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Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 26 — Dec. 26, 2005
  • pp: 10448–10456
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An intelligible explanation of highly-efficient diffraction in deep dielectric rectangular transmission gratings

T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, U. Peschel, A.V. Tishchenko, and O. Parriaux  »View Author Affiliations


Optics Express, Vol. 13, Issue 26, pp. 10448-10456 (2005)
http://dx.doi.org/10.1364/OPEX.13.010448


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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

1. Introduction

Fig. 1. Geometry of the diffraction problem

2. Essentials of the modal method

If a dielectric transmission grating is illuminated by a plane wave as it is shown in Fig. 1, the reflected and the transmitted field can be described as a sum of plane waves: the diffraction orders. The propagation directions of these orders are given by the well-known grating equation

sinφm=sinφin+mλd
(1)

A wave incident upon a grating, or periodic waveguide, also excites discrete modes comparable to the simple case of a slab waveguide. The efficiency of this excitation, the propagation of the modes through the grating region, and how they couple to the diffraction orders of the grating determine the optical properties of the grating. The propagation constants or effective indices of these modes can be derived according to the following procedure (for a more detailed description see [12

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

]):

Let us assume a TE-polarized incident wave with the electric field vector perpendicular to the plane of incidence. The y-invariant electric field Einy writes:

Eyin(x,z)~eik0(xsinφin+zcosφin)
(2)

with the vacuum wave number k0 = 2π/λ. The geometry of the problem permits the separation of the sole electric field component Eygr (x, z) in the grating region into two parts

Eygr(x,z)=u(x)v(z)
(3)

The field inside the grating grooves and the ridges fulfills the Helmholtz-equation for homogeneous media, which is for the z-part:

(2z2+kz2)v(z)=0
(4a)

and for the x-part:

(2x2+ki,x2)u(x)=0
(4b)

where ki,x2=ki2kz2andi={bintheridgeginthegroove ki = nik0 is the wave number in the grooves (in case of air: ng = 1) and ridges, respectively. The propagation constant kz = k 0 neff is equal in both media. Eq. (4b) can be solved in grooves and ridges separately by

ui(x)=Acoski,xx+Bsinki,xx.
(5)

The field continuity conditions at the boundaries between the ridges and grooves lead to a transcendent equation for kz or rather the effective index neff , which can be written in the case of TE-polarized illumination as

cosαd=F(neff2)
(6)

with

F(neff2)=cosβb·cosγgβ2+γ22βγsinβb·sinγg
(7)

where

α=kxin=k0sinφinisthexcomponentofthekvectoroftheincidentwave,
β=kb,x=k0nb2neff2,nb2=εbisthesameintheridges
γ=kg,x=k0ng2neff2,ng2=εgthesameinthegrooves.

vm(z)=Ceik0neffz+Deik0neffz
(8)

If the reflectivity of mode m at the grating-substrate and the grating-air interface is low, the upward propagating part and every effect caused by multiple reflections can be neglected, then vm(z) can be expressed by

vm(z)=Ceik0neffz,
(9)

where C describes the part of the power of the incident wave that has been coupled to the mode. Modes with neff2 < 0 are evanescent, and those with neff2 > 0 propagate along the z-direction. The effective index of the evanescent modes determines how fast their amplitude decreases with increasing groove depth (z-dependence). In cases of shallow gratings, these modes cannot be neglected, whereas they will play a minor role for deep grooves. The diffraction properties of deep gratings are therefore mainly determined by the propagating modes. The x-dependent amplitude um (x) of the mth mode can be calculated by inserting the respective effective index into Eq. (5) and matching the amplitude at the groove-ridge boundaries. According to the Floquet-Bloch Theorem every um (x) fulfills the condition of pseudo-periodicity

um(x+d)=um(x)eiαd,
(10)

where d is the grating period and α=2πλsinφin. In the case of normal incidence (α = 0), um(x) repeats itself every period, and for Littrow-mounting (α= π/d), every two periods. The efficiency of excitation of these modes by the incident wave is, analogous to the waveguide theory, determined by the overlap integral between the field of the incident wave and mode m at the air-grating boundary [13

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

]

Eyin(x,0)um(x)=Eyin(x,0)·um(x)dx2Eyin(x,0)2dx·um(x)2dx,
(11)

as well as by their Fresnel-like transmission coefficient at this interface, which is determined by the change of effective index (“impedance matching”). After propagation through the grating region, the modes are partly reflected at the grating-substrate interface and partly transmitted into the substrate. By doing this, every mode distributes its energy to all possible diffraction orders specified in Eq. (1). The coupling efficiency is again defined by the overlap of the fields and the transmission at this interface caused by the change of propagation constants. The contributions of all grating modes interfere and determine the intensity of one diffraction order.

