Abstract

An analysis of a monochromatic plane wave’s reflection at a nonlinear interface is presented that constitutes a generalization and an extension of previous studies on this subject. We examine the influence of self-focusing and self-defocusing Kerr media on total and partial reflection states. We consider the whole range of angles of incidence and four types of media interfaces. In our model we do not apply the slowly varying amplitude approximation, and we give analytical real positive nonoscillating solutions to the nonlinear differential wave equations in total and partial reflection cases. For each case we also derive relations between the amplitude reflectance and input intensity, which indicate that in some situations the wave’s behavior at a nonlinear interface is governed by the nonlinear critical and the nonlinear characteristic angles. It is pointed out that the reflection states change bistably. We show, for different angles of incidence, the ranges in which the effective permittivity of nonlinear media can be changed via the incident intensity.

© 1998 Optical Society of America

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References

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  1. A. E. Kaplan, “Hysteresis reflection and refraction of light by a nonlinear boundary—new class of effects in nonlinear optics,” JETP Lett. 24, 114–119 (1976).
  2. A. E. Kaplan, “Theory of hysteresis reflection and refraction of light by a nonlinear medium,” Sov. Phys. JETP 45, 896–905 (1977).
  3. B. B. Bojko, N. S. Pietrow, “Odrażenie swieta od usiliwajuszczych i nieliniejnych sred,” Nauka Teck. (1988).
  4. D. Marcuse, “Reflection of a Gaussian beam from a nonlinear interface,” Appl. Opt. 19, 3130–3139 (1980).
    [CrossRef] [PubMed]
  5. P. W. Smith, W. J. Tomlinson, “Nonlinear optical interfaces: switching behavior,” IEEE J. Quantum Electron. QE-20, 30–36 (1984).
    [CrossRef]
  6. R. Boyd, Nonlinear Optics (Academic, New York, 1992).
  7. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  8. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

1988 (1)

B. B. Bojko, N. S. Pietrow, “Odrażenie swieta od usiliwajuszczych i nieliniejnych sred,” Nauka Teck. (1988).

1984 (1)

P. W. Smith, W. J. Tomlinson, “Nonlinear optical interfaces: switching behavior,” IEEE J. Quantum Electron. QE-20, 30–36 (1984).
[CrossRef]

1980 (1)

1977 (1)

A. E. Kaplan, “Theory of hysteresis reflection and refraction of light by a nonlinear medium,” Sov. Phys. JETP 45, 896–905 (1977).

1976 (1)

A. E. Kaplan, “Hysteresis reflection and refraction of light by a nonlinear boundary—new class of effects in nonlinear optics,” JETP Lett. 24, 114–119 (1976).

Bojko, B. B.

B. B. Bojko, N. S. Pietrow, “Odrażenie swieta od usiliwajuszczych i nieliniejnych sred,” Nauka Teck. (1988).

Boyd, R.

R. Boyd, Nonlinear Optics (Academic, New York, 1992).

Kaplan, A. E.

A. E. Kaplan, “Theory of hysteresis reflection and refraction of light by a nonlinear medium,” Sov. Phys. JETP 45, 896–905 (1977).

A. E. Kaplan, “Hysteresis reflection and refraction of light by a nonlinear boundary—new class of effects in nonlinear optics,” JETP Lett. 24, 114–119 (1976).

Marcuse, D.

Pietrow, N. S.

B. B. Bojko, N. S. Pietrow, “Odrażenie swieta od usiliwajuszczych i nieliniejnych sred,” Nauka Teck. (1988).

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

Smith, P. W.

P. W. Smith, W. J. Tomlinson, “Nonlinear optical interfaces: switching behavior,” IEEE J. Quantum Electron. QE-20, 30–36 (1984).
[CrossRef]

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

Tomlinson, W. J.

P. W. Smith, W. J. Tomlinson, “Nonlinear optical interfaces: switching behavior,” IEEE J. Quantum Electron. QE-20, 30–36 (1984).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

P. W. Smith, W. J. Tomlinson, “Nonlinear optical interfaces: switching behavior,” IEEE J. Quantum Electron. QE-20, 30–36 (1984).
[CrossRef]

JETP Lett. (1)

A. E. Kaplan, “Hysteresis reflection and refraction of light by a nonlinear boundary—new class of effects in nonlinear optics,” JETP Lett. 24, 114–119 (1976).

Nauka Teck. (1)

B. B. Bojko, N. S. Pietrow, “Odrażenie swieta od usiliwajuszczych i nieliniejnych sred,” Nauka Teck. (1988).

Sov. Phys. JETP (1)

A. E. Kaplan, “Theory of hysteresis reflection and refraction of light by a nonlinear medium,” Sov. Phys. JETP 45, 896–905 (1977).

