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Coherent transient pulse propagation in two-photon resonance of a homogeneously broadened three-level medium
COHERENT
August 1977
OPTICS COMMUNICATIONS
Volume 22, number 2
TRANSIENT
OF A HOMOGENEOUSLY
PULSE
PROPAGATION
BROADENED
IN TWO-PHOTON
THREE-LEVEL
RESONANCE
MEDIUM
Y. TAKAHASHI, N. TAN-NO and K. YOKOTO of Electronics, Yamagata University, Yonezawa, 992, Japan
Department
Received 21 April 1977
Pulse evolutions with SIT-like propagation are shown in spite of including the optical Stark effect and the self-phase modulation by solving numerically the two-photon Bloch equation and the coupled phase and amplitude equations.
1. Introduction In recent years, there has been a considerable interest in phenomena arising from the two-photon analogy of the well-known one-photon coherent effects, following the initial work of Hartmann [l]. Recently, a new formulation of the adiabatic following model for twophoton transitions has been reported [2,3]. There has been much interest in two-photon coherent effects, both theoretically and experimentally. Such phenomena include the two-photon mutation [4], the twophoton free induction decay [4,5] and the two-photon SIT [6-l 11. In this paper, we represent an evolution of propagating pulses for the two-photon transition of a homogeneously broadened three-level system by solving numerically the two-photon Bloch equation and the coupled phase and amplitude equations. An appearance of SIT-like pulse formations is shown in spite of including the optical Stark effect and the self-phase modulation.
Fig. 1. Energy levels relevant to the two-photon
transition.
tric fields is described by the two-photon Bloch equations [2,7,9]. Similar to the one-photon resonance case, this interaction can be visualized using a vector model, as a precession of a two-photon polarization vector r about an effective field y , dr/dt = yX r
2. Two-photon Bloch-Maxwell equations Coherent transient response of material polarizations play an essential role in propagation of intense, short duration laser pulses through a two-photon resonant medium. This problem can be described using a simplified three-level system shown in fig. 1. The response of the atomic system to the propagating elec-
(1)
where r(U,, V,, p,) is the relevant vector and y(yl, y2,ys) is the torque vector. The precession frequency is given by the magnitude of the y-vector, whose components are y1= --Kqe,,
Y2
=o,
y3 = 6 -&+(AE,-AEJTi, (2)
127
Volume 22, number 2
where ~(i=h,v)
OPTICS COMMUNICATIONS
is the amplitude
of the optical field,
k = 1_1~~,/2fiAw, is the two-photon AE,2 = /.?2 &4tiAw 1
gyroelectric
ratio, and
0
are the optical Stark shifts, and _& is the sum of the instantaneous phase shifts &t @,, and pj is the matrix element of the transition dipole moment. If we define a two-photon area I+b=K
TeheVdt, _m
(3)
this implies a fictitious angle of r from the origin. The equations of motion for the components of r are given bY aP,lat
= Qe,&
- (P,-P$/TI,
XJ,/at
= -[6-+,+(AE^-AE,)/ti]
av,jat
= ]6-&t(~~~-n~,)jfi]
I’,-G/T;,
(4b)
uP (4c)
where T, is the longitudinal relaxation time, and Ti is the transverse relaxation time. On the other hand, under the slowly varying amplitude and phase approximation, we have the reduced wave equations [4],
@a) (5b)
@a)
(6b) i=h,v
where No is the atomic density, and the bracket ( indicates the averaging over the inhomogeneous broadening. 128
In this study, we deal the sharp line case (i.e. zero inhomogeneous broadening). Therefore, we do not carry out the averaging operation. The analysis of the inhomogeneously broadened case is in progress now. Two simultaneously incident optical fields with unequal frequency, which induce two-photon resonant excitation, shift the energy level of the medium according to the magnitude of the field, and cause frequency shift by the self-phase modulation during those propagation. The optical Stark shift is a local effect, while the self-induced phase modulation is a cumulative effect. Under these circumstances of the frequency modulation, we can not say that the two-photon area of pulses corresponds to the incline angle of the dipole vector to its original position. Thus the area theorem does not hold good no longer.
