Abstract

An optical wave equation for scattered light in materials with macroscopic spatial inhomogeneities is presented and analyzed. The equation is applicable to a variety of linear and nonlinear optical interactions in media with intrinsic, induced, or engineered inhomegeneities. While it is valid regardless of the spatial scale of these inhomogeneities, the strongest effect is expected from inhomogeneities of a subwavelength scale size typical for nanostructured optical media.

© 2014 Optical Society of America

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References

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  1. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961, Reprint 1967).
  2. A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, 1991).
  3. G. T. Winch, M. C. Boshoff, C. J. Kok, and A. G. de Torr, “Spectroradiometric and colorimetric characteristics of daylight in the southern hemisphere: Pretoria, South Africa,” J. Opt. Soc. Am. 56, 456–464 (1966).
    [CrossRef]
  4. P. Bousquet, F. Flory, and P. Roche, “Scattering from multilayer thin films: theory and experiment,” J. Opt. Soc. Am. 71, 1115–1123 (1981).
    [CrossRef]
  5. B. I. Stepanov and A. P. Ivanova, eds., Theoretical and Applied Problems of Light Scattering (Nauka & Technika, 1971) (in Russian).
  6. Yu. N. Barabanenkov, Yu. A. Kravtsov, S. M. Rytov, and V. I. Tatarskii, “A state in the theory of the wave propagation in occasionally-inhomogeneous medium,” Sov. Phys. Usp. 13, 551–575 (1971).
    [CrossRef]
  7. V. I. Kovalev and R. G. Harrison, “A new nonlinear-wave-equation formalism for stimulated Brillouin scattering,” Phys. Lett. A 374, 2297–2300 (2010).
    [CrossRef]
  8. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
    [CrossRef]
  9. A. Einstein, “Theorie der Opaleszenz von homogenen Flussigkeiten und Flussigkeitsgemischen in der Nahe des kritischen Zustandes,” Ann. Phys. 33, 1275–1298 (1910).
    [CrossRef]
  10. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Addison–Wesley, 1960).
  11. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).
  12. A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989).
  13. M. Born and E. Wolf, Principles of Optics (Pergamon, 1964).

2010

V. I. Kovalev and R. G. Harrison, “A new nonlinear-wave-equation formalism for stimulated Brillouin scattering,” Phys. Lett. A 374, 2297–2300 (2010).
[CrossRef]

1981

1971

Yu. N. Barabanenkov, Yu. A. Kravtsov, S. M. Rytov, and V. I. Tatarskii, “A state in the theory of the wave propagation in occasionally-inhomogeneous medium,” Sov. Phys. Usp. 13, 551–575 (1971).
[CrossRef]

1966

1962

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

1910

A. Einstein, “Theorie der Opaleszenz von homogenen Flussigkeiten und Flussigkeitsgemischen in der Nahe des kritischen Zustandes,” Ann. Phys. 33, 1275–1298 (1910).
[CrossRef]

Armstrong, J. A.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Barabanenkov, Yu. N.

Yu. N. Barabanenkov, Yu. A. Kravtsov, S. M. Rytov, and V. I. Tatarskii, “A state in the theory of the wave propagation in occasionally-inhomogeneous medium,” Sov. Phys. Usp. 13, 551–575 (1971).
[CrossRef]

Bloembergen, N.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1964).

Boshoff, M. C.

Bousquet, P.

de Torr, A. G.

Ducuing, J.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Einstein, A.

A. Einstein, “Theorie der Opaleszenz von homogenen Flussigkeiten und Flussigkeitsgemischen in der Nahe des kritischen Zustandes,” Ann. Phys. 33, 1275–1298 (1910).
[CrossRef]

Flory, F.

Harrison, R. G.

V. I. Kovalev and R. G. Harrison, “A new nonlinear-wave-equation formalism for stimulated Brillouin scattering,” Phys. Lett. A 374, 2297–2300 (2010).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, 1991).

Kok, C. J.

Korn, G. A.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

Korn, T. M.

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

Kovalev, V. I.

V. I. Kovalev and R. G. Harrison, “A new nonlinear-wave-equation formalism for stimulated Brillouin scattering,” Phys. Lett. A 374, 2297–2300 (2010).
[CrossRef]

Kravtsov, Yu. A.

Yu. N. Barabanenkov, Yu. A. Kravtsov, S. M. Rytov, and V. I. Tatarskii, “A state in the theory of the wave propagation in occasionally-inhomogeneous medium,” Sov. Phys. Usp. 13, 551–575 (1971).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Addison–Wesley, 1960).

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Addison–Wesley, 1960).

Pershan, P. S.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Roche, P.

Rytov, S. M.

Yu. N. Barabanenkov, Yu. A. Kravtsov, S. M. Rytov, and V. I. Tatarskii, “A state in the theory of the wave propagation in occasionally-inhomogeneous medium,” Sov. Phys. Usp. 13, 551–575 (1971).
[CrossRef]

Tatarskii, V. I.

Yu. N. Barabanenkov, Yu. A. Kravtsov, S. M. Rytov, and V. I. Tatarskii, “A state in the theory of the wave propagation in occasionally-inhomogeneous medium,” Sov. Phys. Usp. 13, 551–575 (1971).
[CrossRef]

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961, Reprint 1967).

Thelen, A.

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989).

Winch, G. T.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1964).

Ann. Phys.

A. Einstein, “Theorie der Opaleszenz von homogenen Flussigkeiten und Flussigkeitsgemischen in der Nahe des kritischen Zustandes,” Ann. Phys. 33, 1275–1298 (1910).
[CrossRef]

J. Opt. Soc. Am.

Phys. Lett. A

V. I. Kovalev and R. G. Harrison, “A new nonlinear-wave-equation formalism for stimulated Brillouin scattering,” Phys. Lett. A 374, 2297–2300 (2010).
[CrossRef]

Phys. Rev.