Other (3)

R. Boyd, Nonlinear Optics (Academic, New York, 1992).

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).

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Figures (11)

Fig. 1
Fig. 1

Schematic representation of the nonlinear interface and the coordinate system.

Fig. 2
Fig. 2

Dependence between the squared values of the incident wave’s amplitude EI and the amplitude inside the nonlinear medium E(0) in the case of Δ<0, χ>0, and α[α(L)CNL, αNL]. Curves: A, constant amplitude solutions for a partially reflected wave; D, decaying amplitude solutions for a totally reflected wave; G, level satisfying the condition of α(eff)CNL.

Fig. 3
Fig. 3

Dependence between the squared values of the incident wave’s amplitude EI and the amplitude inside the nonlinear medium E(0) in the case of Δ<0, χ>0, and α[αNL, 90°]. Curves: A, constant amplitude solutions for a partially reflected wave; D, decaying amplitude solutions for a totally reflected wave; F, G, levels satisfying the conditions of the effective interface decay and that of α(eff)CNL, respectively.

Fig. 4
Fig. 4

Areas of incident amplitude related to the different wave reflection states in terms of dependence on the angle of incidence changing above α(L)CNL in the case of Δ<0, χ>0. The patterned areas correspond to the partial (top diagonal lines) and total (bottom diagonal lines) reflection states, and in the cross-hatched area both reflection states are possible. On the curves dividing these areas the angle of incidence becomes αCNL.

Fig. 5
Fig. 5

Dependence between the squared values of the incident wave’s amplitude EI and the amplitude inside the nonlinear medium E(0) in the case of Δ<0, χ<0, and α<α(L)CNL. Curves: A, constant amplitude solutions for a partially reflected wave; C, solutions in a total reflection case approaching the constant level, C1, which fulfills the condition of α(eff)CNL.

Fig. 6
Fig. 6

Areas of incident amplitude related to the different wave reflection states in terms of dependence on the angle of incidence changing below α(L)CNL in the case of Δ<0, χ<0. The patterned areas correspond to the partial (bottom diagonal lines) and total (top diagonal lines) reflection states, and in the cross-hatched area both reflection states are possible. On the curves dividing these areas the angle of incidence becomes αCNL.

Fig. 7
Fig. 7

Square of the real amplitude (C) of a totally reflected wave as a function of the spatial variable z in the case of Δ<0, χ<0, and α<α(L)CNL. Note that the limit value (C1) fulfills the condition of α(eff)CNL and that we have also found solutions of this type in the case of Δ>0, χ<0.

Fig. 8
Fig. 8

Dependence between the squared values of the incident wave’s amplitude EI and the amplitude inside the nonlinear medium E(0) in the case of Δ>0, χ<0, and α[0°, αNL]. Curves: A, constant amplitude solutions for a partially reflected wave; C, solutions in a total reflection case approaching the constant level C1, which fulfills the condition of α(eff)CNL; F, level satisfying the condition of the effective interface decay.

Fig. 9
Fig. 9

Dependence between the squared values of the incident wave’s amplitude EI and the amplitude inside the nonlinear medium E(0) in the case of Δ>0, χ<0, and α[αNL, 90°]. Curves: A, constant amplitude solutions for a partially reflected wave; B, solutions in a partial reflection case approaching the constant level B1; C, solutions in a total reflection case tending to the constant level C1, which fulfills the condition of α(eff)CNL; F, level satisfying the condition of the effective interface decay.

Fig. 10
Fig. 10

Areas of incident amplitude related to the different wave reflection states in terms of dependence on the angle of incidence in the case of Δ>0, χ<0. The patterned areas correspond to the partial (bottom diagonal lines) and total (top diagonal lines) reflection states, and in the cross-hatched area both reflection states are possible. On the curves dividing these areas the angle of incidence becomes αCNL.

Fig. 11
Fig. 11

Square of the real amplitude (B) of a partially reflected wave as a function of the spatial variable z in the case of Δ>0, χ<0, and α>αNL. Note that solutions of this type can exist only in this case and that the limit value (B1) depends on α and EI.

Equations (93)

