Abstract

In order to derive full-wave solutions for electromagnetic wave scattering from rough interfaces between achiral media (free space for instance) and chiral media that satisfy generalized constitutive relations, it is necessary to employ complete modal expansions for the electromagnetic fields above and below the interface. To this end, the familiar Fourier transforms of the fields are expressed as generalized field transforms consisting of the radiation term, the lateral waves, and the surface waves. Maxwell’s equations are converted into generalized telegraphists’ equations [in the companion paper (this issue), J. Opt. Soc. Am. A 30, 335 (2013)] upon the imposition of exact boundary conditions. These telegraphists’ equations are coupled first-order differential equations for the forward- and backward-traveling wave amplitudes associated with all the different species of waves (radiation, lateral, and surface waves) excited at the surface of the chiral medium. The analysis presented here includes the completeness and orthogonal relations of the basis functions associated with the modal expansions. This work is used to distinguish between depolarization due to the chiral properties of the medium and depolarization due to surface irregularities. It has applications in remote sensing and identification of biological and chemical materials based on their optical activity.

© 2013 Optical Society of America

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References

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  1. P. E. Crittenden, “Electromagnetic sensing of chiral materials,” Ph.D. thesis (University of Nebraska Lincoln, 2002).
  2. P. E. Crittenden and E. Bahar, “A modal solution for reflection and transmission at a chiral-chiral interface,” Can. J. Phys. 83, 1265–1288 (2005).
    [CrossRef]
  3. S. A. Schelkunoff, “Generalized telegraphists’ equations for waveguides,” Bell Syst. Tech. J. 31, 784–801 (1952).
  4. S. A. Schelkunoff, “Conversions of Maxwell’s equations into generalized telegraphists’ equations,” Bell Syst. Tech. J. 34, 995–1045 (1955).
  5. C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
    [CrossRef]
  6. C. F. Bohren, “Scattering of electromagnetic waves by an optically active spherical shell,” J. Chem. Phys. 62, 1566–1571 (1975).
    [CrossRef]
  7. A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics Series 335 (Springer, 1989).
  8. M. Silverman, “Reflection and refraction at the surfaces of achiral medium: comparison of gyrotropic consituative relations, invariant or non-invariant under duality transformations,” J. Opt. Soc. Am. A 3, 830–837 (1986).
    [CrossRef]
  9. E. Bahar, “Reflection and transmission at a free-space-chiral interface based on the invariant constitutive relations and for gyrotropic media and the Drude–Born–Fedorov constitutive relations,” J. Opt. Soc. Am. A 26, 1834–1838 (2009).
    [CrossRef]
  10. E. Bahar, “Mueller matrices for waves reflected and transmitted through chiral materials, waveguide modal solutions and applications,” J. Opt. Soc. Am. B 24, 1610–1619 (2007).
    [CrossRef]
  11. M. Silverman, Waves and Grains (Princeton University, 1998).
  12. M. Silverman and T. Black, “Experimental method to detect chiral asymmetry in specular light scattering from a naturally optically active medium,” Phys. Lett. A 126, 171–176 (1987).
    [CrossRef]
  13. M. Silverman, N. Ritchie, G. Cushman, and B. Fisher, “Experimental configurations using optical phase modulation to measure chiral asymmetries in light specularly reflected from a naturally gyrotropic medium,” J. Opt. Soc. Am. A 5, 1852–1862 (1988).
    [CrossRef]
  14. E. Bahar, “Generalized Fourier transform for stratified media,” Can. J. Phys. 50, 3123–3131 (1972).
  15. M. L. Boas, Mathematical Methods in Physical Sciences, 3rd ed. (Wiley, 2006).
  16. L. M. Brekhovskikh, Waves in Layered Media (Academic, 1960), 234–302.
  17. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).
  18. J. R. Wait, Electromagnetic Waves in Stratified Media(Pergamon, 1962).
  19. E. Bahar, “Cross polarization of lateral waves propagating along a free-space-chiral planar interface: application to identification of optically active materials,” J. Opt. Soc. Am. B 28, 1194–1199 (2011).
    [CrossRef]
  20. E. Bahar, “Guided surface waves over a free-space-chiral interface: applications to identification of optically active materials,” J. Opt. Soc. Am. B 28, 868–872 (2011).
    [CrossRef]
  21. E. Bahar, “Comparison of polarimetric techniques for the identification of biological and chemical materials using Mueller matrices, lateral waves and surface waves,” J. Opt. Soc. Am. 28, 2139–2147 (2011).
    [CrossRef]
  22. E. Bahar and P. E. Crittenden, “Electromagnetic wave scattering from a rough interface above a chiral medium: generalized telegraphists’ transforms,” J. Opt. Soc. Am. A30, 335–341 (2012).

