Abstract

The full polarization property of holographic volume-grating enhanced second-harmonic diffraction (SHD) is investigated theoretically. The nonlinear coefficient is derived from a simple atomic model of the material. By using a simple volume-grating model, the SHD fields and Mueller matrices are first derived. The SHD phase-mismatching effect for a thick sample is analytically investigated. This theory is justified by fitting with published experimental SHD data of thin-film samples. The SHD of an existing polymethyl methacrylate (PMMA) holographic 2-mm-thick volume-grating sample is investigated. This sample has two strong coupling linear diffraction peaks and five SHD peaks. The splitting of SHD peaks is due to the phase-mismatching effect. The detector sensitivity and laser power needed to measure these peak signals are quantitatively estimated.

© 2006 Optical Society of America

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References

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  1. T.-W. Nee and S.-M. F. Nee, "Polarization of holographic grating diffraction I. General theory," J. Opt. Soc. Am. A 21, 523-531 (2004).
    [CrossRef]
  2. T.-W. Nee, S.-M. F. Nee, M. Kleinschmit, and S. Shahriar, "Polarization of holographic grating diffraction II. Experiment," J. Opt. Soc. Am. A 21, 532-539 (2004).
    [CrossRef]
  3. M. S. Shahriar, J. Riccobono, and W. Weathers, "Holographic beam combiner," in Proceedings of IEEE International Conference on Microwaves and Optics (IEEE, 1999), pp. 10-14.
  4. M. S. Shahriar, J. Riccobono, M. Kleinschmit, and J. T. Shen, "Coherent and incoherent beam combination using thick holographic substrates," Opt. Commun. 220, 75-83 (2003).
    [CrossRef]
  5. X. D. Zhu and Y. R. Shen, "Generation and detection of a monolayer grating by laser desorption and second-harmonic generation: CO on Ni(111)," Opt. Lett. 14, 503-505 (1989).
    [CrossRef] [PubMed]
  6. T. Suzuki and T. F. Heinz, "Second-harmonic diffraction from a monolayer grating," Opt. Lett. 14, 1201-1203 (1989).
    [CrossRef] [PubMed]
  7. R. D. Schaller, R. J. Saykally, Y. R. Shen, and F. Lagugne-Labarthet, "Poled polymer thin-film gratings studied with far-field optical diffraction and second-harmonic near-field microscopy," Opt. Lett. 28, 1296-1298 (2003).
    [CrossRef] [PubMed]
  8. F. Lagugne-Labarthet, C. Sourisseau, R. D. Schaller, R. J. Saykally, and P. Rochon, "Chromophore orientations in a nonlinear optical azopolymer diffraction grating: even and odd order parameters from far-field Raman and near-field second-harmonic near-field microscopy," Opt. Lett. 28, 1296-1298 (2003).
    [PubMed]
  9. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).
  10. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, 1976).
  11. T. W. Nee, "Surface-irregularity-enhanced subband resonance of seminconductors I. General theory," Phys. Rev. B 29, 3225-3238 (1984).
    [CrossRef]
  12. J. D. Jackson, "Classical electrodynamics" (Wiley, 1962).
  13. S.-M. F. Nee, "Polarization measurement," in The Measurement, Instrumentation and Sensors Handbook, J.G.Webster, ed. (CRC, and IEEE, 1999), pp. 60.1-60.24.

2004 (2)

2003 (3)

1989 (2)

1984 (1)

T. W. Nee, "Surface-irregularity-enhanced subband resonance of seminconductors I. General theory," Phys. Rev. B 29, 3225-3238 (1984).
[CrossRef]

Heinz, T. F.

Jackson, J. D.

J. D. Jackson, "Classical electrodynamics" (Wiley, 1962).

Kleinschmit, M.

