Abstract

Effects in composite volume Bragg gratings (VBGs) are studied theoretically and experimentally. The mathematics of reflection is formulated with a unified account of Fresnel reflections by the boundaries and of VBG reflection. We introduce the strength S of reflection by an arbitrary lossless element such that the intensity of reflection is R=tanh2S. We show that the ultimate maximum/minimum of reflection by a composite lossless system corresponds to addition/subtraction of relevant strengths of the sequential elements. We present a new physical interpretation of standard Fresnel reflection: Strength for TE or for TM reflection is given by addition or by subtraction of two contributions. One of them is an angle-independent contribution of the impedance step, while the other is an angle-dependent contribution of the step of propagation speed. We study an assembly of two VBG mirrors with a thin immersion layer between them that constitutes a Fabry–Perot spectral filter. The transmission wavelength of the assembly depends on the phase shift between the two VBGs. Spectral resolution Δλ(FWHM)=25pm at λ=1063.4nm is achieved with the device of small total physical thickness 2L=5.52mm.

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References

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  1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), pp. 38-49.
  2. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984).
  3. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).
  4. L. M. Brekhovskikh, Waves in Layered Media (Academic, 1980).
  5. P. Yeh, Optical Waves in Layered Media (Wiley, 1988).
  6. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909-2945 (1969).
  7. R. J. Collier, Ch. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, 1971).
  8. B. Ya. Zel'dovich, A. V. Mamaev, and V. V. Shkunov, Speckle-Wave Interactions in Application to Holography and Nonlinear Optics (CRC Press, 1992).
  9. L. B. Glebov, V. I. Smirnov, C. M. Stickley, and I. V. Ciapurin, “New approach to robust optics for HEL systems,” Proc. SPIE 4724, 101-109 (2002).
    [CrossRef]
  10. O. M. Efimov, L. B. Glebov, and V. I. Smirnov “High efficiency volume diffractive elements in photo-thermo-refractive glass,” U.S. Patent No. 6,673,497 B2 (January 6, 2004).
  11. G. Town, K. Sugden, J. Williams, I. Bennion, and S. Poole, “Wide-band Fabry-Perot-like filters in optical fiber,” IEEE Photon. Technol. Lett. 7, 78-80 (1995).
    [CrossRef]
  12. G. Tremblay and Y. Sheng, “Effects of the phase shift split on phase-shifted fiber Bragg gratings,” J. Opt. Soc. Am. A 23, 1511-1516 (2006).
    [CrossRef]
  13. H. A. Haus and C. V. Shank, “Antisymmetric taper of distributed feedback lasers,” IEEE J. Quantum Electron. 12, 532-539 (1976).
    [CrossRef]
  14. R. V. Schmidt, D. C. Flanders, C. V. Shank, and R. D. Standley, “Narrow-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 25, 651-652 (1974).
    [CrossRef]
  15. H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327-2335 (1972).
    [CrossRef]
  16. V. I. Kopp, R. Bose, and A. Z. Genack, “Transmission through chiral twist defects in anisotropic periodic structures,” Opt. Lett. 28, 349-351 (2003).
    [CrossRef] [PubMed]
  17. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).
  18. J. Lumeau, L. B. Glebov, and V. Smirnov, “Tunable narrowband filter based on a combination of Fabry-Perot etalon and volume Bragg grating,” Opt. Lett. 31, 2417-2419 (2006).
    [CrossRef] [PubMed]
  19. A. V. Vinogradov and B. Ya. Zeldovich, “X-ray and far uv multilayer mirrors: principles and possibilities,” Appl. Opt. 16, 89-93 (1977).
    [CrossRef] [PubMed]
  20. P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, 1985), pp. 304-349.