2011

E. Bahar, “Guided surface waves over a free-space-chiral interface: applications to identification of optically active materials,” J. Opt. Soc. Am. B 28, 868–872 (2011).
[CrossRef]

E. Bahar, “Comparison of polarimetric techniques for the identification of biological and chemical materials using Mueller matrices, lateral waves and surface waves,” J. Opt. Soc. Am. 28, 2139–2147 (2011).
[CrossRef]

E. Bahar, “Cross polarization of lateral waves propagating along a free-space-chiral planar interface: application to identification of optically active materials,” J. Opt. Soc. Am. B 28, 1194–1199 (2011).
[CrossRef]

2009

2007

2005

P. E. Crittenden and E. Bahar, “A modal solution for reflection and transmission at a chiral-chiral interface,” Can. J. Phys. 83, 1265–1288 (2005).
[CrossRef]

1988

1987

M. Silverman and T. Black, “Experimental method to detect chiral asymmetry in specular light scattering from a naturally optically active medium,” Phys. Lett. A 126, 171–176 (1987).
[CrossRef]

1986

1975

C. F. Bohren, “Scattering of electromagnetic waves by an optically active spherical shell,” J. Chem. Phys. 62, 1566–1571 (1975).
[CrossRef]

1974

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
[CrossRef]

1972

E. Bahar, “Generalized Fourier transform for stratified media,” Can. J. Phys. 50, 3123–3131 (1972).

1955

S. A. Schelkunoff, “Conversions of Maxwell’s equations into generalized telegraphists’ equations,” Bell Syst. Tech. J. 34, 995–1045 (1955).

1952

S. A. Schelkunoff, “Generalized telegraphists’ equations for waveguides,” Bell Syst. Tech. J. 31, 784–801 (1952).

Bahar, E.

E. Bahar, “Guided surface waves over a free-space-chiral interface: applications to identification of optically active materials,” J. Opt. Soc. Am. B 28, 868–872 (2011).
[CrossRef]

E. Bahar, “Comparison of polarimetric techniques for the identification of biological and chemical materials using Mueller matrices, lateral waves and surface waves,” J. Opt. Soc. Am. 28, 2139–2147 (2011).
[CrossRef]

E. Bahar, “Cross polarization of lateral waves propagating along a free-space-chiral planar interface: application to identification of optically active materials,” J. Opt. Soc. Am. B 28, 1194–1199 (2011).
[CrossRef]

E. Bahar, “Reflection and transmission at a free-space-chiral interface based on the invariant constitutive relations and for gyrotropic media and the Drude–Born–Fedorov constitutive relations,” J. Opt. Soc. Am. A 26, 1834–1838 (2009).
[CrossRef]

E. Bahar, “Mueller matrices for waves reflected and transmitted through chiral materials, waveguide modal solutions and applications,” J. Opt. Soc. Am. B 24, 1610–1619 (2007).
[CrossRef]

P. E. Crittenden and E. Bahar, “A modal solution for reflection and transmission at a chiral-chiral interface,” Can. J. Phys. 83, 1265–1288 (2005).
[CrossRef]

E. Bahar, “Generalized Fourier transform for stratified media,” Can. J. Phys. 50, 3123–3131 (1972).

E. Bahar and P. E. Crittenden, “Electromagnetic wave scattering from a rough interface above a chiral medium: generalized telegraphists’ transforms,” J. Opt. Soc. Am. A30, 335–341 (2012).

Black, T.

M. Silverman and T. Black, “Experimental method to detect chiral asymmetry in specular light scattering from a naturally optically active medium,” Phys. Lett. A 126, 171–176 (1987).
[CrossRef]

Boas, M. L.

M. L. Boas, Mathematical Methods in Physical Sciences, 3rd ed. (Wiley, 2006).

Bohren, C. F.

C. F. Bohren, “Scattering of electromagnetic waves by an optically active spherical shell,” J. Chem. Phys. 62, 1566–1571 (1975).
[CrossRef]

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
[CrossRef]

Brekhovskikh, L. M.

L. M. Brekhovskikh, Waves in Layered Media (Academic, 1960), 234–302.

Crittenden, P. E.

