Abstract

An analytic theory describing soft x-ray diffraction by Lamellar Multilayer Gratings (LMG) has been developed. The theory is derived from a coupled waves approach for LMGs operating in the single-order regime, where an incident plane wave can only excite a single diffraction order. The results from calculations based on these very simple analytic expressions are demonstrated to be in excellent agreement with those obtained using the rigorous coupled-waves approach. The conditions for maximum reflectivity and diffraction efficiency are deduced and discussed. A brief investigation into p-polarized radiation diffraction is also performed.

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  1. I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of XUV multilayer optics,” Phys. Scr. T17, 137–145 (1987).
    [CrossRef]
  2. A. Sammar, J.-M. André, and B. Pardo, “Diffraction and scattering by lamellar amplitude multilayer gratings in the X-UV region,” Opt. Commun. 86(2), 245–254 (1991).
    [CrossRef]
  3. A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. A 333(2-3), 599–606 (1993).
    [CrossRef]
  4. A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. Andre, R. Rivoira, C. K. Malek, F. R. Ladan, and P. Guerin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating,” J. Opt. 24(1), 37–41 (1993).
    [CrossRef]
  5. R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Meth. Res. A 541(3), 590–597 (2005).
    [CrossRef]
  6. I. V. Kozhevnikov, R. van der Meer, H. M. J. Bastiaens, K.-J. Boller, and F. Bijkerk, “High-resolution, high-reflectivity operation of lamellar multilayer amplitude gratings: identification of the single-order regime,” Opt. Express 18(15), 16234–16242 (2010).
    [CrossRef] [PubMed]
  7. R. Benbalagh, “Monochromateurs Multicouches à bande passante étroite et à faible fond continu pour le rayonnement X-UV,” PhD Thesis, University of Paris VI, Paris, 2003.
  8. L. I. Goray, “Numerical analysis of the efficiency of multilayer-coated gratings using integral method,” Nucl. Instrum. Meth. Res. A 536(1-2), 211–221 (2005).
    [CrossRef]
  9. R. Petit, Electromagnetic Theory of Gratings, (Springer-Verlag, 1980.
  10. A. V. Vinogradov and B. Ya. Zeldovich, “X-ray and far uv multilayer mirrors: principles and possibilities,” Appl. Opt. 16(1), 89–93 (1977).
    [CrossRef] [PubMed]
  11. A. V. Vinogradov and B. Ya. Zeldovich, “Multilayer mirrors for x-ray and far-ultraviolet radiation,” Opt. Spektrosk. 42, 404–407 (1977).
  12. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980.

2010

2005

L. I. Goray, “Numerical analysis of the efficiency of multilayer-coated gratings using integral method,” Nucl. Instrum. Meth. Res. A 536(1-2), 211–221 (2005).
[CrossRef]

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Meth. Res. A 541(3), 590–597 (2005).
[CrossRef]

1993

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. A 333(2-3), 599–606 (1993).
[CrossRef]

A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. Andre, R. Rivoira, C. K. Malek, F. R. Ladan, and P. Guerin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating,” J. Opt. 24(1), 37–41 (1993).
[CrossRef]

1991

A. Sammar, J.-M. André, and B. Pardo, “Diffraction and scattering by lamellar amplitude multilayer gratings in the X-UV region,” Opt. Commun. 86(2), 245–254 (1991).
[CrossRef]

1987

I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of XUV multilayer optics,” Phys. Scr. T17, 137–145 (1987).
[CrossRef]

1977

A. V. Vinogradov and B. Ya. Zeldovich, “X-ray and far uv multilayer mirrors: principles and possibilities,” Appl. Opt. 16(1), 89–93 (1977).
[CrossRef] [PubMed]

A. V. Vinogradov and B. Ya. Zeldovich, “Multilayer mirrors for x-ray and far-ultraviolet radiation,” Opt. Spektrosk. 42, 404–407 (1977).

Agafonov, Yu. A.

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. A 333(2-3), 599–606 (1993).
[CrossRef]

Andre, J.-M.

A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. Andre, R. Rivoira, C. K. Malek, F. R. Ladan, and P. Guerin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating,” J. Opt. 24(1), 37–41 (1993).
[CrossRef]

André, J.-M.

