Abstract

We study nonspecular phenomena experienced by two-dimensional ordinary and extraordinary beams propagating in a uniaxial crystal and reflected at a flat uniaxial–isotropic interface. We extend to uniaxial crystals the aberrationless approach used for beams in isotropic media. Analytical expressions for the geometrical nonspecular effects (lateral shift, focal shift, angular shift, and beam waist modification) are given and compared with those predicted by the stationary phase approximation. The theory is applied to the reflection of beams at a flat interface between TiO2 and a vacuum.

© 1997 Optical Society of America

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  1. F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflektion,” Ann. Phys. (Leipzig) 1, 333–346 (1947).
    [CrossRef]
  2. J. Greffet, C. Baylard, “Nonspecular reflection from a lossy dielectric,” Opt. Lett. 18, 1129–1131 (1993).
    [CrossRef] [PubMed]
  3. E. E. Kriezis, P. K. Pandelakis, A. G. Papagiannakis, “Diffraction of a Gaussian beam from a periodic planar screen,” J. Opt. Soc. Am. A 11, 630–636 (1994).
    [CrossRef]
  4. W. Nasalski, “Modified reflectance and geometrical deformation of Gaussian beams reflected at a dielectric interface,” J. Opt. Soc. Am. A 6, 1447–1454 (1989).
    [CrossRef]
  5. W. Nasalski, “Longitudinal and transverse effects of nonspecular reflection,” J. Opt. Soc. Am. A 13, 172–181 (1996).
    [CrossRef]
  6. R. A. Depine, N. E. Bonomo, “Spatial modifications of Gaussian beams reflected at isotropic–uniaxial interfaces,” J. Mod. Opt. 42, 2401–2412 (1995).
    [CrossRef]
  7. R. A. Depine, N. E. Bonomo, “Goos–Hänchen lateral shift for Gaussian beams reflected at achiral–chiral interfaces,” Optik (Stuttgart) 103, 37–41 (1996).
  8. R. Riesz, R. Simon, “Reflection of a Gaussian beam from a dielectric slab,” J. Opt. Soc. Am. A 2, 1809–1817 (1985).
    [CrossRef]
  9. T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. Am. A 3, 558–565 (1986).
    [CrossRef]
  10. S. Zhang, T. Tamir, “Spatial modifications of Gaussian beams diffracted by reflection gratings,” J. Opt. Soc. Am. A 6, 1368–1381 (1989).
    [CrossRef]
  11. J. Greffet, C. Baylard, “Nonspecular astigmatic reflection of a 3D Gaussian beam on an interface,” Opt. Commun. 93, 271–276 (1992).
    [CrossRef]
  12. Song Peng, G. Michael Morris, “Resonant scattering from two-dimensional gratings,” J. Opt. Soc. Am. A 13, 993–1005 (1996).
    [CrossRef]
  13. K. Artmann, “Berechnung der Sietenversetzung des total reflektierten Strahles,” Ann. Phys. (Leipzig) 2, 87–102 (1948).
    [CrossRef]
  14. N. Bonomo, M. Gigli, R. Depine, “Lateral displacement of a beam incident from a uniaxial medium onto a metal,” J. Mod. Opt. 44, 1393–1408 (1997).
    [CrossRef]
  15. H. C. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach (McGraw-Hill, New York, 1983).
  16. R. A. Depine, M. L. Gigli, “Total reflection, total transmission and Brewster angle conditions for planar boundaries between isotropic and uniaxial media with different magnetic permeabilities,” Optik (Stuttgart) 97, 135–141 (1994).
  17. I. Gradshteyn, I. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1979), p. 307.

1997 (1)

N. Bonomo, M. Gigli, R. Depine, “Lateral displacement of a beam incident from a uniaxial medium onto a metal,” J. Mod. Opt. 44, 1393–1408 (1997).
[CrossRef]

1996 (3)

1995 (1)

R. A. Depine, N. E. Bonomo, “Spatial modifications of Gaussian beams reflected at isotropic–uniaxial interfaces,” J. Mod. Opt. 42, 2401–2412 (1995).
[CrossRef]

1994 (2)

E. E. Kriezis, P. K. Pandelakis, A. G. Papagiannakis, “Diffraction of a Gaussian beam from a periodic planar screen,” J. Opt. Soc. Am. A 11, 630–636 (1994).
[CrossRef]

R. A. Depine, M. L. Gigli, “Total reflection, total transmission and Brewster angle conditions for planar boundaries between isotropic and uniaxial media with different magnetic permeabilities,” Optik (Stuttgart) 97, 135–141 (1994).

