Abstract

A new wave-vector-space method of finding electromagnetic wave propagation in bounded media without use of boundary conditions is applied to finding surface polaritons on an optically active anaxial crystal. Because a proper constitutive derivation of the optical-activity tensor shows that the quadrupolar interaction plays an important role, the continuity of the tangential component of the H field is no longer a valid boundary condition. In its absence the use of the wave-vector-space method that uses no boundary conditions is essential. Another unique aspect of the wave-vector-space method is that it derives a surface-nonlocality tensor that accounts for the altered nonlocal interaction of optical activity near the surface. The dispersion relation of the surface polariton is found to be independent of both the bulk optical-activity parameter and the surface-nonlocality parameters. The electric field profile is dependent on the bulk optical-activity parameter to first order. This dependence causes the surface polariton to lose the TM-mode character that it has in a nonoptically active crystal. Surprisingly, the surface-nonlocality parameters also disappear from the field profile to first order. This complete disappearance to first order of the nonlocality parameters is a surprising and physically unexplained result because it does not happen in the transmission and reflection problem or in the optically active waveguide problem.

© 2000 Optical Society of America

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References

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  1. D. F. Nelson, “Mechanisms and dispersion of crystalline optical activity,” J. Opt. Soc. Am. B 6, 1110–1116 (1989).
    [CrossRef]
  2. E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
    [CrossRef]
  3. B. Chen and D. F. Nelson, “Wave-vector-space method for wave propagation in bounded media,” Phys. Rev. B 48, 15365–15371 (1993).
    [CrossRef]
  4. B. Chen and D. F. Nelson, “Wave propagation of exciton polaritons by a wave-vector-space method,” Phys. Rev. B 48, 15372–15389 (1993).
    [CrossRef]
  5. D. F. Nelson, “Deriving the transmission and reflection of an optically active medium without using boundary conditions,” Phys. Rev. E 51, 6142–6153 (1995).
    [CrossRef]
  6. D. F. Nelson, “Surface nonlocality effects on an optically active waveguide,” J. Appl. Phys. 86, 5348–5355 (1999).
    [CrossRef]
  7. D. F. Nelson and A. L. Ivanov, “Alternative explanation of specular optical activity,” Opt. Lett. 23, 86–88 (1998).
    [CrossRef]
  8. V. M. Agranovich and V. L. Ginzburg, “Phenomenological electrodynamics of gyrotropic media,” Sov. Phys. JETP 36, 440–443 (1973).
  9. V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons, 2nd ed. (Springer, Berlin, 1984), pp. 96–97.
  10. D. N. Pattanayak and J. L. Birman, “Wave propagation in optically active and magnetoelectric media of arbitrary geometry,” Phys. Rev. B 24, 4271–4278 (1981).
    [CrossRef]
  11. A. Puri, D. N. Pattanayak, and J. L. Birman, “Surface waves in bounded gyrotropic media,” J. Opt. Soc. Am. 72, 938–942 (1982).
    [CrossRef]
  12. D. F. Nelson, “Deriving surface polaritons without using boundary conditions,” J. Opt. Soc. Am. B 13, 1956–1960 (1996).
    [CrossRef]
  13. M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon, Oxford, 1954), pp. 336–338.
  14. See Sect. IV C of Ref. 4.
  15. It was remarked in Ref. 6 that s appears linearly in the dispersion relation. That is true if the expansion in s is done alone. If an expansion in both s and g is done simultaneously, then it is found that terms linear in s are also linear in g, that is, bilinear in small quantities and thus negligible.

1999 (1)

D. F. Nelson, “Surface nonlocality effects on an optically active waveguide,” J. Appl. Phys. 86, 5348–5355 (1999).
[CrossRef]

1998 (1)

1996 (1)

1995 (1)

D. F. Nelson, “Deriving the transmission and reflection of an optically active medium without using boundary conditions,” Phys. Rev. E 51, 6142–6153 (1995).
[CrossRef]

1993 (2)

B. Chen and D. F. Nelson, “Wave-vector-space method for wave propagation in bounded media,” Phys. Rev. B 48, 15365–15371 (1993).
[CrossRef]

B. Chen and D. F. Nelson, “Wave propagation of exciton polaritons by a wave-vector-space method,” Phys. Rev. B 48, 15372–15389 (1993).
[CrossRef]

1989 (1)

1982 (1)

1981 (1)

D. N. Pattanayak and J. L. Birman, “Wave propagation in optically active and magnetoelectric media of arbitrary geometry,” Phys. Rev. B 24, 4271–4278 (1981).
[CrossRef]

1973 (1)

V. M. Agranovich and V. L. Ginzburg, “Phenomenological electrodynamics of gyrotropic media,” Sov. Phys. JETP 36, 440–443 (1973).

