Abstract

We consider a boundary value problem of reflection of a powerful light wave from a dense resonant medium that exhibits intrinsic optical bistability because of the dipole–dipole interaction of two-level atoms. On the basis of analytical solutions and numerical simulations we establish that intrinsic optical bistability results in the dependence of the reflection coefficient on the adiabatically varying intensity of an external field. The reflection coefficient is an oscillating function of the intensity because of the additional reflectivity that is due to a light-induced sharp discontinuity in the nonlinear permittivity of a dense resonant medium.

© 1998 Optical Society of America

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  1. F. A. Hopf, C. M. Bowden, and W. Louisell, Phys. Rev. A 29, 2591 (1984).
    [CrossRef]
  2. Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Phys. Rev. A 34, 3917 (1986).
    [CrossRef] [PubMed]
  3. Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Opt. Commun. 61, 147 (1987).
    [CrossRef]
  4. J. W. Haus, L. Wang, M. Scalora, and C. M. Bowden, Phys. Rev. A 38, 4043 (1988).
    [CrossRef] [PubMed]
  5. R. Inguva and C. M. Bowden, Phys. Rev. A 41, 1670 (1990).
    [CrossRef] [PubMed]
  6. F. G. Bass and M. Ya. Granovsky, Fiz. Tverd. Tela (Leningrad) 16, 1882 (1974).
  7. F. G. Bass and Yu. G. Gurevich, Hot Electrons and Strong Electromagnetic Waves in Plasmas of Semiconductors and Gas Discharge (Nauka, Moscow, 1975).
  8. B. A. Samson and W. Gawlik, Phys. Rev. A 52, 4352 (1995).
    [CrossRef]
  9. A. A. Afanas’ev and B. A. Samson, Phys. Rev. A 53, 591 (1996).
    [CrossRef]
  10. B. B. Boiko, I. Z. Dzhilavdary, and N. S. Petrov, Zh. Prikl. Spektrosk. 23, 888 (1975).
  11. A. E. Kaplan, Zh. Eksp. Teor. Fiz. 72, 1710 (1977).
  12. In experimental investigation of hysteresis phenomena use is often made of input intensity modulation in triangular law. See, for example, H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, Sydney, 1985).
  13. A. N. Oraevsky, D. J. Jones, and D. K. Bandy, Opt. Commun. 111, 163 (1994).
    [CrossRef]
  14. S. M. Zaharov and E. A. Manykin, Zh. Eksp. Teor. Fiz. 115, 1053 (1994).
  15. L. D. Landau and E. M. Lifshits, Electrodynamics of Continuous Media (Fizmatgiz, Moscow, 1959).
  16. E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen (Akademische Verlagsgesellschaft, Leipzig, 1959), Vol. 1, Sec. 6.
  17. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1964).

1996

A. A. Afanas’ev and B. A. Samson, Phys. Rev. A 53, 591 (1996).
[CrossRef]

1995

B. A. Samson and W. Gawlik, Phys. Rev. A 52, 4352 (1995).
[CrossRef]

1994

A. N. Oraevsky, D. J. Jones, and D. K. Bandy, Opt. Commun. 111, 163 (1994).
[CrossRef]

S. M. Zaharov and E. A. Manykin, Zh. Eksp. Teor. Fiz. 115, 1053 (1994).

1990

R. Inguva and C. M. Bowden, Phys. Rev. A 41, 1670 (1990).
[CrossRef] [PubMed]

1988

J. W. Haus, L. Wang, M. Scalora, and C. M. Bowden, Phys. Rev. A 38, 4043 (1988).
[CrossRef] [PubMed]

1987

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Opt. Commun. 61, 147 (1987).
[CrossRef]

1986

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Phys. Rev. A 34, 3917 (1986).
[CrossRef] [PubMed]

1984

F. A. Hopf, C. M. Bowden, and W. Louisell, Phys. Rev. A 29, 2591 (1984).
[CrossRef]

1977

A. E. Kaplan, Zh. Eksp. Teor. Fiz. 72, 1710 (1977).

1975

B. B. Boiko, I. Z. Dzhilavdary, and N. S. Petrov, Zh. Prikl. Spektrosk. 23, 888 (1975).

1974

F. G. Bass and M. Ya. Granovsky, Fiz. Tverd. Tela (Leningrad) 16, 1882 (1974).

Afanas’ev, A. A.