1993 (1)

1992 (1)

J. Greffet, C. Baylard, “Nonspecular astigmatic reflection of a 3D Gaussian beam on an interface,” Opt. Commun. 93, 271–276 (1992).
[CrossRef]

1989 (2)

1986 (1)

1985 (1)

1948 (1)

K. Artmann, “Berechnung der Sietenversetzung des total reflektierten Strahles,” Ann. Phys. (Leipzig) 2, 87–102 (1948).
[CrossRef]

1947 (1)

F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflektion,” Ann. Phys. (Leipzig) 1, 333–346 (1947).
[CrossRef]

Artmann, K.

K. Artmann, “Berechnung der Sietenversetzung des total reflektierten Strahles,” Ann. Phys. (Leipzig) 2, 87–102 (1948).
[CrossRef]

Baylard, C.

J. Greffet, C. Baylard, “Nonspecular reflection from a lossy dielectric,” Opt. Lett. 18, 1129–1131 (1993).
[CrossRef] [PubMed]

J. Greffet, C. Baylard, “Nonspecular astigmatic reflection of a 3D Gaussian beam on an interface,” Opt. Commun. 93, 271–276 (1992).
[CrossRef]

Bonomo, N.

N. Bonomo, M. Gigli, R. Depine, “Lateral displacement of a beam incident from a uniaxial medium onto a metal,” J. Mod. Opt. 44, 1393–1408 (1997).
[CrossRef]

Bonomo, N. E.

R. A. Depine, N. E. Bonomo, “Goos–Hänchen lateral shift for Gaussian beams reflected at achiral–chiral interfaces,” Optik (Stuttgart) 103, 37–41 (1996).

R. A. Depine, N. E. Bonomo, “Spatial modifications of Gaussian beams reflected at isotropic–uniaxial interfaces,” J. Mod. Opt. 42, 2401–2412 (1995).
[CrossRef]

Chen, H. C.

H. C. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach (McGraw-Hill, New York, 1983).

Depine, R.

N. Bonomo, M. Gigli, R. Depine, “Lateral displacement of a beam incident from a uniaxial medium onto a metal,” J. Mod. Opt. 44, 1393–1408 (1997).
[CrossRef]

Depine, R. A.

R. A. Depine, N. E. Bonomo, “Goos–Hänchen lateral shift for Gaussian beams reflected at achiral–chiral interfaces,” Optik (Stuttgart) 103, 37–41 (1996).

R. A. Depine, N. E. Bonomo, “Spatial modifications of Gaussian beams reflected at isotropic–uniaxial interfaces,” J. Mod. Opt. 42, 2401–2412 (1995).
[CrossRef]

R. A. Depine, M. L. Gigli, “Total reflection, total transmission and Brewster angle conditions for planar boundaries between isotropic and uniaxial media with different magnetic permeabilities,” Optik (Stuttgart) 97, 135–141 (1994).

Gigli, M.

N. Bonomo, M. Gigli, R. Depine, “Lateral displacement of a beam incident from a uniaxial medium onto a metal,” J. Mod. Opt. 44, 1393–1408 (1997).
[CrossRef]

Gigli, M. L.

R. A. Depine, M. L. Gigli, “Total reflection, total transmission and Brewster angle conditions for planar boundaries between isotropic and uniaxial media with different magnetic permeabilities,” Optik (Stuttgart) 97, 135–141 (1994).

Goos, F.

F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflektion,” Ann. Phys. (Leipzig) 1, 333–346 (1947).
[CrossRef]

Gradshteyn, I.

I. Gradshteyn, I. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1979), p. 307.

Greffet, J.

J. Greffet, C. Baylard, “Nonspecular reflection from a lossy dielectric,” Opt. Lett. 18, 1129–1131 (1993).
[CrossRef] [PubMed]

J. Greffet, C. Baylard, “Nonspecular astigmatic reflection of a 3D Gaussian beam on an interface,” Opt. Commun. 93, 271–276 (1992).
[CrossRef]

Hänchen, H.

F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflektion,” Ann. Phys. (Leipzig) 1, 333–346 (1947).
[CrossRef]

Kriezis, E. E.

Morris, G. Michael

Nasalski, W.

Pandelakis, P. K.

Papagiannakis, A. G.

Peng, Song

Riesz, R.

Ryzhik, I.