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εNL(E˜)=ε(L)NL+χ˜|E˜|2,
ε(L)NL=εLsin2 α(L)CNL.
EI(x˜, z˜, t)=eyE˜I exp[i(k˜xx˜+k˜zz˜+ΦI-ωt)],
ER(x˜, z˜, t)=eyE˜R exp[i(k˜xx˜-k˜zz˜+ΦR-ωt)],
E(x˜, z˜, t)=eyE˜(z˜)exp[i(k˜xx˜+Φ(z˜)-ωt)],
curlcurlE=-μ0ε0 2t2[ε(L)NLE+χ˜|E|2E].
E=|χ˜|E˜,
a=[ω/c]a˜fora=x, z,
d2E(z)dz2-dΦ(z)dz2E(z)+[k(L)zNL]2E(z)+χE3(z)=0,
ddzdΦ(z)dzE2(z)=0,
RSz=R(E×H)z=1μ0ωdΦ(z˜)dz˜E˜2(z˜),
d2E(z)dz2-C0E3(z)+k(L)zNL2E(z)+χE3(z)=0.
C0=dΦ(z)dzE2(z)2.
E(0)=EI1+R2+2R cos θ,
dEdz(0)=-EI 2kzR sin θ1+R2+2R cos θ,
C0=EI4kz2(1-R2)2,
ρ=E˜RE˜Iexp[i(ΦR-ΦI)]=R exp(-iθ).
dΦ(z)dz=C0E2(z)
Φ(0)=ΦI+arccos1+R cos θ1+R2+2R cos θ.
dE(z)dz=±i 22[χE4(z)+2k(L)zNL2E2(z)-2C1]1/2,
C1=dE(0)dz2+k(L)zNL2E2(0)+χ2E4(0).
E(0)=EI2(1+cos θ),
dEdz(0)=-EI 2kz sin θ2(1+cos θ)
Φ=ΦI+arccos1+cos θ2.
dE2(z)dz=±i2[χE6(z)+2k(L)zNL2E4(z)-2C2E2(z)+2C0]1/2,
C2=dEdz(0)2+k(L)zNL2E2(0)+χ2E4(0)+C0E2(0).
r=R cos θ.
E(0)=EI(1+r),
dEdz(0)=0,
C0=EI4kz2(1-r2)2
Φ(0)=ΦI.
εNL(E[α(eff)CNL])=εLsin2 α(eff)CNL.
E2=kz2(1-r)2-k(L)zNL2(1+r)2χ(1+r)2.
Φ(z)=ΦI+C0E2z,
β=arctan1+r1-rtan α.
r-1, kz(α)-k(L)zNL(α)kz(α)+k(L)zNL(α),
r-1, kz(α)-k(L)zNL(α)kz(α)+k(L)zNL(α).
E2(z)=8k(L)zNL2χ1cosh2[|k(L)zNL|z-½ ln|d2|].
EI,maxTR2(α)
=k(L)zNL2(α)2χforα(L)CNL<α<αNL[kz2(α)-k(L)zNL2(α)]28|χ|kz2(α)forαNLα<90°.
αNL=arccos|Δ|2εL>α(L)CNL.
θ0, arccos k(L)zNL2(α)+kz2(α)kz2(α)-k(L)zNL2(α).
θarccos 3kz2(α)+k(L)zNL2(α)kz2(α)-k(L)zNL2(α),arccos k(L)zNL2(α)+kz2(α)kz2(α)-k(L)zNL2(α),
E(decay)2=|Δ||χ|,
EI,minPR2(α)=k(L)zNL2(α)4χ.
EI,maxPR2(α)=2 [kz2(α)-k(L)zNL2(α)]3{kz2(α)+2k(L)zNL2(α)-kz[kz2(α)+8k(L)zNL2(α)]1/2}χkz2(α){3kz2(α)-kz(α)[kz2(α)+8k(L)zNL2(α)]1/2}4.
rkz(α)-k(L)zNL(α)kz(α)+k(L)zNL(α),
2kz2(α)+k(L)zNL2(α)-kz(α)[kz2(α)+8k(L)zNL2(α)]1/2kz2(α)-k(L)zNL2(α).
E2(z)=ETR2+ETR2sinh222k(L)zNLz+12ln|d3|forE(0)>k(L)zNL|χ|ETR2-ETR2cosh222k(L)zNLz-12ln|d3|forE(0)<k(L)zNL|χ|,
ETR2(α)=k(L)zNL2(α)χ.
EI,minTR2(α)=k(L)zNL4(α)8|χ|kz2(α).
E2(z)=8k(L)zNL2χ1sinh2[|k(L)zNL|z-½ ln|d1|].
θarccos kz2(α)+k(L)zNL2(α)kz2(α)-k(L)zNL2(α), π,
EI,maxPR2(α)=2 [kz2(α)-k(L)zNL2(α)]3{kz2(α)+2k(L)zNL2(α)-kz[kz2(α)+8k(L)zNL2(α)]1/2}χkz2(α){3kz2(α)-kz(α)[kz2(α)+8k(L)zNL2(α)]1/2}4.