(4a)
_ +,e,., - G/T; >
NOni~k& Pi= ~E~cP~Aw~ ’
August 1977
)
3. Numerical pulse evolution In order to demonstrate the two-photon pulse propagation expressed by the coupled equations (4), (5) and (6), we numerically analyze these equations by the use of dimensionless variables t^= t/T1, i = pz, wherefi=a,Pi=q Ei,~i=~iT1,A~i= i\Ei T#i, s^7 6 T,. Therefore, ql = - G,S,, T3 = 6 - 6, + (AE, - A&). Figs. 2 (a), (b) show evolutions of gaussian pulses arising from passage through a medium-of L =40-l length with the following parameters: 6 = 0, pj =-I, cy= T1/T; = 0.5,~ = a = 1.05, /Jo//+, = 1.4. The input area defined by eq. (3) is 6.0 rr, and the input pulse width (full width at half-maximum) is + T, . The ratio E,/E~ is set equal to 1 .O. Similar to the results calculated using the conventional two-photon SIT formula (without the optical-Stark effect and the self-phase modulation), there are three remarkable effects; (1) the pulse breakup, (2) the peak amplification and (3) the pulse compression. However, although the input pulse area is 6.0 77,the input pulse area is reduced to about 5.25 7~.It is due to a relaxation effect, and dominantly the optical-Stark effect and the selfphase modulation. This implies that the two-photon area theorem does not hold good no longer in this case. The calculated self-phase modulation caused by the passage through the medium is shown in figs. 2 (c), (d). At the same point of i, peaks coinside each other. The self-phase modulation is significant for one weak
Volume
22. number
August
OPTICS COMMUNICATIONS
2
0 mO ii 6 i!
6 5 5
I 0
I I I t 0.2 0.4 0 DIMENSIONLESS
I 0.2 TIME
I
I 0.4
”
”
0.2 TIME
’
0.4 i
I_
i
Fig. 2. Numerical solutions of transient pulse behavior for two-photon pulse propagation and self-phase modulation. (a), (b): Input gaussian pulses and the calculated pulses at each point of the medium, (i = 1 f, 2,3,4). An input area is set to be 6.0n, and the pulse width is 7 Tr (fwhm). The ratio R = eV/eh is set equal to 1.0. (c), (d): Calculated self-phase modulation at each point of the medium. At the entrance of the medium, there is no chirping.
field associated with the large transition dipole moment compared with the other strong input field. The macroscopic two-photon polarization vector r precesses about the effective field y. The vector y consists of the components of y1 and y3. Therefore, y1 and y3 describe the time behavior of the effective field y at each point of the medium. The offset frequency r3 includes both the sum of self-phase modulations of propagating waves and the optical-Stark shift of the two-photon resonance. The calculated offset frequency j3 is shown in fig. 3(e). The two-photon Rabi frequency ihEv is shown in fig. 3(f). IV behavior is similar to that of the restricted two-photon SIT. Thus, at the entrance (i = 0) of the medium, the time dependence of T3 is entirely due to the Stark shift. If we select the ratio e,/eh to be equal to the reciprocal of the ratio I-(,,/P~, we can neglect the offset frequency. However, the self-phase modulations get larger through the propagation. At the end of the input
0.4 0 DIMENSIONLESS
Fig. 3. Numerical results of transient behavior of the vector along with the two-photon pulse propagation of fig. 2 (a) and (b). (e): The offset frequency ‘y^sat the each point of the medium. +s includes both the self-phase modulation of the propagating waves and the optical-Stark shift of the two-photon resonance. (f): Evolution of the two-photon Rabi frequency 51 = &Cv.
w ?I
2
0.2
1911
medium, the optical-Stark shift and the self-phase modulation become comparable in their contribution to”/3.