I. Gradshteyn, I. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1979), p. 307.

Simon, R.

Tamir, T.

Zhang, S.

Ann. Phys. (Leipzig) (2)

F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflektion,” Ann. Phys. (Leipzig) 1, 333–346 (1947).
[CrossRef]

K. Artmann, “Berechnung der Sietenversetzung des total reflektierten Strahles,” Ann. Phys. (Leipzig) 2, 87–102 (1948).
[CrossRef]

J. Mod. Opt. (2)

N. Bonomo, M. Gigli, R. Depine, “Lateral displacement of a beam incident from a uniaxial medium onto a metal,” J. Mod. Opt. 44, 1393–1408 (1997).
[CrossRef]

R. A. Depine, N. E. Bonomo, “Spatial modifications of Gaussian beams reflected at isotropic–uniaxial interfaces,” J. Mod. Opt. 42, 2401–2412 (1995).
[CrossRef]

J. Opt. Soc. Am. A (7)

Opt. Commun. (1)

J. Greffet, C. Baylard, “Nonspecular astigmatic reflection of a 3D Gaussian beam on an interface,” Opt. Commun. 93, 271–276 (1992).
[CrossRef]

Opt. Lett. (1)

Optik (Stuttgart) (2)

R. A. Depine, M. L. Gigli, “Total reflection, total transmission and Brewster angle conditions for planar boundaries between isotropic and uniaxial media with different magnetic permeabilities,” Optik (Stuttgart) 97, 135–141 (1994).

R. A. Depine, N. E. Bonomo, “Goos–Hänchen lateral shift for Gaussian beams reflected at achiral–chiral interfaces,” Optik (Stuttgart) 103, 37–41 (1996).

Other (2)

H. C. Chen, Theory of Electromagnetic Waves: A Coordinate-Free Approach (McGraw-Hill, New York, 1983).

I. Gradshteyn, I. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1979), p. 307.

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Figures (6)

Fig. 1
Fig. 1

Reflection matrix elements for a TiO2–vacuum interface versus α/k0. The parameters are λ0=0.59 µm, =6.84, =8.43, =1, μc=μ=1, and (cx, cy, cz)=(0.88, 0.34, 0.32). (a) |Roo| (dashed curve) and |Reo| (solid curve), (b) |Ree| (dashed curve) and |Roe| (solid curve), (c) phases of Roo (dashed curve) and Reo (solid curve), (d) phases of Ree (dashed curve) and Roe (solid curve).

Fig. 2
Fig. 2

Lateral shifts versus angle of incidence for the ordinary and extraordinary reflected beams when the incident beam is (a) ordinary or (b) extraordinary. Dashed curves: co-polarized beam; solid curves: cross-polarized beam. The incident beams have waists δo(1)+=δe(1)+=59 µm. The interface parameters are as in Fig. 1.

Fig. 3
Fig. 3

Focal shifts versus angle of incidence for the ordinary and extraordinary reflected beams when the incident beam is (a) ordinary or (b) extraordinary. Dashed curves: co-polarized beam; solid curves: cross-polarized beam. The incident beams have waists δo(1)+=δe(1)+=59 µm, and the interface parameters are as in Fig. 1.

Fig. 4
Fig. 4

Angular distance [βoez--βo(1)-] between the direction of the z component of the reflected oe beam and the direction corresponding to a reflected plane wave with propagation constant α0, as a function of βe(1)+. The interface parameters are as in Fig. 1.

Fig. 5
Fig. 5

Field intensity at the maximum of the z distribution on the interface as a function of βe(1)+.

Fig. 6
Fig. 6

[δo(1)--δoez-] for the z component in the reflected oe beam as a function of βe(1)+.

Tables (1)

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Table 1 Definitions for Expression (14)

Equations (54)

