 6 0 mW/cm 2 b y frequency chirping the laser across the lower half of the doppler contour o f the resonance line [14]. Subsequent to the cooling, trapping of the vapor in space can be realized with single mode gaussian laser beams intersecting the vapor and tuned into the lower half o f the natural linewidth [13]. The required power for trapping the radiatively cooled vapor is given b y
1 (~k3/(2ff)2) E2 ~ h 7 N which corresponds to a single mode power density of ~ 6 0 mW/cm 2 for the case o f the sodium D line. This trapping density corresponds to the saturation power for the refractive index nonlinearities. However, since the electrostrictive nonlinear index depends on inten* The hyperfine splitting of the resonance lines of sodium is large compared to the doppler linewidth so the foregoing assumes interaction with only a single hyperfine transition. In this connection we note that the D1 manifold as 5896 A cannot be used due to depletion of the ground state caused by hyperfine optical pumping as discussed in ref. [7].
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July 1979
sity gradients nonlinear optics effects can still take place in the presence of the trapping fields provided the intensity gradients o f the trapping fields are kept small in comparison to those o f the nonlinear optics pump fields. The intensity dependent polarizability nonlinearity will, on the other hand, be saturated out b y the trapping field. The author wishes to thank Drs. J.F. Lam and S. Wandzura for helpful discussions.
References [1] B.Y. Zel'dovich, V.I. Popovickev, V.V. Ragul'skii and F.S. Fagullov, JETP Lett. 15 (1972) 109. [2] V. Wang and C.R. Giuliano, Opt. Lett. 2 (1978) 4. [3] R.W. Hellwarth, J. Opt. Soc. Am. 15 (1972) 109. [4] S.L. Jensen and R.W. Hellwarth, Appl. Phys. Lett. 32 (1978) 166. [5] A. Yariv and D.M. Pepper, Opt. Lett. 1 (1977) 16. [6] D.M. Bloom, P.F. Liao and N.P. Economu, Opt. Lett. 2 (1978) 158. [7] P.F. Liao, D.M. Bloom and N.P. Economou, Appl. Phys. Lett. 32 (1978) 813. [8] R.L. Abrams and R.C. Lind, Opt. Lett. 2 (1978) 94. [9] A. Javan and P.L. Kelley, IEEE, J.Q.E. EQ-2 (1966) 470. [10] A. Yariv, Optics Comm. 25 (1978) 23. [11] T.W. Hansch and A.L. Schawlow, Optics Comm. 13 (1975) 68. [12] V.S. Letokhov, V.G. Minogin and B.D. Pavlik, Optics Comm. 19 (1976) 72. [13] A. Ashkin, Phys. Rev. Lett. 40 (1978) 729. [14] S. Stenholm, V.G. Minogin and V.S. Letokhov, Optics Comm. 25 (1978) 107. [15] D.J. Wineland, R.E. Drullinger and F.L. Wails, Phys. Rev. Lett. 40 (1978) 1639. [16] W. Neuhauser, M. Hohenstatt, P.E. Toschek and H.G. Dehmelt, Appl. Phys. 17 (1978) 123. [17] F. Block, Phys. Rev. 70 (1946) 460. [18] K. Huang, Statistical Mechanics (Wiley & Sons, New York, York-London, 1963).
OPTICS COMMUNICATIONS
July 1979
NONLINEAR OPTICS IN R A D I A T W E L Y COOLED VAPORS A.J. PALMER Hughes Research Laboratories, Malibu, CA 90265, USA
Received 26 February 1979 Revised manuscript received 9 April 1979
We consider the use of a radiatively cooled resonant vapor as a media for 3rd order nonlinear optics processes. The 3rd order nonlinear refractive index coefficient due to both steady state saturation of the atomic polarizability and electrostrictive induced density changes are computed for a radiatively cooled vapor with sodium as an example. For pump detuning on the order of the natural linewidth the electrostrictive and saturation induced nonlinear refractive index coefficients are comparable and several orders of magnitude greater than the highest values yet reported. At pump detunings larger than the natural linewidth the electrostrictive mechanism is shown to dominate in the steady state.