2006

G. Tremblay and Y. Sheng, “Effects of the phase shift split on phase-shifted fiber Bragg gratings,” J. Opt. Soc. Am. A 23, 1511-1516 (2006).
[CrossRef]

J. Lumeau, L. B. Glebov, and V. Smirnov, “Tunable narrowband filter based on a combination of Fabry-Perot etalon and volume Bragg grating,” Opt. Lett. 31, 2417-2419 (2006).
[CrossRef] [PubMed]

2003

2002

L. B. Glebov, V. I. Smirnov, C. M. Stickley, and I. V. Ciapurin, “New approach to robust optics for HEL systems,” Proc. SPIE 4724, 101-109 (2002).
[CrossRef]

1995

G. Town, K. Sugden, J. Williams, I. Bennion, and S. Poole, “Wide-band Fabry-Perot-like filters in optical fiber,” IEEE Photon. Technol. Lett. 7, 78-80 (1995).
[CrossRef]

1977

1976

H. A. Haus and C. V. Shank, “Antisymmetric taper of distributed feedback lasers,” IEEE J. Quantum Electron. 12, 532-539 (1976).
[CrossRef]

1974

R. V. Schmidt, D. C. Flanders, C. V. Shank, and R. D. Standley, “Narrow-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 25, 651-652 (1974).
[CrossRef]

1972

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327-2335 (1972).
[CrossRef]

1969

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909-2945 (1969).

Appl. Opt.

Appl. Phys. Lett.

R. V. Schmidt, D. C. Flanders, C. V. Shank, and R. D. Standley, “Narrow-band grating filters for thin-film optical waveguides,” Appl. Phys. Lett. 25, 651-652 (1974).
[CrossRef]

Bell Syst. Tech. J.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909-2945 (1969).

IEEE J. Quantum Electron.

H. A. Haus and C. V. Shank, “Antisymmetric taper of distributed feedback lasers,” IEEE J. Quantum Electron. 12, 532-539 (1976).
[CrossRef]

IEEE Photon. Technol. Lett.

G. Town, K. Sugden, J. Williams, I. Bennion, and S. Poole, “Wide-band Fabry-Perot-like filters in optical fiber,” IEEE Photon. Technol. Lett. 7, 78-80 (1995).
[CrossRef]

J. Appl. Phys.

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327-2335 (1972).
[CrossRef]

J. Opt. Soc. Am. A

G. Tremblay and Y. Sheng, “Effects of the phase shift split on phase-shifted fiber Bragg gratings,” J. Opt. Soc. Am. A 23, 1511-1516 (2006).
[CrossRef]

Opt. Lett.

Proc. SPIE

L. B. Glebov, V. I. Smirnov, C. M. Stickley, and I. V. Ciapurin, “New approach to robust optics for HEL systems,” Proc. SPIE 4724, 101-109 (2002).
[CrossRef]

Other

O. M. Efimov, L. B. Glebov, and V. I. Smirnov “High efficiency volume diffractive elements in photo-thermo-refractive glass,” U.S. Patent No. 6,673,497 B2 (January 6, 2004).

R. J. Collier, Ch. B. Burckhardt, and L. H. Lin, Optical Holography (Academic, 1971).

B. Ya. Zel'dovich, A. V. Mamaev, and V. V. Shkunov, Speckle-Wave Interactions in Application to Holography and Nonlinear Optics (CRC Press, 1992).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), pp. 38-49.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984).

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

L. M. Brekhovskikh, Waves in Layered Media (Academic, 1980).

P. Yeh, Optical Waves in Layered Media (Wiley, 1988).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd ed. (Academic, 1985), pp. 304-349.

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Figures (6)

Fig. 1
Fig. 1

Notations for the incident a ( z ) and reflected b ( z ) waves in the approximation of infinitely wide plane beams with account of the reflections from both boundaries, z = 0 and z = L , as well as of the reflection by a VBG.

Fig. 2
Fig. 2

Various graphs describing Fresnel reflection at the air–glass boundary versus incidence angle θ = θ air ; see explanations in the text.

Fig. 3
Fig. 3

Reflectivity R of a VBG with account of interference of reflection by the VBG proper with two extra contributions: from the two boundaries of the specimen, for all possible phase combinations. Values of R are between the dashed curves for Fresnel 4% reflections from bare boundaries, and are between the dotted curves for antireflection coatings (ARC) at 0.3% each.

Fig. 4
Fig. 4

Experimental setup for the coherent combination of two VBGs in PTR glass.

Fig. 5
Fig. 5

Experimental transmission of two π-shifted VBGs.