1937 (1)

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Agranovich, V. M.

V. M. Agranovich and V. L. Ginzburg, “Phenomenological electrodynamics of gyrotropic media,” Sov. Phys. JETP 36, 440–443 (1973).

Birman, J. L.

A. Puri, D. N. Pattanayak, and J. L. Birman, “Surface waves in bounded gyrotropic media,” J. Opt. Soc. Am. 72, 938–942 (1982).
[CrossRef]

D. N. Pattanayak and J. L. Birman, “Wave propagation in optically active and magnetoelectric media of arbitrary geometry,” Phys. Rev. B 24, 4271–4278 (1981).
[CrossRef]

Chen, B.

B. Chen and D. F. Nelson, “Wave-vector-space method for wave propagation in bounded media,” Phys. Rev. B 48, 15365–15371 (1993).
[CrossRef]

B. Chen and D. F. Nelson, “Wave propagation of exciton polaritons by a wave-vector-space method,” Phys. Rev. B 48, 15372–15389 (1993).
[CrossRef]

Condon, E. U.

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Ginzburg, V. L.

V. M. Agranovich and V. L. Ginzburg, “Phenomenological electrodynamics of gyrotropic media,” Sov. Phys. JETP 36, 440–443 (1973).

Ivanov, A. L.

Nelson, D. F.

D. F. Nelson, “Surface nonlocality effects on an optically active waveguide,” J. Appl. Phys. 86, 5348–5355 (1999).
[CrossRef]

D. F. Nelson and A. L. Ivanov, “Alternative explanation of specular optical activity,” Opt. Lett. 23, 86–88 (1998).
[CrossRef]

D. F. Nelson, “Deriving surface polaritons without using boundary conditions,” J. Opt. Soc. Am. B 13, 1956–1960 (1996).
[CrossRef]

D. F. Nelson, “Deriving the transmission and reflection of an optically active medium without using boundary conditions,” Phys. Rev. E 51, 6142–6153 (1995).
[CrossRef]

B. Chen and D. F. Nelson, “Wave propagation of exciton polaritons by a wave-vector-space method,” Phys. Rev. B 48, 15372–15389 (1993).
[CrossRef]

B. Chen and D. F. Nelson, “Wave-vector-space method for wave propagation in bounded media,” Phys. Rev. B 48, 15365–15371 (1993).
[CrossRef]

D. F. Nelson, “Mechanisms and dispersion of crystalline optical activity,” J. Opt. Soc. Am. B 6, 1110–1116 (1989).
[CrossRef]

Pattanayak, D. N.

A. Puri, D. N. Pattanayak, and J. L. Birman, “Surface waves in bounded gyrotropic media,” J. Opt. Soc. Am. 72, 938–942 (1982).
[CrossRef]

D. N. Pattanayak and J. L. Birman, “Wave propagation in optically active and magnetoelectric media of arbitrary geometry,” Phys. Rev. B 24, 4271–4278 (1981).
[CrossRef]

Puri, A.

J. Appl. Phys. (1)

D. F. Nelson, “Surface nonlocality effects on an optically active waveguide,” J. Appl. Phys. 86, 5348–5355 (1999).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

Opt. Lett. (1)

Phys. Rev. B (3)

B. Chen and D. F. Nelson, “Wave-vector-space method for wave propagation in bounded media,” Phys. Rev. B 48, 15365–15371 (1993).
[CrossRef]

B. Chen and D. F. Nelson, “Wave propagation of exciton polaritons by a wave-vector-space method,” Phys. Rev. B 48, 15372–15389 (1993).
[CrossRef]