Fig. 2. Eigenvalue-relation F(neff2) for d = 800nm

3. Modal Description applied to highly efficient gratings in fused silica

neffair=cosφin=0.749

and the orders transmitted into the substrate

neffsubstrate=n·cosφ˜=1.290,

where φ˜ is the refracted angle in the substrate and n its refractive index. For small fill factors, the modes approach the effective index of air. For large fill factors, they approach the effective index of the substrate material. If the fill factor was 0.45, the two modes would propagate with neff0 = 1.210 and neff1 = 0.864.

Fig. 3. Effective indices of the first two modes as a function of the fill factor

As mentioned above, the excitation efficiency of these modes is determined by the field overlap between the incident wave and the field of the specific mode at the boundary between air and the grating region. According to Eq. (2), the field of the incident wave is given by

Eyin(x,0)~ei(k0sinφin·x)=ei(α·x)=sin(αx)+icos(αx),
(12)

which for Littrow-mounting (α= π/d) is

Eyin(x,0)~ei(πdx)=sin(πdx)+icos(πdx),
(13)

It consists of a “sine-part” and a “cosine-part” with a periodicity of 2d. The field amplitudes of the first four grating modes are illustrated in Fig. 4. Since the x-dependent part of the Helmholtz-operator defined by Eq. (4b) is linear and self-adjoint, its eigenfunctions and therefore the fields of the modes are either odd or even functions. The periodicity of the modes is determined by Eq. (10), assuming that α is π/d. The modes repeat their amplitudes after two grating periods and whole-numbered multiples of 2d. The field of the 0th grating mode in Fig. 4(a) looks very similar to a cosine function with respect to the definition of the coordinate system in Fig. 1. On the other hand, the amplitude of the first mode (Fig. 4(b)) nearly follows a sine-function. Their periodicity is the same as that of the incident wave, so it is obvious that the incident wave excites these two propagating modes very well. Higher order modes, though they exist, are hardly excited. The exact values of the overlap are 0.4905 and 0.4994 for the propagating modes and 0.0054 and 0.0046 for the two illustrated evanescent modes (Fig. 4(c), 4(d)).

Fig. 4. Amplitudes of the first four TE modal fields in the grating region
(a) mode 0 (propagating) dashed line: cosine function
(b) mode 1 (propagating) dashed line: sine function
(c) mode 2 (evanescent)
(d) mode 3 (evanescent)

These two excited modes, each of which carries nearly half of the energy of the incident wave, propagate along the z-direction. Since their effective indices are different, they accumulate a phase difference. In the case of a fused silica grating, the reflection of the two modes at the grating-substrate interface is low, so most of their energy is coupled out to the transmitted diffraction orders defined in Eq. (1). For Littrow-mounting, the two propagating diffraction orders (the 0th and the -1st order) have symmetrical field distributions. So the coupling efficiency between a grating mode and each one of the two diffraction orders is the same as the coupling between the incident wave and the modes. The intensity in one of the two orders is determined by the interference of the corresponding parts of both modes.

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

This diffraction process is very similar to a Mach-Zehnder Interferometer (Fig. 5). In such an interferometer the incident wave is split into two equal parts that afterwards propagate through different media or different optical paths. After passing the second beam splitter, the light leaves the interferometer either through port 1 or 2 or partly through both of them. The intensity in one port is determined by the interference of the two beams, so it changes sinusoidally with the path difference. This is analogous to the propagation of the two grating modes with different effective indices. If the difference of the optical paths of the two modes is zero, for example if the groove depth is zero, then all light propagates in the 0th transmitted order (the light does not see a grating). For increasing path differences, the intensity of the 0th order decreases continuously until the two modes have accumulated a phase difference of π (resp. a path length difference of λ/2). In this case, all light is diffracted into the -1st order. Since the diffraction is based on a two-beam interference mechanism the diffraction efficiency changes sinusoidally when the groove depth is increased. Fig. 6 shows the numerical calculation of this correlation for a fill factor of 0.5. Both graphs indeed show a nearly sinusoidal trend. Small deviations are caused by the reflection effects, which have been neglected here.

