Abstract

We present a detailed analysis of second-harmonic-generation Maker fringes in biaxially birefringent materials that takes into account the ordinary as well as the extraordinary polarized bound and free waves. This becomes relevant for elliptically polarized second-harmonic-generation signals. To keep the formulas easy to survey, the calculation is performed in a compact 4×4 matrix technique. For a test of the analysis we apply it to measurements in birefringent single crystals of the asymmetrically substituted diacetylene NP/4-MPU.

© 1998 Optical Society of America

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References

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  1. P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
    [CrossRef]
  2. J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
    [CrossRef]
  3. N. Okamoto, Y. Hirano, and O. Sugihara, “Precise estimation of nonlinear-optical coefficients for anisotropic nonlinear films with C∞v symmetry,” J. Opt. Soc. Am. B 9, 2083–2087 (1992).
    [CrossRef]
  4. W. N. Herman and L. M. Hayden, “Maker fringes revisited: second-harmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B 12, 416–427 (1995).
    [CrossRef]
  5. D. S. Bethune, “Optical harmonic generation and mixing in multilayer media: analysis using optical transfer matrix techniques,” J. Opt. Soc. Am. B 6, 910–916 (1989).
    [CrossRef]
  6. D. S. Bethune, “Optical harmonic generation and mixing in multilayer media: extension of optical transfer matrix approach to include anisotropic materials,” J. Opt. Soc. Am. B 8, 367–373 (1991).
    [CrossRef]
  7. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  8. N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
    [CrossRef]
  9. M. Braun, F. Bauer, Th. Vogtmann, and M. Schwoerer, “Precise second-harmonic generation Maker fringe measurements in single crystals of the diacetylene NP/4-MPU and evaluation by a second-harmonic generation theory in 4×4 matrix formulation and ray tracing,” J. Opt. Soc. Am. B 14, 1699–1706 (1997).
    [CrossRef]
  10. P. Strohriegel, H. Schultes, D. Heindl, P. Gruner-Bauer, V. Enkelmann, and E. Dormann, “Preparation, crystal structure and dielectric properties of the new unsymmetricallysubstituted diacetylenes NP/R2,” Ber. Bunsenges. Phys. Chem. 91, 918–924 (1987).
    [CrossRef]
  11. P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics, Vol. 9 of Cambridge Studies in Modern Optics (Cambridge U. Press, Cambridge, UK, 1990).

1997 (1)

1995 (1)

1992 (1)

1991 (1)

1989 (1)

1987 (1)

P. Strohriegel, H. Schultes, D. Heindl, P. Gruner-Bauer, V. Enkelmann, and E. Dormann, “Preparation, crystal structure and dielectric properties of the new unsymmetricallysubstituted diacetylenes NP/R2,” Ber. Bunsenges. Phys. Chem. 91, 918–924 (1987).
[CrossRef]

1970 (1)

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
[CrossRef]

1962 (2)

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
[CrossRef]

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Bauer, F.

Bethune, D. S.

Bloembergen, N.

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
[CrossRef]

Braun, M.

Dormann, E.

P. Strohriegel, H. Schultes, D. Heindl, P. Gruner-Bauer, V. Enkelmann, and E. Dormann, “Preparation, crystal structure and dielectric properties of the new unsymmetricallysubstituted diacetylenes NP/R2,” Ber. Bunsenges. Phys. Chem. 91, 918–924 (1987).
[CrossRef]

Enkelmann, V.

P. Strohriegel, H. Schultes, D. Heindl, P. Gruner-Bauer, V. Enkelmann, and E. Dormann, “Preparation, crystal structure and dielectric properties of the new unsymmetricallysubstituted diacetylenes NP/R2,” Ber. Bunsenges. Phys. Chem. 91, 918–924 (1987).
[CrossRef]

Gruner-Bauer, P.

P. Strohriegel, H. Schultes, D. Heindl, P. Gruner-Bauer, V. Enkelmann, and E. Dormann, “Preparation, crystal structure and dielectric properties of the new unsymmetricallysubstituted diacetylenes NP/R2,” Ber. Bunsenges. Phys. Chem. 91, 918–924 (1987).
[CrossRef]

Hayden, L. M.

Heindl, D.

P. Strohriegel, H. Schultes, D. Heindl, P. Gruner-Bauer, V. Enkelmann, and E. Dormann, “Preparation, crystal structure and dielectric properties of the new unsymmetricallysubstituted diacetylenes NP/R2,” Ber. Bunsenges. Phys. Chem. 91, 918–924 (1987).
[CrossRef]

Herman, W. N.

Hirano, Y.

Jerphagnon, J.

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
[CrossRef]

Kurtz, S. K.

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
[CrossRef]

Maker, P. D.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Nisenoff, M.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Okamoto, N.

Pershan, P. S.

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
[CrossRef]

Savage, C. M.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Schultes, H.