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Meth. Res. A 541(3), 590–597 (2005).
[CrossRef]

A. Sammar, J.-M. André, and B. Pardo, “Diffraction and scattering by lamellar amplitude multilayer gratings in the X-UV region,” Opt. Commun. 86(2), 245–254 (1991).
[CrossRef]

Barchewitz, R.

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Meth. Res. A 541(3), 590–597 (2005).
[CrossRef]

A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. Andre, R. Rivoira, C. K. Malek, F. R. Ladan, and P. Guerin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating,” J. Opt. 24(1), 37–41 (1993).
[CrossRef]

Bastiaens, H. M. J.

Benbalagh, R.

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Meth. Res. A 541(3), 590–597 (2005).
[CrossRef]

Bijkerk, F.

Boller, K.-J.

Brunel, M.

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. A 333(2-3), 599–606 (1993).
[CrossRef]

Erko, A. I.

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. A 333(2-3), 599–606 (1993).
[CrossRef]

Filatova, E. O.

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Meth. Res. A 541(3), 590–597 (2005).
[CrossRef]

Goray, L. I.

L. I. Goray, “Numerical analysis of the efficiency of multilayer-coated gratings using integral method,” Nucl. Instrum. Meth. Res. A 536(1-2), 211–221 (2005).
[CrossRef]

Guerin, P.

A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. Andre, R. Rivoira, C. K. Malek, F. R. Ladan, and P. Guerin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating,” J. Opt. 24(1), 37–41 (1993).
[CrossRef]

Jonnard, P.

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Meth. Res. A 541(3), 590–597 (2005).
[CrossRef]

Julié, G.

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Meth. Res. A 541(3), 590–597 (2005).
[CrossRef]

Kozhevnikov, I. V.

Ladan, F. R.

A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. Andre, R. Rivoira, C. K. Malek, F. R. Ladan, and P. Guerin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating,” J. Opt. 24(1), 37–41 (1993).
[CrossRef]

Malek, C. K.

A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. Andre, R. Rivoira, C. K. Malek, F. R. Ladan, and P. Guerin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating,” J. Opt. 24(1), 37–41 (1993).
[CrossRef]

Marmoret, R.

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Meth. Res. A 541(3), 590–597 (2005).
[CrossRef]

Martynov, V. V.

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. A 333(2-3), 599–606 (1993).
[CrossRef]

Mollard, L.

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Meth. Res. A 541(3), 590–597 (2005).
[CrossRef]

Ouahabi, M.

A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. Andre, R. Rivoira, C. K. Malek, F. R. Ladan, and P. Guerin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating,” J. Opt. 24(1), 37–41 (1993).
[CrossRef]

Pardo, B.

A. Sammar, J.-M. André, and B. Pardo, “Diffraction and scattering by lamellar amplitude multilayer gratings in the X-UV region,” Opt. Commun. 86(2), 245–254 (1991).
[CrossRef]

Rémond, C.

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Meth. Res. A 541(3), 590–597 (2005).
[CrossRef]

Rivoira, R.

A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. Andre, R. Rivoira, C. K. Malek, F. R. Ladan, and P. Guerin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating,” J. Opt. 24(1), 37–41 (1993).
[CrossRef]

Rolland, G.

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Meth. Res. A 541(3), 590–597 (2005).
[CrossRef]

Roschupkin, D. V.

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. A 333(2-3), 599–606 (1993).
[CrossRef]

Sammar, A.

A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. Andre, R. Rivoira, C. K. Malek, F. R. Ladan, and P. Guerin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating,” J. Opt. 24(1), 37–41 (1993).
[CrossRef]

A. Sammar, J.-M. André, and B. Pardo, “Diffraction and scattering by lamellar amplitude multilayer gratings in the X-UV region,” Opt. Commun. 86(2), 245–254 (1991).
[CrossRef]

Troussel, P.

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Meth. Res. A 541(3), 590–597 (2005).
[CrossRef]

van der Meer, R.

Vidal, B.

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. A 333(2-3), 599–606 (1993).
[CrossRef]

Vincent, P.