The complex diffraction process is therefore reduced to a two-beam interference of two grating modes that propagate along the z-direction with different effective indices. For the given grating period, fused silica grating ridges and Littrow-mounting, the values of the effective indices are only affected by the fill factor of the grating. The accumulated phase difference and therefore the diffraction efficiency of the 0th or -1st order is determined by the propagation distance, and therefore the groove depth of the grating. A grating is highly efficient if the two modes carry nearly the same energy and interfere at the grating-substrate interface with a phase difference of an odd-numbered multiple of π. The groove depth h, which is necessary to accumulate this phase shift, depends on the difference of the effective index of both modes. According to Eq. (9), a phase shift of π is achieved at:

h=λ2neff0neff1
(14)

Figure 7(a) shows this groove depth and its odd-numbered multiples as a function of the fill factor. For a fill factor of 0.45 a phase shift of π is reached at a groove depth of 1.55μm, which is pretty near to the value found in [10

10 . T. Clausnitzer , J. Limpert , K. ZÖllner , H. Zellmer , H.-J. Fuchs , E.-B. Kley , A. Tünnermann , M. Jupé , and D. Ristau , “ Highly-efficient transmission gratings in fused silica for chirped pulse amplification systems ,” Appl. Opt. 42 , 6934 – 6938 ( 2003 ) [CrossRef] [PubMed]

]. Figure 7(a) furthermore illustrates, that it is possible to assign a groove depth to every fill factor. So if the fill factor of a fabricated grating deviates from the desired value, it can be compensated by the proper groove depth. Figure7(b) shows the rigorously calculated diffraction efficiency as a function of the profile parameters, as it was shown in [10

10 . T. Clausnitzer , J. Limpert , K. ZÖllner , H. Zellmer , H.-J. Fuchs , E.-B. Kley , A. Tünnermann , M. Jupé , and D. Ristau , “ Highly-efficient transmission gratings in fused silica for chirped pulse amplification systems ,” Appl. Opt. 42 , 6934 – 6938 ( 2003 ) [CrossRef] [PubMed]

]. The light areas illustrate for every fill factor, which groove depth is necessary for a high diffraction efficiency. The comparison of Fig. 7(a) and 7(b) show that the model presented here makes impressively good predictions of the performance of a fused silica grating. However, the calculation of the groove depth using the effective indices does not regard the amplitude condition explained above and the reflection, which also depends on the fill factor. So to find the global maximum of the diffraction efficiency, these effects also have to be considered.

Fig. 7(a). Groove depth to accumulate a phase difference of π, 3π and 5π
Fig. 7(b). Rigorously calculated diffraction efficiency from ref. [10]

4. Some remarks on unpolarized light

The formulas given in section 2 are only valid for TE-polarized light. However, the derivation for TM-polarized light can be made analogous, by using the sole magnetic component Hy . The eigenvalue equation is then in the form

F(neff2)=cosβb·cosγgεg2β2+εb2γ22εbεgβγsinβb·sinγg
(15)

5. Summary

Acknowledgments

This work was funded by the German Research Association (DFG) within the Sonderforschungsbereich SFB / TR7.

References and links

1 .

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

2 .

A. Taflove and 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 and 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 , and 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 , and 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 , and A. Tünnermann , “ Highly efficient polarization independent transmission gratings for pulse stretching and compression ,” in Optical Fabrication, Testing and Metrology , R. Geyl , D. Rimmer , and L. Wang , eds., Proc. SPIE 5252 , 174 – 182 ( 2003 ) [CrossRef]

12 .

P. Sheng , R.S. Stepleman , and 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]

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(050.1960) Diffraction and gratings : Diffraction theory

ToC Category:
Research Papers

Citation
T. Clausnitzer, T. Kämpfe, E.-B. Kley, A. Tünnermann, U. Peschel, A. V. Tishchenko, and O. Parriaux, "An intelligible explanation of highly-efficient diffraction in deep dielectric rectangular transmission gratings," Opt. Express 13, 10448-10456 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-26-10448


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References

  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]

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