Fig. 6
Fig. 6

Spectral shift of resonant transmission due to phase shift Δ γ between two grating modulations; see the text for details.

Equations (139)

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n ( z ) = n 0 + n 2 ( z ) + n 1 ( z ) cos [ Q z + γ ( z ) ] + i n 1 ( z ) cos [ Q z + δ ( z ) ] ,
R = r 2 = tanh 2 S .
v = c n , Z = μ ε , n = ε μ ε vac μ vac .
S VBG ( S 1 + S 2 ) S S VBG + ( S 1 + S 2 ) ,
ε ( r ) E t = × H , μ ( r ) H t = × E .
H ( r , t ) = h ( r , t ) 1 Z ( r ) , E ( r , t ) = e ( r , t ) Z ( r ) ,
g = 1 2 ln Z ( r ) ,
n c h t = × e + g × e , n c e t = × h + g × h ,
A real ( r , t ) = 1 2 [ A ( r ) exp ( i ω t ) + A * ( r ) exp ( i ω t ) ] ,
k air = x ̂ k x + z ̂ k air , z , k x = ω c n air sin θ air ,
k air , z = ω c n air cos θ air .
TE : e ( r ) = y ̂ u y ( z ) exp ( i k x x ) ,
h ( r ) = [ x ̂ w x ( z ) + z ̂ w z ( z ) ] exp ( i k x x ) ;
TM : e ( r ) = [ x ̂ u x ( z ) + z ̂ u z ( z ) ] exp ( i k x x ) ,
h ( r ) = y ̂ w y ( z ) exp ( i k x x ) .
k ( z ) = ω c n ( z ) , p ( z ) = k 2 ( z ) k x 2 ,
g ( z ) = 1 2 d d z ln Z ( z ) , f ( z ) = 1 2 d d z ln p ( z ) k ( z ) .
i k u y = z w x i k x w z + g w x , i k w x = z u y + g u y , i k w z = i k x u y .
z u y = g u y + i k w x , z w x = i p 2 k u y g w x .
a TE ( z ) = 1 8 [ p k u y ( z ) + k p w x ( z ) ] e i k air , z z , b TE ( z ) = 1 8 [ p k u y ( z ) k p w x ( z ) ] e i k air , z z .
a TE ( z ) E y 8 Z + Z 8 H x , b TE ( z ) E y 8 Z Z 8 H x .
S z = a TE 2 b TE 2 .
d d z [ a TE ( z ) b TE ( z ) ] = V ̂ TE [ a TE ( z ) b TE ( z ) ] ,
V ̂ TE = [ i [ p ( z ) k air , z ] [ g ( z ) + f ( z ) ] e 2 i k air , z z [ g ( z ) + f ( z ) ] e 2 i k air , z z i [ p ( z ) k air , z ] ] .
i k u x = z w y g w y , i k u z = i k x w y , i k w y = i k x u z z u x + g u x z u x = g u x + i p 2 k , z w y = i k u x g w y ,
a TM ( z ) = 1 8 [ p k w y ( z ) + k p u x ( z ) ] e i k air , z z ,
b TM ( z ) = 1 8 [ p k w y ( z ) + k p u x ( z ) ] e i k air , z z .
d d z [ a TM ( z ) b TM ( z ) ] = V ̂ TM [ a TM ( z ) b TM ( z ) ] ,
V ̂ TM = [ i [ p ( z ) k air , z ] [ g ( z ) f ( z ) ] e 2 i k air , z z [ g ( z ) f ( z ) ] e 2 i k air , z z i [ p ( z ) k air , z ] ] .