There has been a resurgence of interest in third order nonlinear optics processes recently due in large measure to newly discovered adaptive optics applications which exploit the phase conjugation properties of these processes [ 1 - 1 0 ] . One problem in the practical implementation of these processes is the high required pump power. In this note we identify a radiatively cooled resonant vapor as the ultimate low power media for these processes and compute the nonlinear refractive index coefficient for two separate mechanisms operating within such a vapor. Nonlinear optics processes are governed b y a field dependent susceptibility. The susceptibility is equal to the product o f the atomic or molecular polarizability times the number density. Third order nonlinear optics effects can occur via either an intensity dependent polarizability or an intensity dependent density. The mechanism operating in the recent four wave mixing experiments in atomic vapors belong to the former category and is brought about b y saturation effects [ 6 - 8 ] . An intensity dependent density contribution to the nonlinear susceptibility will also be present in resonant atomic vapors due to influence of electrostrictive pressure gradients on the vapor density. While the contribution of this latter mechanism to the susceptibility in resonant vapors is small compared with the nonlinear polarizability contribution under normal equilibrium temperatures it can dominate under steady state conditions in a radiatively cooled resonant vapor as 104
will be seen below. The nonlinear susceptibility due to both mechanisms can be increased dramatically by reducing the translational temperature of the vapor. For the former mechanism this temperature dependence is due to the reduced doppler bandwidth, yielding a decreased saturation flux for the transition. In the latter case it results from a higher effective compressibility of the vapor at lower temperatures. The technique o f radiative cooling of resonant vapors has been discussed theoretically b y several authors [ 1 1 - 1 4 ] and has recently been demonstrated experimentally [ 15,16]. It has been pointed out that cooling of translational temperatures can be achieved to [14]:
r ~ h'rN/k,
(1)
where h is Plank's constant, k is Bolzmann's constant and 7N is the natural linewidth of the resonance line. This is achieved through damping of the atom velocities b y radiation pressure forces. Such forces will be exerted b y counter propagating laser beams tuned into the lower half o f the doppler contour o f the resonance line of an atomic vapor. For the sodium 5890 A resonance line the temperature defined by eq. (1) is 10 - 4 K. The minimum cooling time occurs at saturation power density for the resonance line and is estimated to be ~ 10 - 5 sec for sodium [14]. A vapor which has been cooled to the limit defined by eq. (1) is left with a naturally broadened linewidth. Near this resonance line the nonlinear refractive index
Volume 30, number 1
OPTICS COMMUNICATIONS
due to the steady state intensity dependent polarizability is given by [8,9] n2sat = 2rrN Oo~/aE2 = 21rN a2 /h (V-Vo) ,
(2)
where the polarizability is given by a -
)~3
('YN/2) (V--VO)
(3)
(2rr) 3 (v--v0)2 + ("/N/2) 2 Here X is the radiation wavelength, "YN is the natural linewidth of the resonance line and (V-Vo) is the detuning frequency from the resonance line center frequency, v0 . Since collisions are absent we have equated the transverse relaxation time to (rr),N)-I [7]. Consider now the contribution of the electrostrictively induced density changes to the nonlinear susceptibility of a resonant vapor. The electrostrictive pressure acting on the vapor is [13] :
Ps _- 1-¢ N°tE2,
(4)
where a is again the atomic polarizability and E is the peak electric field amplitude. For third order nonlinear processes such as degenerate four-wave mixing and self-focusing which transfer no momentum to the medium, the steady state density change caused by the electrostrictive pressure is determined simply by the effective compressibility of the vapor. If we restrict the induced density change to be less than the average density of the vapor, and assume that the vapor is collisionless and has a thermal distribution of velocities, then it is elementary to show from the collisionless linearized Boltzmann equation that the effective compressibility of the vapor is given by [18]:
aN/aPs IT = 1/kT.
(5)
The corresponding nonlinear refractive index coefficient is: n2stric t . 21ra . ~N . . 2 n a.S N aPs ~E 2 ~P ~E 2
nN ~-~. a2
(6)
The ratio of the striction induced to saturation induced nonlinear refractive index coefficients is therefore: r/2strict/n2sat _1 - ~- h(v_Vo)/kT"
n2strict/n2sat =-~ (V--V0)')'N,
July 1979 (8)
thus electrostriction induced density changes will dominate the nonlinear refractive index coefficient in a radiatively cooled vapor for detunings larger than twice the natural linewidth. Note that n2sat changes sign as one goes through resonance while n 2 striction does not. This feature can be used as a signature of the mechanism operating in a given third order nonlinear optics experiment in resonant vapors. We also note that relatively long transient times will be associated with the electrostriction mechanism operating within a radiatively cooled vapor due to the slow atomic motions. This will be especially so, for example, for nearly collinear pump and object beams in four wave mixing where the spatial period of the established density grating is long. The spatial gain coefficient of a typical third order nonlinear optics process such as degenerate four wave mixing or self-focusing is characteristically on the order of * : g ~ (27r/X) (n 2E2).
(9)
It is of interest to compute the conditions under which the spatial gain will exceed the attenuation due to linear absorption in a radiatively cooled resonant vapor. The linear absorption coefficient can be written in terms of the polarizability as: 13= (27r/X) a (~/N/2)/(v-v 0).