A. A. Afanas’ev and B. A. Samson, Phys. Rev. A 53, 591 (1996).
[CrossRef]

Bandy, D. K.

A. N. Oraevsky, D. J. Jones, and D. K. Bandy, Opt. Commun. 111, 163 (1994).
[CrossRef]

Bass, F. G.

F. G. Bass and M. Ya. Granovsky, Fiz. Tverd. Tela (Leningrad) 16, 1882 (1974).

Ben-Aryeh, Y.

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Opt. Commun. 61, 147 (1987).
[CrossRef]

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Phys. Rev. A 34, 3917 (1986).
[CrossRef] [PubMed]

Boiko, B. B.

B. B. Boiko, I. Z. Dzhilavdary, and N. S. Petrov, Zh. Prikl. Spektrosk. 23, 888 (1975).

Bowden, C. M.

R. Inguva and C. M. Bowden, Phys. Rev. A 41, 1670 (1990).
[CrossRef] [PubMed]

J. W. Haus, L. Wang, M. Scalora, and C. M. Bowden, Phys. Rev. A 38, 4043 (1988).
[CrossRef] [PubMed]

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Opt. Commun. 61, 147 (1987).
[CrossRef]

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Phys. Rev. A 34, 3917 (1986).
[CrossRef] [PubMed]

F. A. Hopf, C. M. Bowden, and W. Louisell, Phys. Rev. A 29, 2591 (1984).
[CrossRef]

Dzhilavdary, I. Z.

B. B. Boiko, I. Z. Dzhilavdary, and N. S. Petrov, Zh. Prikl. Spektrosk. 23, 888 (1975).

Englund, J. C.

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Opt. Commun. 61, 147 (1987).
[CrossRef]

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Phys. Rev. A 34, 3917 (1986).
[CrossRef] [PubMed]

Gawlik, W.

B. A. Samson and W. Gawlik, Phys. Rev. A 52, 4352 (1995).
[CrossRef]

Granovsky, M. Ya.

F. G. Bass and M. Ya. Granovsky, Fiz. Tverd. Tela (Leningrad) 16, 1882 (1974).

Haus, J. W.

J. W. Haus, L. Wang, M. Scalora, and C. M. Bowden, Phys. Rev. A 38, 4043 (1988).
[CrossRef] [PubMed]

Hopf, F. A.

F. A. Hopf, C. M. Bowden, and W. Louisell, Phys. Rev. A 29, 2591 (1984).
[CrossRef]

Inguva, R.

R. Inguva and C. M. Bowden, Phys. Rev. A 41, 1670 (1990).
[CrossRef] [PubMed]

Jones, D. J.

A. N. Oraevsky, D. J. Jones, and D. K. Bandy, Opt. Commun. 111, 163 (1994).
[CrossRef]

Kaplan, A. E.

A. E. Kaplan, Zh. Eksp. Teor. Fiz. 72, 1710 (1977).

Louisell, W.

F. A. Hopf, C. M. Bowden, and W. Louisell, Phys. Rev. A 29, 2591 (1984).
[CrossRef]

Manykin, E. A.

S. M. Zaharov and E. A. Manykin, Zh. Eksp. Teor. Fiz. 115, 1053 (1994).

Oraevsky, A. N.

A. N. Oraevsky, D. J. Jones, and D. K. Bandy, Opt. Commun. 111, 163 (1994).
[CrossRef]

Petrov, N. S.

B. B. Boiko, I. Z. Dzhilavdary, and N. S. Petrov, Zh. Prikl. Spektrosk. 23, 888 (1975).

Samson, B. A.