T.-W. Nee, S.-M. F. Nee, M. Kleinschmit, and S. Shahriar, "Polarization of holographic grating diffraction II. Experiment," J. Opt. Soc. Am. A 21, 532-539 (2004).
[CrossRef]

M. S. Shahriar, J. Riccobono, M. Kleinschmit, and J. T. Shen, "Coherent and incoherent beam combination using thick holographic substrates," Opt. Commun. 220, 75-83 (2003).
[CrossRef]

Lagugne-Labarthet, F.

Nee, S.-M. F.

Nee, T. W.

T. W. Nee, "Surface-irregularity-enhanced subband resonance of seminconductors I. General theory," Phys. Rev. B 29, 3225-3238 (1984).
[CrossRef]

Nee, T.-W.

Riccobono, J.

M. S. Shahriar, J. Riccobono, M. Kleinschmit, and J. T. Shen, "Coherent and incoherent beam combination using thick holographic substrates," Opt. Commun. 220, 75-83 (2003).
[CrossRef]

M. S. Shahriar, J. Riccobono, and W. Weathers, "Holographic beam combiner," in Proceedings of IEEE International Conference on Microwaves and Optics (IEEE, 1999), pp. 10-14.

Rochon, P.

Saykally, R. J.

Schaller, R. D.

Shahriar, M. S.

M. S. Shahriar, J. Riccobono, M. Kleinschmit, and J. T. Shen, "Coherent and incoherent beam combination using thick holographic substrates," Opt. Commun. 220, 75-83 (2003).
[CrossRef]

M. S. Shahriar, J. Riccobono, and W. Weathers, "Holographic beam combiner," in Proceedings of IEEE International Conference on Microwaves and Optics (IEEE, 1999), pp. 10-14.

Shahriar, S.

Shen, J. T.

M. S. Shahriar, J. Riccobono, M. Kleinschmit, and J. T. Shen, "Coherent and incoherent beam combination using thick holographic substrates," Opt. Commun. 220, 75-83 (2003).
[CrossRef]

Shen, Y. R.

Sourisseau, C.

Suzuki, T.

Weathers, W.

M. S. Shahriar, J. Riccobono, and W. Weathers, "Holographic beam combiner," in Proceedings of IEEE International Conference on Microwaves and Optics (IEEE, 1999), pp. 10-14.

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, 1976).

Zhu, X. D.

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

Opt. Commun. (1)

M. S. Shahriar, J. Riccobono, M. Kleinschmit, and J. T. Shen, "Coherent and incoherent beam combination using thick holographic substrates," Opt. Commun. 220, 75-83 (2003).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. B (1)

T. W. Nee, "Surface-irregularity-enhanced subband resonance of seminconductors I. General theory," Phys. Rev. B 29, 3225-3238 (1984).
[CrossRef]

Other (5)

J. D. Jackson, "Classical electrodynamics" (Wiley, 1962).

S.-M. F. Nee, "Polarization measurement," in The Measurement, Instrumentation and Sensors Handbook, J.G.Webster, ed. (CRC, and IEEE, 1999), pp. 60.1-60.24.

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, 1984).

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart & Winston, 1976).

M. S. Shahriar, J. Riccobono, and W. Weathers, "Holographic beam combiner," in Proceedings of IEEE International Conference on Microwaves and Optics (IEEE, 1999), pp. 10-14.

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Figures (13)

Fig. 1
Fig. 1

Reflected, transmitted, and diffracted beams of the holographic grating sample.

Fig. 2
Fig. 2

Calculated SHD transmission coefficient T 2 ( Θ 2 ) spectrum of the 2-mm-thick volume-grating sample (Ref. [2]). A 1064 nm laser of 1 × 10 8 W cm 2 power is incident at angle 48.34 ° .

Fig. 3
Fig. 3

Calculated linear transmission coefficient T 1 ( Θ 1 ) spectra of the 2-mm-thick sample (Ref. [2]). A 1064 nm light is incident at angle 48.34 ° .

Fig. 4
Fig. 4

SHD T 2 a (source a), T 2 b (source b) spectra for a1 and a2 peaks.

