Abstract

The energy conservation of grating diffraction is analyzed in a particular condition of incidence in which two incident waves reach a symmetrical grating from the two sides of the grating normal at the first-order Littrow mounting. In such a situation the incident waves generate an interference pattern with the same period as the grating. Thus in each direction of diffraction, interference occurs between two consecutive diffractive orders of the symmetrical incident waves. By applying only energy conservation and the geometrical symmetry of the grating profile to this problem it is possible to establish a general constraint for the phases and amplitudes of the diffracted orders of the same incident wave. Experimental and theoretical results are presented confirming the obtained relations.

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References

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    [CrossRef]
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    [CrossRef] [PubMed]

2003

1999

1998

V. Kettunen and F. Wyrowski, "Reflection mode phase retardation by dielectric gratings," Opt. Commun. 158, 41-44 (1998).
[CrossRef]

1997

1980

L. C. Botten, J. L. Adams, R. C. McPhedran, and G. H. Derrick, "Symmetry properties of lossless diffraction gratings," J. Opt. 11, 43-52 (1980).
[CrossRef]

1978

L. C. Botten, "A new formalism for transmission gratings," Opt. Acta 25, 481-499 (1978).
[CrossRef]

1972

D. L. Staebler and J. J. Amodei, "Coupled-wave analysis of holographic storage in LiNbO3," J. Appl. Phys. 43, 1042-1049 (1972).
[CrossRef]

1969

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).

1965

J. L. Uretsky, "The scattering of plane waves from periodic surfaces," Ann. Phys. 33, 400-427 (1965).
[CrossRef]

Adams, J. L.

L. C. Botten, J. L. Adams, R. C. McPhedran, and G. H. Derrick, "Symmetry properties of lossless diffraction gratings," J. Opt. 11, 43-52 (1980).
[CrossRef]

Amodei, J. J.

D. L. Staebler and J. J. Amodei, "Coupled-wave analysis of holographic storage in LiNbO3," J. Appl. Phys. 43, 1042-1049 (1972).
[CrossRef]

Botten, L. C.

L. C. Botten, J. L. Adams, R. C. McPhedran, and G. H. Derrick, "Symmetry properties of lossless diffraction gratings," J. Opt. 11, 43-52 (1980).
[CrossRef]

L. C. Botten, "A new formalism for transmission gratings," Opt. Acta 25, 481-499 (1978).
[CrossRef]

Cescato, L.

Chandezon, J.

Cordeiro, C. M. B.

Derrick, G. H.

L. C. Botten, J. L. Adams, R. C. McPhedran, and G. H. Derrick, "Symmetry properties of lossless diffraction gratings," J. Opt. 11, 43-52 (1980).
[CrossRef]

Freschi, A.

Goodmam, J. W.

J. W. Goodmam, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Granet, G.

Kettunen, V.

V. Kettunen and F. Wyrowski, "Reflection mode phase retardation by dielectric gratings," Opt. Commun. 158, 41-44 (1998).
[CrossRef]

Kikuta, H.

Kogelnik, H.

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).

Li, L.

McPhedran, R. C.

L. C. Botten, J. L. Adams, R. C. McPhedran, and G. H. Derrick, "Symmetry properties of lossless diffraction gratings," J. Opt. 11, 43-52 (1980).
[CrossRef]

Ohira, Y.

Plumey, F.-P.

Staebler, D. L.

D. L. Staebler and J. J. Amodei, "Coupled-wave analysis of holographic storage in LiNbO3," J. Appl. Phys. 43, 1042-1049 (1972).
[CrossRef]

Uretsky, J. L.

J. L. Uretsky, "The scattering of plane waves from periodic surfaces," Ann. Phys. 33, 400-427 (1965).
[CrossRef]

Wyrowski, F.

V. Kettunen and F. Wyrowski, "Reflection mode phase retardation by dielectric gratings," Opt. Commun. 158, 41-44 (1998).
[CrossRef]

Ywata, K.

Ann. Phys.

J. L. Uretsky, "The scattering of plane waves from periodic surfaces," Ann. Phys. 33, 400-427 (1965).
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).

J. Appl. Phys.

D. L. Staebler and J. J. Amodei, "Coupled-wave analysis of holographic storage in LiNbO3," J. Appl. Phys. 43, 1042-1049 (1972).
[CrossRef]

J. Opt.

L. C. Botten, J. L. Adams, R. C. McPhedran, and G. H. Derrick, "Symmetry properties of lossless diffraction gratings," J. Opt. 11, 43-52 (1980).
[CrossRef]

Opt. Acta

L. C. Botten, "A new formalism for transmission gratings," Opt. Acta 25, 481-499 (1978).
[CrossRef]

Opt. Commun.