A. A. Afanas’ev and B. A. Samson, Phys. Rev. A 53, 591 (1996).
[CrossRef]

B. A. Samson and W. Gawlik, Phys. Rev. A 52, 4352 (1995).
[CrossRef]

Scalora, M.

J. W. Haus, L. Wang, M. Scalora, and C. M. Bowden, Phys. Rev. A 38, 4043 (1988).
[CrossRef] [PubMed]

Wang, L.

J. W. Haus, L. Wang, M. Scalora, and C. M. Bowden, Phys. Rev. A 38, 4043 (1988).
[CrossRef] [PubMed]

Zaharov, S. M.

S. M. Zaharov and E. A. Manykin, Zh. Eksp. Teor. Fiz. 115, 1053 (1994).

Fiz. Tverd. Tela (Leningrad)

F. G. Bass and M. Ya. Granovsky, Fiz. Tverd. Tela (Leningrad) 16, 1882 (1974).

Opt. Commun.

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Opt. Commun. 61, 147 (1987).
[CrossRef]

A. N. Oraevsky, D. J. Jones, and D. K. Bandy, Opt. Commun. 111, 163 (1994).
[CrossRef]

Phys. Rev. A

J. W. Haus, L. Wang, M. Scalora, and C. M. Bowden, Phys. Rev. A 38, 4043 (1988).
[CrossRef] [PubMed]

R. Inguva and C. M. Bowden, Phys. Rev. A 41, 1670 (1990).
[CrossRef] [PubMed]

F. A. Hopf, C. M. Bowden, and W. Louisell, Phys. Rev. A 29, 2591 (1984).
[CrossRef]

Y. Ben-Aryeh, C. M. Bowden, and J. C. Englund, Phys. Rev. A 34, 3917 (1986).
[CrossRef] [PubMed]

B. A. Samson and W. Gawlik, Phys. Rev. A 52, 4352 (1995).
[CrossRef]

A. A. Afanas’ev and B. A. Samson, Phys. Rev. A 53, 591 (1996).
[CrossRef]

Zh. Eksp. Teor. Fiz.

S. M. Zaharov and E. A. Manykin, Zh. Eksp. Teor. Fiz. 115, 1053 (1994).

A. E. Kaplan, Zh. Eksp. Teor. Fiz. 72, 1710 (1977).

Zh. Prikl. Spektrosk.

B. B. Boiko, I. Z. Dzhilavdary, and N. S. Petrov, Zh. Prikl. Spektrosk. 23, 888 (1975).

Other

F. G. Bass and Yu. G. Gurevich, Hot Electrons and Strong Electromagnetic Waves in Plasmas of Semiconductors and Gas Discharge (Nauka, Moscow, 1975).

In experimental investigation of hysteresis phenomena use is often made of input intensity modulation in triangular law. See, for example, H. M. Gibbs, Optical Bistability: Controlling Light with Light (Academic, Sydney, 1985).

L. D. Landau and E. M. Lifshits, Electrodynamics of Continuous Media (Fizmatgiz, Moscow, 1959).

E. Kamke, Differentialgleichungen. Lösungsmethoden und Lösungen (Akademische Verlagsgesellschaft, Leipzig, 1959), Vol. 1, Sec. 6.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1964).

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Figures (5)

Fig. 1
Fig. 1

Regions of bistability on the Iδ plane.

Fig. 2
Fig. 2

Dependence of population difference N and the imaginary (Im χ) and real (Re χ) parts of the resonant susceptibility on the radiation intensity I at b=16 and δ=-2.

Fig. 3
Fig. 3

Dependence of reflection coefficient R on incident wave intensity I0 at 0=4.5 and δ=-0.5. The solid curves are calculated from Eq. (29). The dotted curves are obtained from numerical simulations of Eq. (18).

Fig. 4
Fig. 4

Reflection coefficient R and layer thickness ξ0 versus adiabatic increase (decrease) of external field intensity I0 at 0=4.5, b=6, and δ=-0.5.