[ a ( z 2 ) b ( z 2 ) ] = K ̂ ( ( p k air , z ) ( z 2 z 1 ) ) [ a ( z 1 ) b ( z 1 ) ] .
K ̂ ( φ ) = [ e i φ 0 0 e i φ ] .
[ a ( z 2 ) b ( z 2 ) ] = M ̂ ( z 2 , z 1 ) [ a ( z 1 ) b ( z 1 ) ] , M ̂ ( z 2 , z 1 ) = [ M a a M a b M b a M b b ] .
d M ̂ ( z , z 1 ) d z = V ̂ ( z ) M ̂ ( z , z 1 ) , M ̂ ( z 1 , z 1 ) = 1 ̂ ,
det M ̂ = 1 .
r ( b a ) = b ( z 1 ) = M b a ( z 2 , z 1 ) M b b ( z 2 , z 1 ) ,
t ( a a ) = a ( z 2 ) = det M ̂ ( z 2 , z 1 ) M b b ( z 2 , z 1 ) = 1 M b b ( z 2 , z 1 ) ,
r ( a b ) = a ( z 2 ) = + M a b ( z 2 , z 1 ) M b b ( z 2 , z 1 ) ,
t ( b b ) t ( a a ) = b ( z 1 ) = 1 M b b ( z 2 , z 1 ) .
a 2 b 2 = const ( no - loss assumption ) .
M ̂ * T η ̂ M ̂ = η ̂ ,
η ̂ = [ 1 0 0 1 ] , M a a 2 M b a 2 = 1 = M b b 2 M a b 2 ;
M a b * M a a M b b * M b a = 0 .
M ̂ = [ e i α 0 0 e i α ] Σ ̂ ( S ) [ e i β 0 0 e i β ] K ̂ ( α ) Σ ̂ ( S ) K ̂ ( β ) [ e i ( α β ) cosh S e i ( α + β ) sinh S e i ( α + β ) sinh S e i ( β α ) cosh S ] ,
Σ ̂ ( S ) = [ cosh S sinh S sinh S cosh S ] .
[ a ( + ) b ( ) ] = U ̂ [ a ( ) b ( + ) ] , U ̂ = [ t ( a a ) r ( a b ) r ( b a ) t ( b b ) ] .
t ( a a ) = e i ( α β ) 1 cosh S , r ( a b ) = e 2 i α tanh S ,
r ( b a ) = e 2 i β tanh S , t ( b b ) = t ( a a ) .
S = arctanh R , T = 1 R ( no - loss assumption ) .
M ̂ = Σ ̂ ( S ) = [ cosh S sinh S sinh S cosh S ] , t = 1 cosh S ,
r = tanh S , t 2 + r 2 = 1 .
d S z d z 0 1 M a a 2 + M b a 2 0 , M b b 2 M a b 2 1 0 ,
M a b * M a a M b b * M b a ( 1 M a a 2 + M b a 2 ) ( M b b 2 M a b 2 1 ) ,
all at z 0 .
V ̂ = [ i ( p k air , z ) α loss 2 cos θ 0 0 α loss 2 cos θ i ( p k air , z ) ] ,
d M ̂ d z = V ̂ M ̂ , M ̂ ( z ) = K ̂ ( ( p k air , z ) z ) L ̂ ( Y ) ,
L ̂ ( Y ) = [ e Y 0 0 e Y ] , Y = α loss z 2 cos θ .
r TE r ( E y E y ) = cos θ 1 Z 1 cos θ 2 Z 2 cos θ 1 Z 1 + cos θ 2 Z 2 ,
r TM r ( E x E x ) = Z 1 cos θ 1 Z 2 cos θ 2 Z 1 cos θ 1 + Z 2 cos θ 2 .
n 1 sin θ 1 = n 2 sin θ 2 .
r TE r ( E y E y ) = r TM r ( E x E x ) = cos θ 1 cos θ 2 cos θ 1 + cos θ 2 tanh S Δ n , S Δ n = 1 2 ln ( cos θ 2 cos θ 1 ) .
r TE r ( E y E y ) = r TM r ( E x E x ) = Z 2 Z 1 Z 2 + Z 1 tanh S Δ Z , S Δ Z = 1 2 ln ( Z 1 Z 2 ) .
d a d z = [ g ( z ) ± f ( z ) ] b , d b d z = [ g ( z ) ± f ( z ) ] a , ( + ) TE , ( ) TM.