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ko+=αxˆ+γo+yˆ,ke+=αxˆ+γe+yˆ.
Ea+(r)=-+Ca+(α)exp[ika+(α)·r]eˆa+(α)dα,
a=o, e,r=(x, y),
Eb-(r)=-+Cb-(α)exp[ikb-(α)·r]eˆb-(α)dα,
b=o, e,
Co-Ce-=RooRoeReoReeCo+Ce+.
kb-=αxˆ+γb-yˆ,
γ0±=(k02μc-α2)1/2,
γe±=(-)cxcyαΓ1/2+(-)cy2,
Γ=k02μc(-)cy2+k02μc2+α2[(cz2-1)-cz2],
eo±=(eo±·eo±*)1/2eˆo±=ko±×cˆ,
ee±=(ee±·ee±*)1/2eˆe±=k02μccˆ-(ke±·cˆ)ke±.
Ca+(α)=exp-(α-α0)2ωa2,a=o, e.
Ea+(x, y)=-+ exp-(α-α0)2ωa2 × exp{i[αx+γa+(α)y]}eˆa+(α)dα,
Eb-(x, y)=Ebo-(x, y)+Ebe-(x, y)=aEba-(x, y),
Eba-(x, y)=-+Rba(α)exp-(α-α0)2ωa2
×exp{i[αx+γb-(α)y]}eˆb-(α)dα.
En(x, y)=-+Gn(α)exp-(α-α0)2ωn2 × exp{i[αx+γn(α)y]}eˆn(α)dα,
n=1, , 6,
Enj(x, y)=-+ expln[rnj(α)]-(α-α0)2ωn2+i[αx+γn(α)y]dα,j=x, y, z,
rnj(α)=Gn(α)enj(α)=exp{ln[rnj(α)]}.
Enj(x, y)=rnj(α0)exp{i[α0x+γn(α0)y]} × -+ exp-(α-α0)2σnj2(y)4+i(x+Mny-Lnj)dα,
Mn=γn(α)αα0=Mn+iMn,
Lnj=irnj(α0)rnj(α)αα0=Lnj+iLnj,
σnj2(y)=4/ωn2+i2Nn(y-Fnj),
Nn=-2γn(α)α2α0=Nn+iNn,
Fnj=-iNnα1rnj(α)rnj(α)αα0=Fnj+iFnj,
Enj(x, y)=2π1/2rnj(α0)σnj(y)exp-(x+Mny-Lnj)2σnj2(y) × exp{i[α0x+γn(α0)y]},
j=x, y, z,n=1, , 6.
tan βnj=|NnLnj/Pnj-Mn|,
Pnj=2/ωn2+NnFnj,
Nn=Mn=0.
(xFnj, yFnj)=(Lnj-MnFnj, Fnj),
δnj(y=yFnj)=2(cos2 βnj)Pnj2Pnj-2Nn2 sin2 βnj cos2 βnj1/2,
|Enj(xFnj, yFnj)|2=2π|rnj(α0)|2|Pnj|expLnj2Pnj.
xLnj=Lnj+Lnj(FnjNn-FnjNn)Pnj+FnjNn.
|Enj(xLnj, 0)|2
=2π|rnj(α0)|2[(Pnj+FnjNn)2+(FnjNn-FnjNn)2]1/2
×expLnj2Pnj+FnjNn.
Laj+=0,
Laj+=1|eaj+(α0)||eaj+(α)|αα0,
Faj+=0,
Faj+=2γa+(α)α2α0-1 1|eaj+(α0)| × 2|eaj+(α)|α2α0-1|eaj+(α0)||eaj+(α)|αα02,
Lbaj-=-φ(Rba(α))αα0+φ(ebj-(α))αα0,
Lbaj-=1|Rba(α0)||Rba(α)|αα0+1|ebj-(α0)||ebj-(α)|αα0,
Fbaj-=2γb-(α)α2α02+2γb-(α)α2α02-1-2γb-(α)α2α02φ(Rba(α))α2α0+2φ(ebj-(α))α2α0+2γb-(α)α2α0 × 1|Rba(α0)|2|Rba(α)|α2α0-1|Rba(α0)||Rba(α)|αα02+1|ebj-(α0)|2|ebj-(α)|α2α0-1|ebj-(α0)||ebj-(α)|αα02,
Fbaj-=2γb-(α)α2α02+2γb-(α)α2α02-12γb-(α)α2α02φ(Rba(α))α2α0+2φ(ebj-(α))α2α0+2γb-(α)α2α0 × 1|Rba(α0)|2|Rba(α)|α2α0-1|Rba(α0)||Rba(α)|αα02+1|ebj-(α0)|2|ebj-(α)|α2α0-1|ebj-(α0)||ebj-(α)|αα02,
yFbaj=-2γb-(α)α2α0-12φ(Rba(α))α2α0,
f(v)=-+g(α)exp[iΦ(v, α)]dα,
Φ(vM, α)αα=α0=0,
xLnjSPM=Lnj,
tan βnjSPM=|-Mn|,
δnjδn(1)=2 cos βn(1)ωn,
tan βn(1)=|-Mn|.

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