rkz(α)-k(L)zNL(α)kz(α)+k(L)zNL(α),
2kz2(α)+k(L)zNL2(α)-kz(α)[kz2(α)+8k(L)zNL2(α)]1/2kz2(α)-k(L)zNL2(α).
E2(z)=EPR2-3ξcosh232|χ|ξz,
ξ=-12EI2(1+r)2+2k(L)zNL23χ
EPR2=-12EI2(1+r)2-k(L)zNL2χ.
EI2=2 -2kz2(1-r)2-k(L)zNL2(1+r)2+2kz(1-r)[kz2(1-r)2+k(L)zNL2(1+r)2]1/2χ(1+r)4.
EI2(α)k(L)zNL2(α)[3kz(α)+k(L)zNL(α)]218|χ|kz2(α),[kz2(α)+k(L)zNL2(α)]28|χ|kz2(α),
r3kz2(α)-k(L)zNL2(α)kz2(α)+k(L)zNL2(α), 3kz(α)-k(L)zNL(α)3kz(α)+k(L)zNL(α).
E,maxPR2=|Δ||χ|,
Φ(z)=ΦI+C0EPR2z+C0EPR2arctan3|ξχ|2|k(L)zNL2-χEPR2|1/2 tanh32|χξ|z[|k(L)zNL2-χEPR2|]1/2.
β=arctanEPR2EI2(1-r2)tan α,
Sx(z)=1μ0ωk˜xE˜2(z)=|χ|ε0μ0 kxEPR2-3ξcosh232|χ|ξz.
EI,minTR2(α)=2k(L)zNL2(α)-kz2(α)8|χ|,
θ0, arccos 2k(L)zNL2(α)-3kz2(α)2k(L)zNL2(α)-kz2(α).
E(0)E(z) dE[χE4(z)+2k(L)zNL2E2(z)-2C1]1/2=±i 22z.
E2(z)=8k(L)zNL2χd1 exp[-2|k(L)zNL|z]{1-d1 exp[-2|k(L)zNL|z]}2,
d1=[|χ|E2(0)+2|k(L)zNL2|]1/2-[2|k(L)zNL2|]1/2[|χ|E2(0)+2|k(L)zNL2|]1/2+[2|k(L)zNL2|]1/2;
E2(z)=8k(L)zNL2χ d2 exp[±2|k(L)zNL|z]{1+d2 exp[±2|k(L)zNL|z]}2,
d2=[|2k(L)zNL2|]1/2-[|2k(L)zNL2|-E2(0)|χ|]1/2[|2k(L)zNL2|]1/2+[|2k(L)zNL2|-E2(0)|χ|]1/2.
2k(L)zNL2χ2+8 C1χ=0.
E2(z)=k(L)zNL2χ1+d3 exp[±2k(L)zNLz]1-d3 exp[±2k(L)zNLz]2,
d3=|χ|E(0)+k(L)zNL|χ|E(0)-k(L)zNL.
EI2=-k(L)zNL2(1+cos θ)+kz2(1-cos θ)χ(1+cos θ)2,
EI2=kz2(1-cos θ)2|χ|(1+cos θ)21+k(L)zNL2(1+cos θ)kz2(1-cos θ)±1+2 k(L)zNL2(1+cos θ)kz2(1-cos θ)1/2.
dU(z)dz=±i2χ[U3(z)-p2U(z)+p3]1/2,
p2=2C2χ+32k(L)zNL23χ2,
p3=22k(L)zNL2C23χ2+C0χ+2k(L)zNL23χ3.
U(0)U(z) dU[U3(z)-p2U(z)+p3]1/2=±i2χz.
D=p322-p233=0,
E2(z)=-20.5p33+30.5p33 ×1+d1 exp[±(6|χ|0.5p33)1/2z]1-d1 exp[±(6|χ|0.5p33)1/2z]2-2k(L)zNL23χ,
d1=E2(0)+2k(L)zNL23χ+20.5p331/2-30.5p33E2(0)+2k(L)zNL23χ+20.5p331/2+30.5p33,
E2(z)=-20.5p33+30.5p33 ×d2±tanh32|χ0.5p33|1/2z1±d2 tanh32|χ0.5p33|1/2z2-2k(L)zNL23χ,
d2=E2(0)+2k(L)zNL23χ+20.5p3330.5p331/2.
E2=0.5p33-2k(L)zNL23χ.
D=-(1+r)1227EI2-kz2(1-r)2χ(1+r)4(1-γ) ×EI2+2 kz2(1-r)2χ(1+r)4(2+γ-21+γ) ×EI2+2 kz2(1-r)2χ(1+r)4(2+γ+21+γ),
γ=k(L)zNL(1+r)kz(1-r)2.
EI2=kz2(1-r)2-k(L)zNL2(1+r)2χ(1+r)4
EI2=2 -2kz2(1-r)2-k(L)zNL2(1+r)2χ(1+r)4±2 2kz(1-r)[kz2(1-r)2+k(L)zNL2(1+r)2]1/2χ(1+r)4,
r-1, 3kz(α)-k(L)zNL(α)3kz(α)+k(L)zNL(α).

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