Fig. 5
Fig. 5

Dependence of reflection coefficient R on I0 for the parameters of Fig. 4.

Equations (53)

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Eˆ(z, t)=E(z)exp(-iωt)+c.c.
dNdt=i 2μ (E*P-EP*)-N-1T1,
dPdt=i μ EN-1T2 [1-i(δ+bN)]P,
b=43 π μ2 N0T2
P=μN0P exp(-iωt)+c.c.=χ(|E|2)E exp(-iωt)+c.c.,
χ=3i4π bN1-i(δ+bN)
1-NN [1+(δ+bN)2]=|E|2,
E(z)=E0expi ωc z+r exp-i ωc z,z0.
d2Erdz2+ω2c2 (0+4πχ)Er=0,z0,
E(-0)=Ed(+0),
dE(-0)dz=dEd(+0)dz,
Ed(z0-0)=Eu(z0+0),
dEd(z0-0)dz=dEu(z0+0)dz.
Eu(z=)=0.
d2Eddξ2+(0+4πχd)Ed=0,0ξξ0,
d2Eudξ2+(0+4πχu)Eu=0,ξ>ξ0,
E0(1+r)=Ed(0),E0(1-r)+idEddξ0=0,
Ed(ξ0)=Eu(ξ0),d(Ed-Eu)dξξ0=0,
Eu(ξ=)=0.
|Eu(ξ0)|=Idown.
d2Erdξ2+(0+4πχ)Er=0,ξ0,
E0(1+r)=Er(0),E0(1-r)+idErdξ0=0,
Er(ξ=)=0.
N1-|Er|21+(δ+b)2.
=¯+Δ|Er|2,
¯=0-3b(δ+b)1+(δ+b)2+i 3b1+(δ+b)2,
Δ=-i3b 1-iδ[1-i(δ+b)]2[1+(δ+b)2].
Er0(ξ)=2E01+¯ exp(i¯ξ).
d2Erdξ2+[¯+σ exp(-2κξ)]Er=0,
Er(ξ)=C+Jνσκ exp(-κξ)+C-J-νσκ exp(-κξ),
C-=0,C+=2E0(1-¯)Jνσκ+iσJν-1σκ.
r=1-¯1+¯+iβ|E0|2,
β=4 Δκ ¯(1+¯)2|1+¯|2 3¯-n|¯|+3iκ¯.
r=(1+d)(d-u)+(1-d)(d+u)exp(-2idξ0)(1-d)(d-u)+(1+d)(d+u)exp(-2idξ0).
d(1+d)(d+u)+(1-d)(d-u)exp(i2dξ0)exp(-Imdξ0)=14 Idown|E0|.
exp(-2 Imdξ0)1.
ξ0=1κd ln F,
F=4|E0|2Idown nd2+κd2[(1+nd)2+κd2][(nd+nu)2+(kd+κu)2]2.
ξmin=1κd ln 2IupIdown nd2+κd2(nd+nu)2+(κd+κu)21/2.
I0=I^d=Idown4 [(1+nu)2+κu2].
I^u=Iup4 [(1+nd)2+κd2].
r1-d1+d 1+4d(1-d) (d-u)(d+u) 1F2×exp2i ndκd ln F.
|ΔE0|2 =|E0|2expπ κdnd-12,
d2Edξ2+[0+4πχˆ(|E|2)]E=0,0<ξ<,
E(0)=2E0+idEdξ0,E(ξ)=0,
Es+1ξξ+[0+4πχˆ(|Es|2)]Es+1=0,
Es+1(ξ1)+Es+1(ξ2)2=2E0+ih-1[Es+1(ξ2)-Es+1(ξ1)],
Es+1(ξK)=0,
Es+1(ξk),k=1, 2, K,
|Es(0)|2 >Iup;
χˆ(|Es|2)=χdfor|Es|2 >Idown,
χˆ(|Es|2)=χufor|Es|2Idown.
φs=α|Es|2+(1-α)φs-1,φ0=0

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