(1 O)
Upon comparing eqs. (10, (9), (6) and (12) one sees that the threshold power density defined by putting g -- ~ is independent of detuning for the electrostriction mechanism while it increases linearly with detuning for the saturation mechanism. For the sodium D 2 line the electrostrictive threshold power density is ~30/aW/cm 2. This power level is roughly four orders of magnitude below that utilized in the resonant vapors of ref. [7] and over eight orders of magnitude below that utilized in the resonant vapors of ref. [6]. [This power level is a couple of orders of magnitude below saturation power for either nonlinearity for detunings on the order of the natural linewidth. For so-
(7)
At the radiatively cooled temperature limit defined by eq. (1) this ratio becomes
We consider the gain due only to the real part of the nonlinear refractive index. In ref. [8] it is shown that a net gain in degenerate four wave mixing can also exist at line center due to the imaginary part of the nonlinear refractive index.
105
Volume 30, number 1
OPTICS COMMUNICATIONS
dium the latter is ~ 6 × 10 - 2 W/cm 2.] The gain and attenuation themselves depend on b o t h the density and the detuning. The maximum value of the gain occurs for (V-Vo) ~ 7N/2. At this detuning in the radiation cooling limit we can write: a ~ X3/(27r) 2,
~3~ ;k2/47r
g ~ n(k2/(2rr) 3 ) N ( E 2 X 3 / h T N ) .
(1 1 , 1 2 ) (13)
For the sodium D 2 line* net gain coefficients on the order o f 1 cm -1 will occur at threshold power densities for densities on the order o f 1 X 1010 cm -3. A trapped one centimeter long cylindrical volume o f radiatively cooled sodium vapor at this density can be prepared with cw single mode dye laser beams. The cooling can be accomplished on a 10/as time scale through the use of counterpropagating single mode cw beams each at a power level o f ~ 6 0 mW/cm 2 b y frequency chirping the laser across the lower half of the doppler contour o f the resonance line [14]. Subsequent to the cooling, trapping of the vapor in space can be realized with single mode gaussian laser beams intersecting the vapor and tuned into the lower half o f the natural linewidth [13]. The required power for trapping the radiatively cooled vapor is given b y
1 (~k3/(2ff)2) E2 ~ h 7 N which corresponds to a single mode power density of ~ 6 0 mW/cm 2 for the case o f the sodium D line. This trapping density corresponds to the saturation power for the refractive index nonlinearities. However, since the electrostrictive nonlinear index depends on inten* The hyperfine splitting of the resonance lines of sodium is large compared to the doppler linewidth so the foregoing assumes interaction with only a single hyperfine transition. In this connection we note that the D1 manifold as 5896 A cannot be used due to depletion of the ground state caused by hyperfine optical pumping as discussed in ref. [7].
106
July 1979
sity gradients nonlinear optics effects can still take place in the presence of the trapping fields provided the intensity gradients o f the trapping fields are kept small in comparison to those o f the nonlinear optics pump fields. The intensity dependent polarizability nonlinearity will, on the other hand, be saturated out b y the trapping field. The author wishes to thank Drs. J.F. Lam and S. Wandzura for helpful discussions.
References [1] B.Y. Zel'dovich, V.I. Popovickev, V.V. Ragul'skii and F.S. Fagullov, JETP Lett. 15 (1972) 109. [2] V. Wang and C.R. Giuliano, Opt. Lett. 2 (1978) 4. [3] R.W. Hellwarth, J. Opt. Soc. Am. 15 (1972) 109. [4] S.L. Jensen and R.W. Hellwarth, Appl. Phys. Lett. 32 (1978) 166. [5] A. Yariv and D.M. Pepper, Opt. Lett. 1 (1977) 16. [6] D.M. Bloom, P.F. Liao and N.P. Economu, Opt. Lett. 2 (1978) 158. [7] P.F. Liao, D.M. Bloom and N.P. Economou, Appl. Phys. Lett. 32 (1978) 813. [8] R.L. Abrams and R.C. Lind, Opt. Lett. 2 (1978) 94. [9] A. Javan and P.L. Kelley, IEEE, J.Q.E. EQ-2 (1966) 470. [10] A. Yariv, Optics Comm. 25 (1978) 23. [11] T.W. Hansch and A.L. Schawlow, Optics Comm. 13 (1975) 68. [12] V.S. Letokhov, V.G. Minogin and B.D. Pavlik, Optics Comm. 19 (1976) 72. [13] A. Ashkin, Phys. Rev. Lett. 40 (1978) 729. [14] S. Stenholm, V.G. Minogin and V.S. Letokhov, Optics Comm. 25 (1978) 107. [15] D.J. Wineland, R.E. Drullinger and F.L. Wails, Phys. Rev. Lett. 40 (1978) 1639. [16] W. Neuhauser, M. Hohenstatt, P.E. Toschek and H.G. Dehmelt, Appl. Phys. 17 (1978) 123. [17] F. Block, Phys. Rev. 70 (1946) 460. [18] K. Huang, Statistical Mechanics (Wiley & Sons, New York, York-London, 1963).