P. E. Crittenden and E. Bahar, “A modal solution for reflection and transmission at a chiral-chiral interface,” Can. J. Phys. 83, 1265–1288 (2005).
[CrossRef]

P. E. Crittenden, “Electromagnetic sensing of chiral materials,” Ph.D. thesis (University of Nebraska Lincoln, 2002).

E. Bahar and P. E. Crittenden, “Electromagnetic wave scattering from a rough interface above a chiral medium: generalized telegraphists’ transforms,” J. Opt. Soc. Am. A30, 335–341 (2012).

Cushman, G.

Fisher, B.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

Lakhtakia, A.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics Series 335 (Springer, 1989).

Ritchie, N.

Schelkunoff, S. A.

S. A. Schelkunoff, “Conversions of Maxwell’s equations into generalized telegraphists’ equations,” Bell Syst. Tech. J. 34, 995–1045 (1955).

S. A. Schelkunoff, “Generalized telegraphists’ equations for waveguides,” Bell Syst. Tech. J. 31, 784–801 (1952).

Silverman, M.

Varadan, V. K.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics Series 335 (Springer, 1989).

Varadan, V. V.

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics Series 335 (Springer, 1989).

Wait, J. R.

J. R. Wait, Electromagnetic Waves in Stratified Media(Pergamon, 1962).

Bell Syst. Tech. J.

S. A. Schelkunoff, “Generalized telegraphists’ equations for waveguides,” Bell Syst. Tech. J. 31, 784–801 (1952).

S. A. Schelkunoff, “Conversions of Maxwell’s equations into generalized telegraphists’ equations,” Bell Syst. Tech. J. 34, 995–1045 (1955).

Can. J. Phys.

E. Bahar, “Generalized Fourier transform for stratified media,” Can. J. Phys. 50, 3123–3131 (1972).

P. E. Crittenden and E. Bahar, “A modal solution for reflection and transmission at a chiral-chiral interface,” Can. J. Phys. 83, 1265–1288 (2005).
[CrossRef]

Chem. Phys. Lett.

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
[CrossRef]

J. Chem. Phys.

C. F. Bohren, “Scattering of electromagnetic waves by an optically active spherical shell,” J. Chem. Phys. 62, 1566–1571 (1975).
[CrossRef]

J. Opt. Soc. Am.

E. Bahar, “Comparison of polarimetric techniques for the identification of biological and chemical materials using Mueller matrices, lateral waves and surface waves,” J. Opt. Soc. Am. 28, 2139–2147 (2011).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Phys. Lett. A

M. Silverman and T. Black, “Experimental method to detect chiral asymmetry in specular light scattering from a naturally optically active medium,” Phys. Lett. A 126, 171–176 (1987).
[CrossRef]

Other

P. E. Crittenden, “Electromagnetic sensing of chiral materials,” Ph.D. thesis (University of Nebraska Lincoln, 2002).

E. Bahar and P. E. Crittenden, “Electromagnetic wave scattering from a rough interface above a chiral medium: generalized telegraphists’ transforms,” J. Opt. Soc. Am. A30, 335–341 (2012).

A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time Harmonic Electromagnetic Fields in Chiral Media, Lecture Notes in Physics Series 335 (Springer, 1989).

M. L. Boas, Mathematical Methods in Physical Sciences, 3rd ed. (Wiley, 2006).

L. M. Brekhovskikh, Waves in Layered Media (Academic, 1960), 234–302.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

J. R. Wait, Electromagnetic Waves in Stratified Media(Pergamon, 1962).

M. Silverman, Waves and Grains (Princeton University, 1998).

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Figures (2)

Fig. 1.
Fig. 1.

Plane stratified chiral media.

Fig. 2.
Fig. 2.

Paths of integration in the complex plane.

Equations (100)