P. Strohriegel, H. Schultes, D. Heindl, P. Gruner-Bauer, V. Enkelmann, and E. Dormann, “Preparation, crystal structure and dielectric properties of the new unsymmetricallysubstituted diacetylenes NP/R2,” Ber. Bunsenges. Phys. Chem. 91, 918–924 (1987).
[CrossRef]

Schwoerer, M.

Strohriegel, P.

P. Strohriegel, H. Schultes, D. Heindl, P. Gruner-Bauer, V. Enkelmann, and E. Dormann, “Preparation, crystal structure and dielectric properties of the new unsymmetricallysubstituted diacetylenes NP/R2,” Ber. Bunsenges. Phys. Chem. 91, 918–924 (1987).
[CrossRef]

Sugihara, O.

Terhune, R. W.

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Vogtmann, Th.

Ber. Bunsenges. Phys. Chem. (1)

P. Strohriegel, H. Schultes, D. Heindl, P. Gruner-Bauer, V. Enkelmann, and E. Dormann, “Preparation, crystal structure and dielectric properties of the new unsymmetricallysubstituted diacetylenes NP/R2,” Ber. Bunsenges. Phys. Chem. 91, 918–924 (1987).
[CrossRef]

J. Appl. Phys. (1)

J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41, 1667–1681 (1970).
[CrossRef]

J. Opt. Soc. Am. B (5)

Phys. Rev. (1)

N. Bloembergen and P. S. Pershan, “Light waves at the boundary of nonlinear media,” Phys. Rev. 128, 606–622 (1962).
[CrossRef]

Phys. Rev. Lett. (1)

P. D. Maker, R. W. Terhune, M. Nisenoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8, 21–22 (1962).
[CrossRef]

Other (2)

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics, Vol. 9 of Cambridge Studies in Modern Optics (Cambridge U. Press, Cambridge, UK, 1990).

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Figures (3)

Fig. 1
Fig. 1

Origin of the Maker fringes. Paths of rays with the wave vectors of the bound (b) and the free ( f ) waves in the crystal. The interference, depending on the angle of incidence φ, between the bound and the free waves generates the well-known Maker fringe oscillation pattern.

Fig. 2
Fig. 2

Incident, transmitted, and reflected beams on the front side (reflex 1), and reflected beam on the rear side (reflex 2). I, Beam waist is much greater than crystal thickness (thin films); II, beam waist is much smaller than crystal thickness (thick crystals). In the dark area the reflected and the transmitted beams overlap; in the light area there is no overlap.

Fig. 3
Fig. 3

Maker fringes of NP/4-MPU in seven configurations; the front plane of the crystal is the aˆcˆ plane. The angle Θ between the rotation axis and the a axis of the crystal is varied from 0° to 90° in steps of 15°. Left, measurement; right, theoretical fit.

Tables (2)

Tables Icon

Table 1 Refractive Indices of NP/4-MPU Single Crystals for Several Wavelengths λ Determined by Interferometry, Reflectometry, and SHG Maker Fringe Measurements

Tables Icon

Table 2 Values (pm V-1) of Modules of the Contracted Matrix d of NP/4-MPU Single Crystals Determined by SHG Maker Fringe Measurementsa

Equations (54)