Fig. 5
Fig. 5

t o 2 and t t 2 spectra of a1 and a2 beams due to a and b sources.

Fig. 6
Fig. 6

a1–a2 beam splitting due to phase mismatching ( n 2 n ) , SHD angles Θ 2 , and transmission coefficients T 2 ( Θ 2 ) .

Fig. 7
Fig. 7

Spectral T 2 ( Θ 2 ) for n 2 = 1.484 (phase matching, solid curve) and 1.494 (phase-mismatching, dashed curve) are shown for Θ 2 = 47.8 ° to 50.8 ° .

Fig. 8
Fig. 8

SHD transmission coefficient T 2 and t t 2 ( Θ 2 = 47.8 to 50.8) for three samples 20 , 500 , and 2000 μ m thick.

Fig. 9
Fig. 9

Thickness dependence of SHD transmission coefficient T 2 for the four peaks a1, b1, a2, and b2 [Fig. 4].

Fig. 10
Fig. 10

Calculated SHD transmission coefficient T 2 ( Θ 2 ) spectra of two thin-film samples 410 and 110 nm thick. A 1064 nm laser of 1 × 10 8 W cm 2 power is normally incident to the sample.

Fig. 11
Fig. 11

Calculated linear transmission coefficient T 1 ( Θ 1 ) spectra of the 410-nm-thick sample for normally incident light of wavelengths 632.8 and 1064 nm .

Fig. 12
Fig. 12

Output intensities I out = I i T 2 of the five peaks a1, b1, a2, b2, and c for incident laser intensity I i = 10 5 to 10 8 W cm 2 .

Fig. 13
Fig. 13

I i -dependent I out and the SHG coefficients D i p and D i s of the b2 peak for three incident laser polarization states.

Tables (2)

Tables Icon

Table 1 Second-Harmonic Diffraction and Polarization Properties of Thick Grating Sample A: λ 2 = 532 nm , n = 1.484 , n 2 = 1.494 , Φ 2 = 0 °

Tables Icon

Table 2 Linear Diffraction and Polarization Properties of Thick Grating Sample A: λ 1 = 1064 nm , n = 1.484 , Φ 1 = 0 °

Equations (70)