D. N. Pattanayak and J. L. Birman, “Wave propagation in optically active and magnetoelectric media of arbitrary geometry,” Phys. Rev. B 24, 4271–4278 (1981).
[CrossRef]

Phys. Rev. E (1)

D. F. Nelson, “Deriving the transmission and reflection of an optically active medium without using boundary conditions,” Phys. Rev. E 51, 6142–6153 (1995).
[CrossRef]

Rev. Mod. Phys. (1)

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Sov. Phys. JETP (1)

V. M. Agranovich and V. L. Ginzburg, “Phenomenological electrodynamics of gyrotropic media,” Sov. Phys. JETP 36, 440–443 (1973).

Other (4)

V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons, 2nd ed. (Springer, Berlin, 1984), pp. 96–97.

M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon, Oxford, 1954), pp. 336–338.

See Sect. IV C of Ref. 4.

It was remarked in Ref. 6 that s appears linearly in the dispersion relation. That is true if the expansion in s is done alone. If an expansion in both s and g is done simultaneously, then it is found that terms linear in s are also linear in g, that is, bilinear in small quantities and thus negligible.

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Equations (88)

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×H-Dt=0,×E+Bt=0,
D0E+P-Q,H1μ0B-M,
Pi=rcirηTr,
Mi=12ijkrsqjkrsηTrη˙s,
Qij=12rsqijrsηTrηTs.
2ηrt2=cjr Ej-Ωr2ηr+sNsqkjsr Ek,j+s(Lksr-Lkrs)η,ks+sLk,ksrηs,
×(×E)+1c22Et2
=-μ0tPt-Qt+×M.
F(k, ω)=1(2π)4F(r, t)×exp[-i(kr-ωt)]drdt,
F(r, t)=F(k, ω)×exp[i(kr-ωt)]dkdω.
m(r)=mθ(z),
θ(z)=1z>00z<0.
p(u)(k)=mδ(kx)δ(ky)2πi(kz-i),
Ii(k, ω)
=[(k2-k02)δij-kikj]Ej(k, ω)-k020
×rcirηr(k, ω)-ikjrsqijsrNrηs(k, ω)
×p(u)(k-k)dk=0,
12πiF(k)p(u)(k-k)dk=F(u)(k),
F(k)=F(u)(k)+F(l)(k).
[(k02-k2)δij+kikj][Ej(u)(k)+Ej(l)(k)]
+k020rcir-iklrsNsqlisrηr(u)(k)=0.
isNsqljsrkl Ej(k)p(u)(k-k)dk
=iklsNsqljsrEj(u)(k)-sNsq3jsr Ej(0)(kx, ky)/2π,
E(0)(kx, ky)E(kx, ky, kz)dkz.
(ω2-Ωr2)ηr(u)+crE(u)+iklsNsqljsr Ej(u)
+ikls(Llsr-Llrs)ηs(u)+12πs(L3rsηs(0)-Nsq3jsr Ej(0))=0.
ηs(u)=csE(u)Ωs2-ω2,ηs(0)=csE(0)Ωs2-ω2.
ηr(u)(k)=cjrΩr2-ω2+iklsNsqljsrΩr2-ω2+(Llsr+Llrs)cjs(Ωr2-ω2)(Ωs2-ω2)Ej(u)(k)+12πsL3rscjs(Ωr2-ω2)(Ωs2-ω2)-Nsq3jsrΩr2-ω2Ej(0)(kx, ky).
