Abstract

This paper describes in a very easy and intelligible way, how the diffraction efficiencies of binary dielectric transmission gratings depend on the geometrical groove parameters and how a high efficiency can be obtained. The phenomenological explanation is based on the modal method. The mechanism of excitation of modes by the incident wave, their propagation constants and how they couple into the diffraction orders helps to understand the diffraction process of such gratings and enables a grating design without complicated numerical calculations.

© 2005 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw Hill, New York, 1968)
  2. A. Taflove, S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (rtech House, Norwood, 2000).
  3. A. V. Tishchenko, "Generalized source method: New possibilities for waveguide and grating problems," Opt. Quantum Electron. 32, 971-980 (2000)
    [CrossRef]
  4. J. Turunen, "Diffraction theory of dielectric surface relief gratings," in Micro-optics, H.P. Herzig ed. (Taylor&Francis Inc., 1997)
  5. M. G. Moharam, T. K. Gaylord, "Diffraction analysis of dielectric surface-relief gratings," J. Opt. Soc. Am. 72, 1385-1391 (1982)
    [CrossRef]
  6. R. E. Collin, "Reflection and transmission at a slotted dielectric interface," Canad. J. of Phys. 34, 398-411 (1956)
    [CrossRef]
  7. S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956)
  8. I. C. Botten, M.S. Craig, R. C. McPhedran, J. L. Adams, J.R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta. 28, 413-428 (1981)
    [CrossRef]
  9. A. V. Tishchenko, "Phenomenological representation of deep and high contrast lamellar gratings by means of the modal method," Opt. Quantum Electron. 37, 309-330 (2005)
    [CrossRef]
  10. T. Clausnitzer, J. Limpert, K. Zöllner, H. Zellmer, H.-J. Fuchs, E.-B. Kley, A. Tünnermann, M. Jupé, D. Ristau, "Highly-efficient transmission gratings in fused silica for chirped pulse amplification systems," Appl. Opt. 42, 6934-6938 (2003)
    [CrossRef] [PubMed]
  11. T. Clausnitzer, E.-B. Kley, H.-J. Fuchs, A. Tünnermann, "Highly efficient polarization independent transmission gratings for pulse stretching and compression," in Optical Fabrication, Testing and Metrology, R. Geyl, D. Rimmer, L. Wang, eds., Proc. SPIE 5252, 174-182 (2003)
    [CrossRef]
  12. P. Sheng, R.S. Stepleman, P.N. Sanda, "Exact eigenfunctions for square-wave gratings - Application to diffraction and surface-plasmon calculations," Phys. Rev. B 26, 2907-2916 (1982)
    [CrossRef]
  13. A. Yariv, "Coupled mode theory for guided-wave optics," IEEE J. Quantum Electron. 9, 919-933 (1973)
    [CrossRef]

Appl. Opt. (1)

Canad. J. of Phys. (1)

R. E. Collin, "Reflection and transmission at a slotted dielectric interface," Canad. J. of Phys. 34, 398-411 (1956)
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Yariv, "Coupled mode theory for guided-wave optics," IEEE J. Quantum Electron. 9, 919-933 (1973)
[CrossRef]

J. Opt. Soc. Am. (1)

Micro-optics 1997 (1)

J. Turunen, "Diffraction theory of dielectric surface relief gratings," in Micro-optics, H.P. Herzig ed. (Taylor&Francis Inc., 1997)

Opt. Acta. (1)

I. C. Botten, M.S. Craig, R. C. McPhedran, J. L. Adams, J.R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta. 28, 413-428 (1981)
[CrossRef]

Opt. Quantum Electron. (2)

A. V. Tishchenko, "Phenomenological representation of deep and high contrast lamellar gratings by means of the modal method," Opt. Quantum Electron. 37, 309-330 (2005)
[CrossRef]

A. V. Tishchenko, "Generalized source method: New possibilities for waveguide and grating problems," Opt. Quantum Electron. 32, 971-980 (2000)
[CrossRef]

Phys. Rev. B (1)

P. Sheng, R.S. Stepleman, P.N. Sanda, "Exact eigenfunctions for square-wave gratings - Application to diffraction and surface-plasmon calculations," Phys. Rev. B 26, 2907-2916 (1982)
[CrossRef]

Proc. SPIE (1)

T. Clausnitzer, E.-B. Kley, H.-J. Fuchs, A. Tünnermann, "Highly efficient polarization independent transmission gratings for pulse stretching and compression," in Optical Fabrication, Testing and Metrology, R. Geyl, D. Rimmer, L. Wang, eds., Proc. SPIE 5252, 174-182 (2003)
[CrossRef]

Sov. Phys. JETP (1)

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956)

Other (2)

J. W. Goodman, Introduction to Fourier Optics (McGraw Hill, New York, 1968)

A. Taflove, S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (rtech House, Norwood, 2000).

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Figures (8)

Fig. 1.
Fig. 1.

Geometry of the diffraction problem

Fig. 2.
Fig. 2.

Eigenvalue-relation F(neff2) for d = 800nm

Fig. 3.
Fig. 3.

Effective indices of the first two modes as a function of the fill factor

Fig. 4.
Fig. 4.

Amplitudes of the first four TE modal fields in the grating region

Fig. 5.
Fig. 5.

Analogy to a Mach-Zehnder-Interferometer. The two grating modes propagate through different optical paths; the intensity in port 1 or 2 are determined by their phase difference.

Fig. 6.
Fig. 6.

Numerically calculated diffraction efficiency of the 0th and the -1st order as a function of the groove depth.

Fig. 7(a).
Fig. 7(a).

Groove depth to accumulate a phase difference of π, 3π and 5π

Fig. 7(b).
Fig. 7(b).

Rigorously calculated diffraction efficiency from ref. [10]

Equations (21)

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sin φ m = sin φ in + m λ d
E y in ( x , z ) ~ e i k 0 ( x sin φ in + z cos φ in )
E y gr ( x , z ) = u ( x ) v ( z )
( 2 z 2 + k z 2 ) v ( z ) = 0
( 2 x 2 + k i , x 2 ) u ( x ) = 0
u i ( x ) = A cos k i , x x + B sin k i , x x .
cos α d = F ( n eff 2 )
F ( n eff 2 ) = cos β b · cos γ g β 2 + γ 2 2 β γ sin βb · sin γg
α = k x in = k 0 sin φ in is the x component of the k vector of the incident wave ,
β = k b , x = k 0 n b 2 n eff 2 , n b 2 = ε b is the same in the ridges
γ = k g , x = k 0 n g 2 n eff 2 , n g 2 = ε g the same in the grooves.
v m ( z ) = Ce i k 0 n eff z + De i k 0 n eff z
v m ( z ) = Ce i k 0 n eff z ,
u m ( x + d ) = u m ( x ) e i α d ,
E y in ( x , 0 ) u m ( x ) = E y in ( x , 0 ) · u m ( x ) dx 2 E y in ( x , 0 ) 2 dx · u m ( x ) 2 dx ,
n eff air = cos φ in = 0.749
n eff substrate = n · cos φ ˜ = 1.290 ,
E y in ( x , 0 ) ~ e i ( k 0 sin φ in · x ) = e i ( α · x ) = sin ( α x ) + i cos ( αx ) ,
E y in ( x , 0 ) ~ e i ( π d x ) = sin ( π d x ) + i cos ( π d x ) ,
h = λ 2 n eff 0 n eff 1
F ( n eff 2 ) = cos β b · cos γg ε g 2 β 2 + ε b 2 γ 2 2 ε b ε g β γ sin βb · sin γg

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