4. Concluding
remarks
The model of the two-photon pulse propagation that we have investigated here differs from the previous work [7,8] in the point of taking account of the optical Stark effect and the self-phase modulation. In conclusion, we can not apply the two-photon area theorem to the case of the large input area. However, in spite of including these additional effects, our computer analysis has indicated many similarities between the two cases such as the pulse breakup, the peak amplification and the pulse compression. Especially, for the case of the small input area, which was omitted for the limited paper, pulse evolution have entirely the properties of two-photon SIT described by the area theorem. Finally, we must mention, for the experimental observation, a problem of lasing in one-photon transition to the intermediate state in the three-level system. A gain of the lasing in a inhomogeneously broadened medium such as an atomic vapor reported by Tan-no [8] is generally smaller than that estimated by Grischkowsky et al. [2]. When we consider an effective inverted atomic density No X (T;/r) (No; the 129
Volume 22, number 2
OPTICS COMMUNICATIONS
atomic density, 7; the pulse width, T;; the inhomogenous relaxation time), the exponential gain is reduced to eS2, Thus, we cannot expect the lasing.
Acknowledgement We are grateful to Professor H. Inaba of Tohoku University
for his instruction.
References [l] S.R. Hartmann, IEEE J. Quantum Electron. QE-4 (1968) 802.
130
August 1977
[2] D. Grischkowsky, M.M.T. Loy and P.F. Liao, Phys. Rev. Al2 (1975) 2514. [3] D. Grischkowsky and M.M.T. Loy, Phys. Rev. Al2 (1975) 1117. [4] M.M.T. Loy, Phys. Rev. Letters 36 (1976) 1454. 151 M. Matsuoka, H. Nakatsuka and J. Okada, Phys. Rev. Al2 (1975) 1062. [6] E.M. Belenv and I.A. Poluektov, Sov. Phys. JETP 29 (1969) 754. [7] M. Takatsuji, Phys. Rev. All (1975) 619. [8] N. Tan-no, K. Yokoto and H. Inaba, Phys. Rev. Letters 29 (1972) 1211. [9] N. Tan-no, K. Yokoto and H. Inaba, J. Phys. B8 (1975)
339. [IO] N. Tan-no, K. Yokoto and K. Inaba, J. Phys. B8 (1975) 349. [ll] J.N. Elgin, G.H.C. New and K.E. Orkney, Opt. Comm. 18 (1976) 250.
August 1977
OPTICS COMMUNICATIONS
Volume 22, number 2
TRANSIENT
OF A HOMOGENEOUSLY
PULSE
PROPAGATION
BROADENED
IN TWO-PHOTON
THREE-LEVEL
RESONANCE
MEDIUM
Y. TAKAHASHI, N. TAN-NO and K. YOKOTO of Electronics, Yamagata University, Yonezawa, 992, Japan
Department
Received 21 April 1977
Pulse evolutions with SIT-like propagation are shown in spite of including the optical Stark effect and the self-phase modulation by solving numerically the two-photon Bloch equation and the coupled phase and amplitude equations.
1. Introduction In recent years, there has been a considerable interest in phenomena arising from the two-photon analogy of the well-known one-photon coherent effects, following the initial work of Hartmann [l]. Recently, a new formulation of the adiabatic following model for twophoton transitions has been reported [2,3]. There has been much interest in two-photon coherent effects, both theoretically and experimentally. Such phenomena include the two-photon mutation [4], the twophoton free induction decay [4,5] and the two-photon SIT [6-l 11. In this paper, we represent an evolution of propagating pulses for the two-photon transition of a homogeneously broadened three-level system by solving numerically the two-photon Bloch equation and the coupled phase and amplitude equations. An appearance of SIT-like pulse formations is shown in spite of including the optical Stark effect and the self-phase modulation.