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K = 2 π n K x o = [ K x , K y , K z ] ,
n K = [ cos θ K cos φ K , cos θ K sin φ K , sin θ K ]
E i ( x , t ) = [ E i p cos θ , E i s , E i p sin θ ] exp [ i k ( x sin θ z cos θ ) ] exp [ i ω t ] .
E ( x , t ) = E 1 ( x ) exp ( i ω t ) + E 2 ( x ) exp ( i 2 ω t ) + .
P ( x , t ) = P 1 ( x ) exp ( i ω t ) + P 2 ( x ) exp ( i 2 ω t ) + ,
P 1 ( x ) = α ( ω ) ρ g ( x ) E 1 ( x ) ,
P 2 ( x ) = ρ g ( x ) m 2 e 3 α ( 2 ω ) α 2 ( ω ) [ D ( x ) E 1 ( x ) ] E 1 ( x ) ,
α ( ω ) = e 2 m 1 ω 2 ω 0 2 + i ω γ ,
ε ( x ) = ε o + 4 π α ( ω ) ρ g ( x ) ,
ρ g ( x ) = 1 A u L u Σ l f ( n K x l x o ) , ( l = , 2 , 1 , 0 , 1 , 2 , ) ,
ε ( x ) = ε o + Δ ε Σ l f ( n K x l x o ) .
Δ ε = 4 π α ( ω ) 1 A u L u .
D ( x ) = 3 ω o 2 2 ε ( x ) ε ( x ) .
D ( x ) n K 3 ω o 2 Δ ε 2 ε o Σ l f ( n K x l x o ) .
E 1 ( x ) = E 1 ( z ) exp [ i k ( x sin Θ cos Φ + y sin Θ sin Φ z cos Θ ) ] .
E 1 ( z ) = ( E 1 p ( z ) cos Θ cos Φ E 1 s ( z ) sin Φ E 1 p ( z ) cos Θ sin Φ + E 1 s ( z ) cos Φ E 1 p ( z ) sin Θ ) .
P 2 ( x , y , z ) = exp ( i q x x ) exp ( i q y y ) P 2 ( q x , q y , z ) ,
P 2 ( q x , q y , z ) = P 2 ( o ) ( q x , q y , z ) exp [ i z ( q x 2 k sin Θ cos Φ cos θ K cos φ K sin θ K 2 k cos Θ ) ] ,
P 2 ( o ) ( q x , q y , z ) = Δ ρ W 2 ( q x ) E 1 ( z ) E 1 ( z ) { δ q y , [ ( q x 2 k sin Θ cos Φ ) tan φ K + 2 k sin Θ sin Φ ] ,
Δ ρ = 1 A u L u ,
[ q x , q y , q z ] = k 2 [ sin Θ 2 cos Φ 2 , sin Θ 2 sin Φ 2 , cos Θ 2 ] ,
W 2 ( q x ) = N x F ( q x 2 k sin Θ cos Φ cos θ K cos φ K ) g ( q x 2 k sin Θ cos Φ cos θ K cos φ K x o ) 1 cos θ K cos φ K ,
L u = N x x o .
F ( p ) = i p L u d u f ( u ) exp [ i p u ] .
g ( β ) = Σ l exp ( i l β ) N x = 1 N x sin ( N x β 2 ) sin ( β 2 ) .
E 2 ( x , y , z ) = E 2 ( q x , q y , z ) exp ( i q x x ) exp ( i q y y ) .
( 2 z 2 + 4 k 2 q x 2 q y 2 ) E 2 ( q x , q y , z ) = 4 π ε ( ( q x 2 4 k 2 ) P 2 x ( z ) + q x q y P 2 y ( z ) i q x P 2 z ( z ) q x q y P 2 x ( z ) + ( q y 2 4 k 2 ) P 2 y ( z ) i q y P 2 z ( z ) i q x P 2 x ( z ) i q y P 2 y ( z ) P 2 z ( z ) 4 k 2 P 2 z ( z ) ) .
E 2 ( q x , q y , z ) = exp [ i q z z ] ( E 2 p ( z ) cos Θ 2 cos Φ 2 E 2 s ( z ) sin Φ E 2 p ( z ) cos Θ 2 sin Φ 2 + E 2 s ( z ) cos Φ 2 E 2 p ( z ) sin Θ 2 ) .
( E 2 p ( 0 ) E 2 s ( 0 ) ) = P ( E 1 p ( 0 + ) E 1 s ( 0 + ) ) .
( E 1 p ( 0 ) E 1 s ( 0 ) ) = Q ( E i p E i s ) .