[k02(κij+iklgijl)-k2δij+kikj]Ej(u)(k)
+[(k02-k2)δij+kikj]Ej(l)(k)
-k02 gij3SEj(0)(k)/2π=0,
κij(ω)δij+10rcircjrΩr2-ω2=κji,
gijl(ω)gijlS-gjilS=-gjil(ω),
gijlS(ω)10rsNsgljsr cirΩr2-ω2-cirLlrscjs(Ωr2-ω2)(Ωs2-ω2).
κij(ω)=κ(ω)δij,gijl(ω)=g(ω)ijl.
[k02(κδij+igijlkl)-k2δij+kikj]Ej(u)(k)
+[(k02-k2)δij+kikj]Ej(l)(k)
-k02 gij3SEj(0)(kx, ky)/2π=0,
K(u)E(u)=Λ(u)E(u),
Kij(u)k02(κδi j+igi jlkl)-k2δi j+kikj
k=kxiˆ+iqzkˆ,
Λ(u)=κk02-k2+gk02 k000κk02-k2-gk02 k000κk02,
Eˆ u=(-iqziˆ+ikjˆ+kxkˆ)/2k,
Eˆ l=(-iqziˆ-ikjˆ+kxkˆ)/2k,
Eˆ L=(kxiˆ+iqzkˆ)/k.
E^α(k)E^β*(k*)=δαβ(α,β=u, l, L).
Λ(l)=k02-k2000k02-k2000k02,
κk02-k2±gk02 k=0,
n±k±/k0=12(±gk0+4κ+g2k02)κ±gk0/2.
qz=+β2-k±2+qM±.
qz=-β2-k02-qV.
E(u)(k)=12πia+(kx, ky)Eˆ+(kM+)kz-iqM++a-(kx, ky)Eˆ-(kM-)kz-iqM-,
E(l)(k)=-12πib+(kx, ky)Eˆ+(kV)kz+iqV+b-(kx, ky)E^-(kV)kz+iqV,
kM±[β, 0, iqM±],kV[β, 0, -iqV].
a+=A+δ(ky)δ(kx-β),a-=A-δ(ky)δ(kx-β),
b+=B+δ(ky)δ(kx-β),b-=B-δ(ky)δ(kx-β).
E(0)(kx, ky)=δ(kx-β)δ(ky)×[A+E^+(k+)+A-E^-(k-)+B+E^+(kV)+B-E^-(kV)]/2.
iqM±(κk02-β2-kz2)±gk02 k±kz
=(kz-iqM±)[(qM±)2-iqM±kz±gk02k±],
±k±(kz2+β2-κk02)-k02 g(β2+iqM±kz)
=(kz-iqM±)[±k±(kz+iqM±)-iqM±k02 g],
kz2+β2-κk02k02k±g
=(kz-iqM±)(kz+iqM±),
Sji2k02gjk3SEk(0),
0=qM+ A+k++qM- A-k-+qV(B++B-)k0,
0=k- A++k+ A--k0(B++B-)-S1,
0=A+-A--(B+-B-),
0=gM+k- A+k+-gM-k+ A-k--gV(B+-B-)-S2,
g123S=s+g/2,g213S=s-g/2,
S1=-k022s+g2(A+-A-+B+-B-)=-k02(s+g/2)(A+-A-),
S2=k022s-g2×gM+ A+k++qM- A-k--qVB+k0-qVB-k0=k02s-g2qM+ A+k++qM- A-k-,
0=qM+ A+n++qM- A-n-+qV(B++B-),
0=npA++nmA--(B++B-),
0=A+-A--(B+-B-),
0=qM+np A+n+-qM-nm A-n--qV(B+-B-),
npκ+k0s,nmκ-k0s.
(np+nm)(qM+qM--qV2n+n-)
+(npnm-1)qV(qM+n-+qM-n+)=0,
A-=-n-(qM++qVnpn+)A+/D,
B+=-[qM+n-(nm-1)-qM-n+(np+1)
-qVn+n-(np+nm)]A+/2D,
B-=-[qM+n-(nm+1)-qM-n+(np-1)+qVn+n-(np+nm)]A+/2D,
Dn+(qM-+qVnmn-).
E(r, t)={θ(z)[A+E^+(k+)exp(-qM+z)+A-E^-(k-)×exp(-qM-z)+θ(-z)[B+E^+(kV)+B-E^-(kV)]exp(qVz)}exp[i(βx-ωt)],
β=k0κ(ω)κ(ω)+11/2,
qV=k0-κ(ω)-1,qM±=qM±iα,
qMk0|κ(ω)|-κ(ω)-1,αgk022-κ(ω)-1-κ(ω).
E(x, z, t)=ELexp[i(βx-ωt)]×{θ(z)exp(-qMz)[iˆcos αz+jˆ(k0/β)×sin αz+ikˆ(-κ)-1/2cos αz]+θ(-z)exp(qVz)[iˆ-ikˆ(-κ)1/2]}.

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