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. A 333(2-3), 599–606 (1993).
[CrossRef]

Vinogradov, A. V.

I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of XUV multilayer optics,” Phys. Scr. T17, 137–145 (1987).
[CrossRef]

A. V. Vinogradov and B. Ya. Zeldovich, “X-ray and far uv multilayer mirrors: principles and possibilities,” Appl. Opt. 16(1), 89–93 (1977).
[CrossRef] [PubMed]

A. V. Vinogradov and B. Ya. Zeldovich, “Multilayer mirrors for x-ray and far-ultraviolet radiation,” Opt. Spektrosk. 42, 404–407 (1977).

Zeldovich, B. Ya.

A. V. Vinogradov and B. Ya. Zeldovich, “Multilayer mirrors for x-ray and far-ultraviolet radiation,” Opt. Spektrosk. 42, 404–407 (1977).

A. V. Vinogradov and B. Ya. Zeldovich, “X-ray and far uv multilayer mirrors: principles and possibilities,” Appl. Opt. 16(1), 89–93 (1977).
[CrossRef] [PubMed]

Appl. Opt.

J. Opt.

A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. Andre, R. Rivoira, C. K. Malek, F. R. Ladan, and P. Guerin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating,” J. Opt. 24(1), 37–41 (1993).
[CrossRef]

Nucl. Instrum. Meth. Res. A

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Meth. Res. A 541(3), 590–597 (2005).
[CrossRef]

L. I. Goray, “Numerical analysis of the efficiency of multilayer-coated gratings using integral method,” Nucl. Instrum. Meth. Res. A 536(1-2), 211–221 (2005).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. A

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. A 333(2-3), 599–606 (1993).
[CrossRef]

Opt. Commun.

A. Sammar, J.-M. André, and B. Pardo, “Diffraction and scattering by lamellar amplitude multilayer gratings in the X-UV region,” Opt. Commun. 86(2), 245–254 (1991).
[CrossRef]

Opt. Express

Opt. Spektrosk.

A. V. Vinogradov and B. Ya. Zeldovich, “Multilayer mirrors for x-ray and far-ultraviolet radiation,” Opt. Spektrosk. 42, 404–407 (1977).

Phys. Scr.

I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of XUV multilayer optics,” Phys. Scr. T17, 137–145 (1987).
[CrossRef]

Other

R. Petit, Electromagnetic Theory of Gratings, (Springer-Verlag, 1980.

R. Benbalagh, “Monochromateurs Multicouches à bande passante étroite et à faible fond continu pour le rayonnement X-UV,” PhD Thesis, University of Paris VI, Paris, 2003.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980.

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Figures (5)

Fig. 1
Fig. 1

Schematic of the cross section of an LMG. (a): An incident beam from the left (In), under grazing angle θ0, is reflected from the multilayer and diffracted into multiple orders (Out) by the grating structure. The multilayer is built up from N bi-layers with thickness d. Each bi-layer consists of an absorber material (A) with thickness γd and a spacer material (S) with thickness (1-γ)d. The grating structure of the LMG is defined by the grating period D and lamel width ΓD. (b): The normalized function U(x) is used to describe the lamellar profile.

Fig. 2
Fig. 2

Reflectivity R0 (at E = 183.4 eV) of conventional Mo/B4C multilayer mirror (d = 6 nm, γ = 0.34, N = 50) versus grazing angle. Calculations were performed with the use of exact algorithm (curve 1, red), simple analytic expressions (15) and (16) deduced in the present paper (curve 2, blue), and formulas obtained in Ref. [10,11]. (curve 3, green).

Fig. 3
Fig. 3

Reflectivity (at E = 183.4 eV) versus the grazing angle of an incident beam for a conventional Mo/B4C multilayer mirror (Γ = 1) and for LMGs with different Γ-ratios: Γ = 1/2, 1/3, and 1/10. The multilayer parameters are the same as for Fig. 2, with the exception that the number of bi-layers increases with Γ according to N = 100/ Γ. The lamel width ΓD is fixed at 70nm and so the grating period increases through the relation D = 70 nm/ Γ. The colored curves were calculated with the use of rigorous coupled waves approach (Eq. (7)), where 5 diffraction orders were taken into account, while black dashed curves were calculated via simple analytic expressions (Eqs. (15) and (16)).