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D=ϵ(E+βxE)
B=μ(H+βxH).
J=Jδ(xx)δ(yy)az=Jzaz
K=Kδ(xx)δ(yy)az=Kzaz.
F(r,t)=Re[f(r,ω)exp(jωt)].
F(r,t)=f^(r,ω)exp(jωt)dω.
f^(r,ω)=12πF(r,t)exp(jωt)dt.
J=JζK.
H=Q1+Q2
E=jη(Q1Q2).
2Q1z+γ12Q1z=jζγ12Kz/k
2Q2z+γ22Q2z=0.
γ1=k/(1kβ)
γ2=k/(1+kβ).
J=jζK.
2Q1z+γ22Q1z=0
2Q2z+γ12Q2z=jζγ22Kz/k.
D=ϵEjωgH
B=μH+jωgE.
Qiz(q,y)=Q^iz(x,y)exp(jqx)dx=Q^iz(q,y)exp(jqx),i=1,2,
Q^iz(x,y)=12πQiz(q,y)exp(jqx)dq=12πQ^iz(q,y)exp[jq(xx)]dq.
H^z(q,y)=Kζ0γ1022k0u10{eju10|yy|+R0RReju10(y+y)+R0LRej(u20y+u10y),y>0T0RRej(u11yu10y)+T0LRej(u21yu10y),y<0
E^z(q,y)=jKγ1022k0u10{eju10|yy|+R0RReju10(y+y)R0LRej(u20y+u10y),y>0ζr(T0RRej(u11yu10y)T0LRej(u21yu10y)),y<0.
E^x(q,y)=jωϵH^zyβE^zy
H^x(q,y)=jωμE^zyβH^zy.
H^z(q,y)=Kζ0γ1022k0u20{eju20|yy|+R0LLeju20(y+y)+R0RLej(u10y+u20y),y>0T0LLej(u21yu20y)+T0RLej(u11yu20y),y<0
E^z(q,y)=jKγ2022k0u20{eju20|yy|+R0LLeju20(y+y)R0RLej(u10y+u20y),y>0ζr(T0LLej(u21yu20y)T0RLej(u11yu20y)),y<0.
ErC=[ErRErL]=[R0RRR0RLR0LRR0LL][EiREiL]=RCEiC
EtC=[ErRErL]=[T0RRT0RLT0LRT0LL][EiREiL]=TCEiC.
EC=[EREL]=[1j1j][EVEH]AEL
EL=A1EC=12[11jj]EC.
RL=[RVVRVHRHVRHH]=A1RCA
TL=[TVVTVHTHVTHH]=A1TCA.
RC=RCO+k1β1RCOTC=TCO+k1β1RCO.
R10CO=[R10RR00R10LL]=12T01HHT10VVtan2θ1[1001].
T01VVT10HH=T01HHT10VV=4cosθ0cosθ1(Y0cosθ0+Y1cosθ1)(Z0cosθ0+Z1cosθ1).
Y0=ζ0=(ϵ0/μ0)1/2=1/Z0,Y1=ζ1=(ϵ1/μ1)1/2=1/Z1.
RLO=[RVV00RHH].
R10VV=Z0cosθ0Z1cosθ1Z0cosθ0+Z1cosθ1R10HH=Y0cosθ0Y1cosθ1Y0cosθ0+Y1cosθ1.
k0sinθ0=k1sinθ1.
k0sinθ0=γ1Rsinθ1R=γ1Lsinθ1L.
k1β1A1RCOA=jk1β1[0RLLRRR0]=[0RVHRHV0].
TLO=[1+RVV00Y1Y0(1+RHH)].
1+R10VV=2Z0cosθ0/(Z0cosθ0+Z1cosθ1)
1+R10HH=2Y0cosθ0/(Y0cosθ0+Y1cosθ1).
k1β1TL=jk1β1RLL[01Y1Y00]=[0TVHTHV0].
uij=(γiq2q2)1/2i=1,2,j=0,1
q=γij,
Hz(x,y)=12πH^z(q,y)exp[jq(xx)]dq=12π[u10+u20+u11+u21]H^z(q,y)exp[jq(xx)]dqjRes[H^z(q1p,y)exp[jq1p(xx)]]jRes[H^z(q2p,y)exp[jq2p(xx)]].
H10(x,y)=K20ψ10Ψ10R+exp[jq(xx)]du10.
ψ10(u,y)={1R0RR[exp(ju10y)+R0RRexp(ju10y)+R0LRexp(ju20y)],y>01R0RR[T0RRexp(ju11y)+T0LRexp(ju21y)],y<0
Ψ10R+(u,y)=[exp(ju10y)+R0RRexp(ju10y)]Y10(u),y>0,
Y10(u)=ζ0γ102/2πk0q.
k0[(xx)2+(yy)2]1/2=k0ρd1.
HdR(x,y)=Kζ0γ1024k0πexp[ju10(yy)]exp[jq(xx)]du10q.
HdR(x,y)Kζ0γ102exp(jπ4)exp(jγ10ρd)/2k02πγ10ρd.