V. Kettunen and F. Wyrowski, "Reflection mode phase retardation by dielectric gratings," Opt. Commun. 158, 41-44 (1998).
[CrossRef]

Opt. Lett.

Other

J. W. Goodmam, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

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Figures (4)

Fig. 1
Fig. 1

Interference scheme between the transmitted diffracted orders at symmetrical Littrow mounting. The reflected diffracted orders are omitted for simplicity.

Fig. 2
Fig. 2

Phase difference between the minus first and the zeroth diffracted orders by transmission ( φ 1 T φ 0 T ) as a function of the grating depth for sinusoidal relief gratings in photoresist ( n = 1.645 ) for the TE polarization and λ = 457.9 nm and for three different grating periods 0.4, 0.6, and 0.8 μ m . In the same graphic (right scale) is shown the sum of the diffraction efficiencies of all other existing diffracted orders (excluding the minus first and zeroth).

Fig. 3
Fig. 3

Experimental results for the phase difference between the minus first and zeroth transmitted diffracted orders ( φ 1 T φ 0 T ) for a holographic surface relief photoresist grating of period 0.8 μ m with different grating depth. In the same figure are shown the theoretical phase difference between the first and zeroth transmitted diffracted orders ( φ 1 T φ 0 T ) as a function of the grating depth for three different grating profiles: lamellar (with filling factor = 0.5 ), sinusoidal, and triangular. The inset is the scanning electron micrograph of the cross section of the indicated sample.

Fig. 4
Fig. 4

Experimental measurements and corresponding theoretical curves for the phase difference between the minus first and zeroth diffracted orders ( φ 1 T φ 0 T ) by transmission and by reflection ( φ 1 R φ 0 R ) for a holographic surface relief photoresist grating ( n = 1.645 ) of period 0.4 μ m as a function of the grating depth for the TE polarization and λ = 457.9 nm .

Tables (3)

Tables Icon

Table 1 Diffraction Efficiencies, Phases, and Product Terms for a Sinusoidal Grating of Period 0.4 μ m and Depth 0.2 μ m

Tables Icon

Table 2 Diffraction Efficiencies, Phases, and Product Terms for a Sinusoidal Grating of Period 0.6 μ m and Depth 0.5 μ m

Tables Icon

Table 3 Diffraction Efficiencies, Phases, and Product Terms for a Sinusoidal Grating of Period 0.8 μ m and Depth 0.6 μ m

Equations (15)

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I i = C E i 2 = I r ( i ) + I s ( i 1 ) + 2 I r ( i ) I s ( i 1 ) cos ( ψ + φ r ( i ) φ s ( i 1 ) ) ,
i = N + 1 N [ I r ( i ) T + I s ( i 1 ) T + I r ( i ) T I s ( i 1 ) T cos ( ψ + φ r ( i ) T φ s ( i 1 ) T ) ] + i = M + 1 M [ I r ( i ) R + I s ( i 1 ) R + I r ( i ) R I s ( i 1 ) R cos ( ψ + φ r ( i ) R φ s ( i 1 ) R ) ] = I r + I s .
I r ( i ) R = η r ( i ) R I r ,
I r ( i ) T = η r ( i ) T I r ,
I s ( i ) R = η s ( i ) R I s ,
I s ( i ) T = η s ( i ) T I s .
i = N + 1 N η s ( i 1 ) T + i = M + 1 M η s ( i 1 ) R = 1 ,
i = N + 1 N η r ( i ) T + i = M + 1 M η r ( i 1 ) R = 1 .
i = M + 1 M η r ( i ) R η s ( i 1 ) R cos ( ψ + φ r ( i ) R φ s ( i 1 ) R ) + i = N + 1 N η r ( i ) T η s ( i 1 ) T cos ( ψ + φ r ( i ) T φ s ( i 1 ) T ) = 0 .
φ r ( i ) = φ s ( i ) ,
η r ( i ) = η s ( i ) .
i = 1 M η s ( i ) R η s ( i 1 ) R cos ( φ s ( i ) R φ s ( i 1 ) R ) + i = 1 N η s ( i ) T η s ( i 1 ) T cos ( φ s ( i ) T φ s ( i 1 ) T ) = 0 .
η 1 R η 0 R cos ( φ 1 R φ 0 R ) + η 1 T η 0 T cos ( φ 1 T φ 0 T ) = 0 .
φ 1 R φ 0 R = m + π 2 ,
φ 1 T φ 0 T = m + π 2 .

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