Fig. 4
Fig. 4

Diffraction efficiency Rn (for n = −5,...,0) at E = 183.4 eV versus grazing angle of the incident beam. Parameters of the LMG: Mo/B4C multilayer structure (d = 6 nm, γ = 0.34, N = 300), D = 210 nm, Γ = 1/3. The colored curves were calculated using the rigorous coupled waves approach, where 9 diffraction orders (from + 2nd to −6th) were taken into account. The black dashed curves were calculated via the simple analytical expressions (15) and (22). The agreement is excellent, except for the −3rd order (blue curve).

Fig. 5
Fig. 5

1st order diffraction efficiency at E = 183.4 eV versus grazing angle of incident beam and for different values of Γ. The lamellar width is fixed ΓD = 70 nm and the number of bi-layers is chosen as N = 100/Γ. The remaining parameters of the Mo/B4C multilayer structure are the same as in Fig. 4. The colored curves were calculated using the rigorous coupled waves approach, where 5 diffraction orders were taken into account. The black dashed curves were calculated via the simple analytical expressions (15) and (22).

Equations (29)

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ε ( x , z < 0 ) = 1 ;     ε ( x , 0 z L ) = 1 χ ( z ) U ( x ; Γ , D ) ;     ε ( x ,   z > L ) = ε s u b
U ( x ; Γ , D ) = n = + U n exp ( 2 i π n x / D ) ,     U 0 = Γ ,     U n 0 = [ 1 exp ( 2 i π n Γ ) ] / ( 2 i π n )
E ( x , z ) = n = + F n ( z ) exp ( i q n x ) ; q n = q 0 + 2 π n D ; q 0 = k cos θ 0 ; k = 2 π λ
        d 2 F n ( z ) d z 2 + κ n 2 F n ( z ) = k 2 χ ( z ) m U n m F m ( z ) ; F n ( 0 ) + i κ n F n ( 0 ) = 2 i κ n δ n , 0 ; F n ( L ) i κ n ( s ) F n ( L ) = 0
F n ( z ) = A n ( z ) exp ( i κ n z ) + C n ( z ) exp ( i κ n z )
A n ( z ) exp ( i κ n z ) + C n ( z ) exp ( i κ n z ) = 0
{ d A n ( z ) d z = i k 2 2 κ n χ ( z ) m = + U n m [ A m ( z ) e i ( κ m κ n ) z + C m ( z ) e i ( κ m + κ n ) z ] d C n ( z ) d z = + i k 2 2 κ n χ ( z ) m = + U n m [ A m ( z ) e i ( κ m + κ n ) z + C m ( z ) e i ( κ m κ n ) z ]
A n ( 0 ) = δ n , 0 ;     C n ( L ) = 0
Γ D Δ θ M M < < d
{ d A 0 ( z ) d z = i k 2 2 κ 0 Γ χ ( z ) [ A 0 ( z ) + C 0 ( z ) e 2 i κ 0 z ] d C 0 ( z ) d z = + i k 2 2 κ 0 Γ χ ( z ) [ A 0 ( z ) e 2 i κ 0 z + C 0 ( z ) ]
χ ( z ) = χ S + ( χ A χ S ) u ( z )
u ( z ; γ , d ) = n = + u n exp ( 2 i π n z d ) ,     u 0 = γ ,     u n 0 = 1 2 i π n [ 1 exp ( 2 i π n γ ) ]