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962).
[CrossRef]

Sov. Phys. Usp.

Yu. N. Barabanenkov, Yu. A. Kravtsov, S. M. Rytov, and V. I. Tatarskii, “A state in the theory of the wave propagation in occasionally-inhomogeneous medium,” Sov. Phys. Usp. 13, 551–575 (1971).
[CrossRef]

Other

B. I. Stepanov and A. P. Ivanova, eds., Theoretical and Applied Problems of Light Scattering (Nauka & Technika, 1971) (in Russian).

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961, Reprint 1967).

A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, 1991).

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Addison–Wesley, 1960).

G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, 1968).

A. Thelen, Design of Optical Interference Coatings (McGraw-Hill, 1989).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1964).

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Figures (5)

Fig. 1.
Fig. 1.

(a and b) Spectra, S(λ) that follows from Eq. (33), in relative units (rel. un.), versus λ/λ0 for l=λ0 (1), 7λ0 (2), and 4λ0 (3).

Fig. 2.
Fig. 2.

(a and b) Spectra, S(λ), that follows from Eq. (39), for the parameters similar to those in Fig. 1.

Fig. 3.
Fig. 3.

Measured R(λ) (thick lines) and calculated spectra S(λ) in relative units (thin lines) for mirrors designed for (a) λ0700nm, (b) 480 nm, and (c), (d) 250 nm.

Fig. 4.
Fig. 4.

(a) Measured R(λ) and (b) shapes, ΔR(λ) of the respective spectra that are 10 times stretched vertically. 1, for the TMDC mirror alone; 2, for the gold-coated mirror; 3, for the combined mirror; 4, the calculated S(λ) in relative units for TMDC structures with the twofold difference in the number of layers in (a) and (b).

Fig. 5.
Fig. 5.

Variation of S(q) with q/kp for λp=630nm and (a) l=1 and (b) 3 μm.

Equations (42)

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ε(r)=ε0+δε(r).
×H⃗=1cD⃗t,
×E⃗=1cH⃗t,
D⃗=0,
H⃗=0,
D⃗=εE⃗.
××E⃗+1c22D⃗t2=0,
E⃗=E⃗p+E⃗S.
××E⃗S+1c22D⃗St2=0
D⃗S=ε0E⃗S+δεE⃗p
××E⃗p+1c22D⃗pt2=0
D⃗p=ε0E⃗p.
2E⃗Sε0c22E⃗St2=(E⃗S)+δεc22E⃗pt2,
D⃗S=(ε0E⃗S+δεE⃗p)=0.
E⃗S=1ε0E⃗pδε.
2E⃗Sε0c22E⃗St2=1ε0(E⃗pδε)+δεc22E⃗pt2,
2E⃗Sε0c22E⃗St2=(δεc22E⃗pt2),
E⃗S=D⃗SδεE⃗pε0,
2D⃗S+ε0c2ωS2D⃗S=××[δεE⃗p].
××D⃗S+ε0c22D⃗St2=××[δεE⃗p].
E⃗p(r,t)=E⃗p(r,t)ei(ωptkpr),
××[δεE⃗p]=δε(××E⃗p)+δε×(×E⃗p)E⃗p2δε+(E⃗p·)δε(δε·)E⃗p.
××E⃗p=2E⃗p=ε0c22E⃗pt2.
××[δεE⃗p]=δε2E⃗p+δε×(×E⃗p)E⃗p2δε(δε·)E⃗p+(E⃗p·)δε=ε0c2δε2E⃗pt2+δε×(×E⃗p)E⃗p2δε(δε·)E⃗p+(E⃗p·)δε.
2D⃗Sε0c22D⃗St2=ε0δεc22E⃗pt2+[E⃗p2δεδε×(×E⃗p)+(δε·)E⃗p(E⃗p·)δε].
E⃗p(z,t)=E⃗pei(ωptkpz),
δε(z)=βeiqz.
D⃗S(z,t)=A⃗S(z)ei(ωpt+kpz),
2z2D⃗S+kS2D⃗S=δεE⃗p(kp2+q22qkp)δεE⃗p(qkp)2.
2ikSzA⃗S=βE⃗p(qkp)2eiΔkz,
A⃗S(l)=iβE⃗p(qkp)2kp(q2kp)sin[(q2kp)l2].
E⃗S(l)=A⃗S(l)ε0δεE⃗p(l)ε0=βE⃗pε0{1+i(qkp)2kp(q2kp)sin[(q2kp)l2]}.
S=(βε0)2{1+(qkp)4kp2(q2kp)2sin2[(q2kp)l2]}.
2E⃗Sε0c22E⃗St21ε0××[δεE⃗p]=δεc22E⃗pt2+1ε0[E⃗p2δεδε×(×E⃗p)+(δε·)E⃗p].
2z2E⃗S+kS2E⃗S=δεε0E⃗p(qkp)2.
E⃗S(z,t)=E⃗S(z)ei(ωSt+kSz),
2z2E⃗S+kS2E⃗S=δεε0E⃗pkp2.
E⃗S(l)=iβE⃗pkp(q2kp)sin[(q2kp)l2]
S(l)=(βε0)2kp2(q2kp)2sin2[(q2kp)l2].
2E⃗Sε0c22E⃗St21ε0E⃗p2δε,
2z2E⃗S+kS2E⃗Sδεε0E⃗pq2,
δε^(r,t)=δε^L(r,t)+δε^NL(r,t),

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