HdR(x,y)=Kζ0k0H02(k0ζd)/4.
H20(x,y)=K20ψ20Ψ20R+exp[jq(xx)]du20,x>x,y>0.
ψ20(u,y)={1R0LL[exp(ju20y)+R0LLexp(ju20y)+R0RLexp(ju10y)],y>01R0LL[T0LLexp(ju21y)+T0RLexp(ju11y)],y<0
Ψ20R+(u,y)=γ10γ20R0RLexp(ju10y)Y20(u),y>0.
Y20(u)=ζ0γ202/2πk0q.
H11(x,y)=K20ψ11Ψ11R+exp[jq(xx)]du11.
ψ11(u,y)={1R1RR[T1RRexp(ju10y)+T1LRexp(ju20y)],y>01R1RR[exp(ju11y)+R1RRexp(ju11y)+R1LRexp(ju21y)],y<0
Ψ11R+(u,y)=k1k0γ10γ11T1RRexp(ju10y)Y11(u),y>0.
Y11(u)=ζ1γ112/2πk1q.
H21(x,y)=K20ψ21Ψ21R+exp[jq(xx)]du21,x>x,y>0.
ψ21(u,y)={1R1LL[T1LLexp(ju20y)+T1RLexp(ju10y)],y>01R1LL[exp(ju21y)+R1LLexp(ju21y)+R1RLexp(ju11y)],y<0
Ψ21R+(u,y)=k1k0γ10γ21T1RLexp(ju10y)Y21(u),y>0.
Y21(u)=ζ1γ212/2πk1q.
Hij(x,y)=K20ψijΨijR+exp[jq|xx|]duij,y>0.
Hij(x,y)=K20ψijΨijL+exp[q|xx|]duij.
q1p=qV=k1[ζr21]1/2/[1ϵr2]1/2
q2p=qH=k1[ηr21]1/2/[1μr2]1/2.
jRes[H^(qip,y)]=K2ψipΨip,i=1,2.
ψip(qip,y)={Res(R0RR/u10)exp(ju10py)+Res(R0LR/u10)exp(ju20py),y>0Res(T0RR/u10)exp(ju11py)+Res(T0LR/u10)exp(ju21py),y<0
Ψip(qip,y)=exp(ju10py)Yip,y>0.
Yip=ζ0γ10/k0.
Res(R0L/u0)=(Res(R0VV/u0)Res(R0VH/u0)Res(R0HV/u0)Res(R0HH/u0)).
Res(R0VV/u0)=2u0VqV[1(ϵoϵ1)2]
Res(R0HH/u0)=2u0HqH[1(μoμ1)2].
Res(R0VH/u0)=Res(R0HVu0)=jk1β1(1ηr2)[qV/u0Vϵr21+qH/u0H1μr2].
Res(T0L/u0)=(Res(R0VV/u0)Res(R0VH/u0)Res(R0HV/u0)Y1Y0Res(R0HH/u0)Y1Y0).
u10=u20=u0=(k02q2)1/2,u0V=[k02(qV)2]1/2,u0H=[k02(qH)2]1/2.
k0[(xx)2+(yy)2]1/2k0ζd1.
(2x2+q2)exp[jq|xx|]=2jqδ(xx),
ijΨijR(u,y)YR(u,y)ψijR(u,y)=δ(yy),i=1,2j=0,1,p.
YR(u,y)={Y10y>0Y11y<0.
ijΨijR(u,y)YR(u,y)ψijL(u,y)=0.
ΨijP(u,y)ψiJQ(u,y)=δ(yy)δPQ.
Hz(x,y)=Q1z(x,y)+Q2z(x,y)=P=R,LijH^zij(x,u)ψijP(u,y),
H^zij(x,u)=P=R,LΨijP(u,y)QPz(x,y)YP(u,y)dy.
H^zij(x,u)=P=R,LΨijP(u,y)YP(u,y)mnH^zmn(x,u)ψmnP(u,y)dy=mnH^zmn(x,u)P=R,LΨijP(u,y)ψmnP(u,y)YP(u,y)dy.
P=R,LΨijP(u,y)ψmnP(u,y)YP(u,y)dy=δimδjnΔ(u,u).
Δ(u,u)={δ(uu)uupδu,upu=up,
Ez(x,y)=j[ijH^zij(x,u)ηψijRijH^zij(x,u)ηψijL(u,y)].
Ey(x,y)=ij(E^yij(x,u)ψijR(u,y)qηγ1+E^yij(x,u)ψijL(u,y)qηγ2)
E^yij(x,u)=jψijRγ1qQ1y(x,y)YR(u,y)dy+jψijL(u,y)γ2qQ2y(x,y)YL(u,y)dy.
Q1y(x,y)=jijE^yij(x,u)ψijR(u,y)qγ1
Q2y(x,y)=jijE^yij(x,u)ψijL(u,y)qγ2.
Hy(x,y)=jijE^yij(x,u)ψijR(u,y)qγ1jijE^yij(x,y)ψijLqγ2.

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