{ d A 0 ( z ) d z + i k 2 2 κ 0 [ χ ¯ Γ A 0 ( z ) + ( χ A χ S ) u j Γ C 0 ( z ) e 2 i ( π j / d κ 0 ) z ] = Δ A ( z ) d C 0 ( z ) d z i k 2 2 κ 0 [ χ ¯ Γ C 0 ( z ) + ( χ A χ S ) u j Γ A 0 ( z ) e 2 i ( π j / d κ 0 ) z ] = Δ C ( z )
{ d a 0 ( z ) d z + i ( π j d κ 0 + k 2 2 κ 0 χ ¯ Γ ) a 0 ( z ) + i k 2 2 κ 0 ( χ A χ S ) Γ u j c 0 ( z ) = 0 d c 0 ( z ) d z i ( π j d κ 0 + k 2 2 κ 0 χ ¯ Γ ) c 0 ( z ) i k 2 2 κ 0 ( χ A χ S ) Γ u j a 0 ( z ) = 0
R 0 = | B tanh ( S N d ) b tanh ( S N d ) i B + B b 2 | 2
b = χ ¯ Γ + 2 sin θ 0 ( j λ 2 d sin θ 0 ) ;     B ± = ( χ A χ S ) u ± j Γ ;       S = k 2 sin θ 0 B + B b 2
j λ 2 d = sin θ 0 Γ 2 sin θ 0 [ Re χ ¯ Re ( χ A χ S ) Im ( χ A χ S ) Im χ ¯ sin 2 ( π j γ ) ( π j ) 2 ]
R 0 p e a k = 1 w 1 + w ; w = 1 y 2 1 + f 2 y 2 ; f = Re ( χ A χ S ) Im ( χ A χ S ) ; y = Im ( χ A χ S ) Im χ ¯ sin ( π j γ ) π j
L M S ~ 1 S = λ sin θ 0 π Γ Im χ ¯ ( 1 y 2 ) ( 1 + f 2 y 2 )
tan ( π j γ ) = π j [ γ + Im χ S / Im ( χ A χ S ) ]
{ d a 0 ( z ) d z + i ( π j d κ 0 + κ m 2 + k 2 2 κ 0 χ ¯ Γ ) a 0 ( z ) + i k 2 2 κ 0 ( χ A χ S ) u j U m c m ( z ) = 0 d c m ( z ) d z i ( π j d κ 0 + κ m 2 + k 2 2 κ m χ ¯ Γ ) c m ( z ) i k 2 2 κ m ( χ A χ S ) u j U m a 0 ( z ) = 0
B ± = ( χ A χ S ) u ± j U m ;     S = k 2 sin θ 0 sin θ m B + B b 2 ; b = sin θ 0 + sin θ m 2 sin θ 0 sin θ m χ ¯ Γ + 2 sin θ 0 sin θ m ( j λ 2 d sin θ 0 + sin θ m 2 )
j λ 2 d = sin θ 0 + sin θ m 2 sin θ 0 + sin θ m 4 sin θ 0 sin θ m Γ Re χ ¯ +                          + Re ( χ A χ S ) sin θ 0 + sin θ m Im ( χ A χ S ) Im χ ¯ sin 2 ( π j γ ) ( π j ) 2 sin 2 ( π m Γ ) ( π m ) 2 Γ
y = Im ( χ A χ S ) Im χ ¯ sin ( π j γ ) π j 2 sin θ 0 sin θ m sin θ 0 + sin θ m sin ( π m Γ ) π m Γ
ε ( x , z ) = 1 χ ¯ m U m e 2 i π m x / D ( χ A χ S ) m j 0 U m u j e 2 i π ( m x / D + j z / d )
ε ( x , z ) 1 Γ χ ¯ ( χ A χ S ) U m u j e 2 i π ( m x / D + j z / d )
ε ˜ ( x , z ) ε ( x , z ) + 1 2 k 2 2 ε ( x , z ) ε ( x , z ) 3 4 k 2 [ ε ( x , z ) ] 2 ε 2 ( x , z )
ε ˜ ( x , z ) = 1 χ ¯ m U m e 2 i π m x / D [ 1 1 2 ( m λ D ) 2 ]                   ( χ A χ S ) m j 0 U m u j e 2 i π ( m x / D + j z / d ) [ 1 1 2 ( j λ d ) 2 1 2 ( m λ D ) 2 ]
ε ( x , z ) 1 Γ χ ¯ ( χ A χ S ) U m u j e 2 i π ( m x / D + j z / d ) [ 1 1 2 ( j λ d ) 2 1 2 ( m λ D ) 2 ]                            1 Γ χ ¯ ( χ A χ S ) U m u j e 2 i π ( m x / D + j z / d ) cos ( θ 0 + θ m )

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