Fig. 1. Energy levels relevant to the two-photon
transition.
tric fields is described by the two-photon Bloch equations [2,7,9]. Similar to the one-photon resonance case, this interaction can be visualized using a vector model, as a precession of a two-photon polarization vector r about an effective field y , dr/dt = yX r
2. Two-photon Bloch-Maxwell equations Coherent transient response of material polarizations play an essential role in propagation of intense, short duration laser pulses through a two-photon resonant medium. This problem can be described using a simplified three-level system shown in fig. 1. The response of the atomic system to the propagating elec-
(1)
where r(U,, V,, p,) is the relevant vector and y(yl, y2,ys) is the torque vector. The precession frequency is given by the magnitude of the y-vector, whose components are y1= --Kqe,,
Y2
=o,
y3 = 6 -&+(AE,-AEJTi, (2)
127
Volume 22, number 2
where ~(i=h,v)
OPTICS COMMUNICATIONS
is the amplitude
of the optical field,
k = 1_1~~,/2fiAw, is the two-photon AE,2 = /.?2 &4tiAw 1
gyroelectric
ratio, and
0
are the optical Stark shifts, and _& is the sum of the instantaneous phase shifts &t @,, and pj is the matrix element of the transition dipole moment. If we define a two-photon area I+b=K
TeheVdt, _m
(3)
this implies a fictitious angle of r from the origin. The equations of motion for the components of r are given bY aP,lat
= Qe,&
- (P,-P$/TI,
XJ,/at
= -[6-+,+(AE^-AE,)/ti]
av,jat
= ]6-&t(~~~-n~,)jfi]
I’,-G/T;,
(4b)
uP (4c)
where T, is the longitudinal relaxation time, and Ti is the transverse relaxation time. On the other hand, under the slowly varying amplitude and phase approximation, we have the reduced wave equations [4],
@a) (5b)
@a)
(6b) i=h,v
where No is the atomic density, and the bracket ( indicates the averaging over the inhomogeneous broadening. 128
In this study, we deal the sharp line case (i.e. zero inhomogeneous broadening). Therefore, we do not carry out the averaging operation. The analysis of the inhomogeneously broadened case is in progress now. Two simultaneously incident optical fields with unequal frequency, which induce two-photon resonant excitation, shift the energy level of the medium according to the magnitude of the field, and cause frequency shift by the self-phase modulation during those propagation. The optical Stark shift is a local effect, while the self-induced phase modulation is a cumulative effect. Under these circumstances of the frequency modulation, we can not say that the two-photon area of pulses corresponds to the incline angle of the dipole vector to its original position. Thus the area theorem does not hold good no longer.
(4a)
_ +,e,., - G/T; >
NOni~k& Pi= ~E~cP~Aw~ ’
August 1977
)
3. Numerical pulse evolution In order to demonstrate the two-photon pulse propagation expressed by the coupled equations (4), (5) and (6), we numerically analyze these equations by the use of dimensionless variables t^= t/T1, i = pz, wherefi=a,Pi=q Ei,~i=~iT1,A~i= i\Ei T#i, s^7 6 T,. Therefore, ql = - G,S,, T3 = 6 - 6, + (AE, - A&). Figs. 2 (a), (b) show evolutions of gaussian pulses arising from passage through a medium-of L =40-l length with the following parameters: 6 = 0, pj =-I, cy= T1/T; = 0.5,~ = a = 1.05, /Jo//+, = 1.4. The input area defined by eq. (3) is 6.0 rr, and the input pulse width (full width at half-maximum) is + T, . The ratio E,/E~ is set equal to 1 .O. Similar to the results calculated using the conventional two-photon SIT formula (without the optical-Stark effect and the self-phase modulation), there are three remarkable effects; (1) the pulse breakup, (2) the peak amplification and (3) the pulse compression. However, although the input pulse area is 6.0 77,the input pulse area is reduced to about 5.25 7~.It is due to a relaxation effect, and dominantly the optical-Stark effect and the selfphase modulation. This implies that the two-photon area theorem does not hold good no longer in this case. The calculated self-phase modulation caused by the passage through the medium is shown in figs. 2 (c), (d). At the same point of i, peaks coinside each other. The self-phase modulation is significant for one weak
Volume
22. number
August
OPTICS COMMUNICATIONS
2
0 mO ii 6 i!