P = U [ t 2 p 0 0 t 2 s ] [ cos Θ 2 cos Φ 2 cos Θ 2 sin Φ 2 sin Θ 2 sin Φ 2 cos Φ 2 0 ] × [ sin 2 Θ 2 cos 2 Φ 2 2 1 sin 2 Θ 2 cos Φ 2 sin Φ 2 Q z sin Θ 2 cos Φ 2 sin 2 Θ 2 cos Φ 2 sin Φ 2 sin 2 Θ 2 sin 2 Φ 2 1 Q z sin Θ 2 sin Φ 2 Q Z sin Θ 2 cos Φ 2 sin Θ 2 sin Φ 2 Q Z Q z 2 1 ] [ cos Θ cos Φ sin Φ cos Θ sin Φ cos Φ sin Θ 0 ] ,
U = Δ ε ε o V [ t 1 p D i p ( cos Θ cos Φ cos θ K cos φ K + sin Θ sin θ K ) + t 1 s D i s cos Θ sin Φ cos θ K sin φ K ] ,
D i l = E i l E o , ( 1 = p , s ) ,
E o = L u e ρ o ,
V = 3 ( ε o 1 ) 2 32 π t t t o ( N x , β , u o x o ) ,
t o ( N x , β , u o x o ) = N x β sin ( N x β 2 ) N x sin ( β 2 ) exp ( β 2 u o 2 4 x o 2 ) u o x o ,
t t = 4 π a n 2 λ q z γ + ( 4 π a λ ( n 2 cos Θ 2 n cos Θ ) + β a x o sin θ K ) 1 cos θ K cos φ K ,
β = 4 π x o ( n 2 sin Θ 2 cos Φ 2 n sin Θ cos Φ ) λ cos θ K cos φ K
γ ± ( x ) = exp ( ± i x ) 1 ± i x .
Q z = β λ 4 π n x o sin θ K cos Θ .
f ( u ) = exp ( u 2 u o 2 ) .
E 2 ( x ) = E 2 ( 0 ) exp [ i 2 k ( x sin Θ 2 cos Φ 2 + y sin Θ 2 sin Φ 2 z cos Θ 2 ] .
E 2 ( 0 ) = ( E 2 p ( 0 ) cos Θ 2 cos Φ 2 E 2 s ( 0 ) sin Φ 2 E 2 p ( 0 ) cos Θ 2 sin Φ 2 + E 2 s ( 0 ) cos Φ 2 E 2 p ( 0 ) sin Θ 2 ) .
sin Θ 2 = n 2 sin Θ 2 .
( E 2 p ( 0 ) E 2 s ( 0 ) ) = J 2 ( E i p E i s ) .
J 2 = P Q .
A pol = α ( ω ) ( A u x o ) .
M 2 = M 2 a + M 2 b .
T 2 = M 2 ( 1 , 1 ) .
T 2 = M 2 a ( 1 , 1 ) + M 2 b ( 1 , 1 ) = T 2 a + T 2 b .
4 π x o ( sin Θ 2 cos Φ 2 sin Θ cos Φ ) λ cos θ K cos φ K = β = m π , ( m = ± 1 , ± 2 , ) .
n 2 cos Θ 2 n cos Θ = β λ 4 π x o sin θ K .
V ( r ) = e Q r + l 2 2 m e r 2 ,
r o l 2 m 2 e Q .
x ( t ) = x 1 ( t ) + x 2 ( t ) δ 1 ( t ) + δ 2 ( t ) .
d 2 d t 2 x = ω o 2 x ,
ω o 2 = 1 2 m e [ V ( r o ) + V ( r o ) ] = e Q m e r o 3 .
Q j = Q ε j ( j = 1 , 2 ) .
d 2 d t 2 x = ω o 2 x D x 2 .
ω o 2 1 2 m e [ V ( r o ) + V ( r o ) ] = e Q 2 m e r o 3 ε 1 + ε 2 ε 1 ε 2 ,
D 1 8 m e [ V ( r o ) V ( r o ) ] 3 e Q 4 m e r o 4 [ ε 1 ε 2 ε 1 ε 2 ]
D ω o 2 3 2 r o ε 1 ε 2 ε 1 + ε 2 3 Δ x Δ ε ( x ) 2 ε ( x ) 3 2 ε ( x ) ε ( x ) .
D ( x ) = 3 ω o 2 2 ε ( x ) ε ( x ) .
S i = ( I i Q i U i V i ) = S po ( E i p * E i p + E i s * E i s E i p * E i p E i s * E i s E i p * E i s + E i s * E i p i ( E i p * E i s E i s * E i p ) ) .
E i p [ I i + Q i 2 S po ] 1 2 , E i s = [ I i + Q i 2 S po ] 1 2 exp ( i φ 1 ) .
φ 1 = tan 1 ( V i U i ) .
S 2 = ( I Q U V ) = M 2 S i .
S o = E o 2 S po = ( N x x o e ρ o ) 2 S p o .
D i p = E i p E o = [ I i + Q i 2 S o ] 1 2 ,
D i s = E i s E o = [ I i Q i 2 S o ] 1 2 exp ( i φ i ) .

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