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××Eω(r, t)-ω2μ=ωEω(r, t)=0,
Eω(r, t)Eωexp[i(kωr-ωt)]
=p^ωQωexp[i(kωr-ωt)],
kω(αω, βω, γω)=[kω,x(0), kω, y(0), γω],
M=ωEω=0M=ω p^ω=0,
M=ωβω2+γω2-mω,xx-αωβω-mω,xy-αωγω-mω,xz-αωβω-mω, yxαω2+γω2-mω, yy-βωγω-mω, yz-αωγω-mω,zx-βωγω-mω,zyαω2+βω2-mω,zz,
mω,ijω2μω,ij.
p^ω,i=(νω,i×wω,i)+(wω,i×uω,i)+(uω,i×νω,i)|(νω,i×wω,i)+(wω,i×uω,i)+(uω,i×νω,i)|,
uω,iβω2+γω,i2-mω,xx-αωβω-mω,xy-αωγω,i-mω,xz,
vω,i-αωβω-mω, yxαω2+γω,i2-mω, yy-βωγω-mω, yz,
wω,i-αωγω,i-mω,zx-βωγω-mω,zyαω2+βω2-mω,zz.
Eω=i=14Eω,iexp[i(kω,ir-ωt)]=i=14Qω,ip^ω,iexp[i(kω,ir-ωt)].
Hω,i=cωμkω,i×Eω,i=:hω,i|Eω,i|,
hω,i=cωμkω,i×p^ω,i.
P2ω(2)=d11d12d13d14d15d16d21d22d23d24d25d26d31d32d33d34d35d36Eω,x2Eω, y2Eω,z22Eω, y Eω,z2Eω,z Eω,x2Eω,x Eω, y=d=Eω,x2Eω, y2Eω,z22Eω, y Eω,z2Eω,z Eω,x2Eω,x Eω, y=i=14j=14d=Eω,i,x Eω, j,xEω,i, y Eω, j, yEω,i,z Eω, j,z2Eω,i, y Eω, j,z2Eω,i,z Eω, j,x2Eω,i,x Eω, j, y×exp[i(kω,ir+kω, jr-2ωt)]=:g=110P2ω, g(2)exp(ψg).
P2ω,g(2)=d=Eω, g,x2Eω, g, y2Eω,g,z22Eω, g,y Eω, g,z2Eω, g,z Eω, g,x2Eω, g,x Eω, g,yexp(ψg),
g=1fori=j=12fori=j=23fori=j=34fori=j=4,
P2ω,g(2)=d=2Eω,i, x Eω, j,x2Eω,i, y Eω, j,y2Eω,i,z Eω, j,z2(Eω,i,y Eω, j,z+Eω, j, y Eω,i,z)2(Eω,i,z Eω, j,x+Eω, j,z Eω,i,x)2(Eω,i,x Eω,j,y+Eω,j,x Eω,i,y)exp(ψg),
g=5fori=1, j=26fori=1, j=37fori=1, j=48fori=2, j=39fori=2, j=410fori=3, j=4,
ψg=i(kω,ir+kω, jr-2ωt)=:i(kb,gr-2ωt).
××E2ω(r, t)-(2ω)2μ=2ωE2ω(r, t)=(2ω)2μP2ω(2).
Ef=i=14Ef,iexp[i(kf,ir-2ωt)]=i=14Qf,i p^f,iexp[i(kf,ir-2ωt)],
Hf=i=14Hf,iexp[i(kf,ir-2ωt)]=i=14Qf,ihf,iexp[i(kf,ir-2ωt)],
αf=2αω,βf=βω.
××Eb, g(r, t)-(2ω)2μ=2ωEb, g(r, t)
=(2ω)2μP2ω, g(2)=(2ω)2μP2ω, gexp(ψg),
Eb, g(r, t)Eb, gexp(ψg)=p^b, gQb, gexp[i(kb, gr-2ωt)],
kb, g=:(α2ω, β2ω, γb, g).
M=b, gEb, g(r, t)=(2ω)2μP2ω, g(2)M=b, g p^b, gQb, g=(2ω)2μP2ω, g,
M=b,gβ2ω2+γb,g2-m2ω,xx-α2ωβ2ω-m2ω,xy-α2ωγb,g-m2ω,xz-α2ωβ2ω-m2ω,yxα2ω2+γb,g2-m2ω,yy-β2ωγb,g-m2ω,yz-α2ωγb,g-m2ω,zx-β2ωγb,g-m2ω,zyα2ω2+β2ω2-m2ω,zz,
m2ω,ij(2ω)2μ2ω,ij.
Eb,g=(2ω)2μ[M=b,g(2ω)]-1P2ω,g(2),
Qb,g=|Eb, g |,p^b,g=Eb,g|Eb,g|.
Eb=g=110Qb, g p^b,gexp[i(kb, gr-2ωt)].
E2ω=Ef+Eb=i=14Qf,i p^f,iexp[i(kf,ir-2ωt)]
+g=110Qb, gp^b, gexp[i(kb, gr-2ωt)].
xˆE2ω/ω(s-1)=xˆE2ω/ω(s),
yˆH2ω/ω(s-1)=yˆH2ω/ω(s),
yˆE2ω/ω(s-1)=yˆE2ω/ω(s),
xˆH2ω/ω(s-1)=xˆH2ω/ω(s),
D=f (s-1)Qf (s-1)+D=b(s-1)Qb(s-1)
=D=f (s)P=f (s)Qf(s)+D=b(s)P=b(s)Qb(s),
D=ω(s-1)Qω(s-1)=D=ω(s)P=ω(s)Qω(s),
Qf/ω(s)Qf/ω,1(s)Qf/ω,2(s)Qf/ω,3(s)Qf/ω,4(s),
D=f/ω(s)xˆp^f/ω,1(s)xˆp^f/ω,4(s)yˆhf/ω,1(s)yˆhf/ω,4(s)yˆp^f/ω,1(s)yˆp^f/ω,4(s)xˆhf/ω,1(s)xˆhf/ω,4(s),
P=f/ω(s)exp[iγf/ω,1d(s)]exp[iγf/ω,4d(s)],
Qb(s)Qb,1(s)Qb,2(s):Qb,9(s)Qb,10(s),
D=b(s)xˆp^b,1(s)xˆp^b,10(s)yˆhb,1(s)yˆhb,10(s)yˆp^b,1(s)yˆp^b,10(s)xˆhb,1(s)xˆhb,10(s),
P=b(s)exp[iγb,1d(s)]00exp[iγb,10d(s)], 
D=f(1)Qf(1)=D=f (2)P=f (2)Qf (2)
+D=b(2)P=b(2)Qb(2),
D=f (3)Qf (3)=D=f (2)Qf(2)+D=b(2)Qb(2).
Qf,1(1)=Qf,3(1)=Qf,2(3)=Qf,4(3)=0.
d==0000da,ac0000db,bc00dc,aadc,bbdc,cc000,

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