6 5 5
I 0
I I I t 0.2 0.4 0 DIMENSIONLESS
I 0.2 TIME
I
I 0.4
”
”
0.2 TIME
’
0.4 i
I_
i
Fig. 2. Numerical solutions of transient pulse behavior for two-photon pulse propagation and self-phase modulation. (a), (b): Input gaussian pulses and the calculated pulses at each point of the medium, (i = 1 f, 2,3,4). An input area is set to be 6.0n, and the pulse width is 7 Tr (fwhm). The ratio R = eV/eh is set equal to 1.0. (c), (d): Calculated self-phase modulation at each point of the medium. At the entrance of the medium, there is no chirping.
field associated with the large transition dipole moment compared with the other strong input field. The macroscopic two-photon polarization vector r precesses about the effective field y. The vector y consists of the components of y1 and y3. Therefore, y1 and y3 describe the time behavior of the effective field y at each point of the medium. The offset frequency r3 includes both the sum of self-phase modulations of propagating waves and the optical-Stark shift of the two-photon resonance. The calculated offset frequency j3 is shown in fig. 3(e). The two-photon Rabi frequency ihEv is shown in fig. 3(f). IV behavior is similar to that of the restricted two-photon SIT. Thus, at the entrance (i = 0) of the medium, the time dependence of T3 is entirely due to the Stark shift. If we select the ratio e,/eh to be equal to the reciprocal of the ratio I-(,,/P~, we can neglect the offset frequency. However, the self-phase modulations get larger through the propagation. At the end of the input
0.4 0 DIMENSIONLESS
Fig. 3. Numerical results of transient behavior of the vector along with the two-photon pulse propagation of fig. 2 (a) and (b). (e): The offset frequency ‘y^sat the each point of the medium. +s includes both the self-phase modulation of the propagating waves and the optical-Stark shift of the two-photon resonance. (f): Evolution of the two-photon Rabi frequency 51 = &Cv.
w ?I
2
0.2
1911
medium, the optical-Stark shift and the self-phase modulation become comparable in their contribution to”/3.
4. Concluding
remarks
The model of the two-photon pulse propagation that we have investigated here differs from the previous work [7,8] in the point of taking account of the optical Stark effect and the self-phase modulation. In conclusion, we can not apply the two-photon area theorem to the case of the large input area. However, in spite of including these additional effects, our computer analysis has indicated many similarities between the two cases such as the pulse breakup, the peak amplification and the pulse compression. Especially, for the case of the small input area, which was omitted for the limited paper, pulse evolution have entirely the properties of two-photon SIT described by the area theorem. Finally, we must mention, for the experimental observation, a problem of lasing in one-photon transition to the intermediate state in the three-level system. A gain of the lasing in a inhomogeneously broadened medium such as an atomic vapor reported by Tan-no [8] is generally smaller than that estimated by Grischkowsky et al. [2]. When we consider an effective inverted atomic density No X (T;/r) (No; the 129
Volume 22, number 2
OPTICS COMMUNICATIONS
atomic density, 7; the pulse width, T;; the inhomogenous relaxation time), the exponential gain is reduced to eS2, Thus, we cannot expect the lasing.
Acknowledgement We are grateful to Professor H. Inaba of Tohoku University
for his instruction.
References [l] S.R. Hartmann, IEEE J. Quantum Electron. QE-4 (1968) 802.
130
August 1977
[2] D. Grischkowsky, M.M.T. Loy and P.F. Liao, Phys. Rev. Al2 (1975) 2514. [3] D. Grischkowsky and M.M.T. Loy, Phys. Rev. Al2 (1975) 1117. [4] M.M.T. Loy, Phys. Rev. Letters 36 (1976) 1454. 151 M. Matsuoka, H. Nakatsuka and J. Okada, Phys. Rev. Al2 (1975) 1062. [6] E.M. Belenv and I.A. Poluektov, Sov. Phys. JETP 29 (1969) 754. [7] M. Takatsuji, Phys. Rev. All (1975) 619. [8] N. Tan-no, K. Yokoto and H. Inaba, Phys. Rev. Letters 29 (1972) 1211. [9] N. Tan-no, K. Yokoto and H. Inaba, J. Phys. B8 (1975)
339. [IO] N. Tan-no, K. Yokoto and K. Inaba, J. Phys. B8 (1975) 349. [ll] J.N. Elgin, G.H.C. New and K.E. Orkney, Opt. Comm. 18 (1976) 250.