[ a ( z ) b ( z ) ] = [ cosh S ( z ) sinh S ( z ) sinh S ( z ) cosh S ( z ) ] [ a ( 0 ε ) b ( 0 ε ) ] Σ ̂ ( S ( z ) ) [ a ( 0 ε ) b ( 0 ε ) ] , S ( z ) = 0 ε z [ g ( z ) ± f ( z ) ] d z .
g ( z ) = S Δ Z δ ( z ) , S Δ Z = ln Z 1 Z 2 , f ( z ) = S Δ n δ ( z ) , S Δ n = ln cos θ 2 cos θ 1 ,
[ a ( + 0 ) b ( + 0 ) ] = Σ ̂ ( S ) [ a ( 0 ) b ( 0 ) ] ,
S TE = S Δ Z + S Δ n , S TM = S Δ Z S Δ n .
TE : E y ( z ) = 2 Z ( z ) k ( z ) p ( z ) [ a TE ( z ) e i k air , z z + b TE ( z ) e i k air , z z ] , TM : E x ( z ) = 2 Z ( z ) p ( z ) k ( z ) [ a TM ( z ) e i k air , z z + b TM ( z ) e i k air , z z ] ,
r TE r ( E y E y ) = tanh S TE ,
r TM r ( E x E x ) = tanh S TM .
n ( z ) = n 0 + n 2 ( z ) + n 1 ( z ) cos [ Q z + γ ( z ) ] + i n 1 ( z ) cos [ Q z + δ ( z ) ] ,
Z ( z ) = Z vac n ( z ) Z vac n 0 2 ( n 0 n 2 ( z ) n 1 ( z ) cos [ Q z + γ ( z ) ] i n 1 ( z ) cos [ Q z + δ ( z ) ] ) .
f ( z ) = 1 2 n air 2 sin θ air 2 n 2 ( z ) n air 2 sin θ air 2 d ln n ( z ) d z , g ( z ) = 1 2 d ln n ( z ) d z ,
d ln n ( z ) d z Q n 1 ( z ) sin [ Q z + γ ( z ) ] + i n 1 ( z ) sin [ Q z + δ ( z ) ] n 0 .
Q 2 ω c n 0 cos θ in , cos θ in = 1 n air 2 sin θ air 2 n 0 2 .
g ( z ) + f ( z ) ω c n 1 ( z ) sin [ Q z + γ ( z ) ] + i n 1 ( z ) sin [ Q z + δ ( z ) ] cos θ in ,
g ( z ) f ( z ) [ g ( z ) + f ( z ) ] cos 2 θ in .
sin [ Q z + γ ( z ) ] = 1 2 i [ e i Q z + i γ ( z ) e i Q z i γ ( z ) ] ,
d d z M ̂ ( z ) = V ̂ TE , TM ( z ) M ̂ ( z ) ,
V ̂ ( z ) = [ i [ p ( z ) k air , z ] i κ + ( z ) e i Q z 2 i k air , z z i κ ( z ) e i Q z + 2 i k air , z z i [ p ( z ) k air , z ] ] ,
TE : κ + ( z ) = ω c n 1 ( z ) e i γ ( z ) + i n 1 ( z ) e i δ ( z ) 2 cos θ in ,
κ ( z ) = ω c n 1 ( z ) e i γ ( z ) + i n 1 ( z ) e i δ ( z ) 2 cos θ in .
M ̂ ( z ) = K ̂ ( ( Q 2 k air , z ) z ) P ̂ ( z ) ,
d P ̂ ( z ) d z = W ̂ ( z ) P ̂ ( z ) ,
W ̂ ( z ) = [ i [ p ( z ) Q 2 ] i κ + ( z ) i κ ( z ) i [ p ( z ) Q 2 ] ] .
M ̂ ( z 2 = z 1 + L , z 1 ) = K ̂ ( k air , z z 1 ) M ̂ ( L ) K ̂ ( k air , z z 1 ) .
p = ω c ( n 0 + n 2 ) 2 n air 2 sin θ air 2 ω c ( n 0 cos θ in + n 2 cos θ in ) = p + i p , p = α loss 2 cos θ in .
d P ̂ ( z ) d z = W ̂ P ̂ ( z ) , W ̂ = [ i Δ i κ + i κ i Δ ]
P ̂ ( L ) = e W ̂ L = 1 ̂ cosh G + W ̂ L sinh G G ,
Δ = p + i p Q 2 , G = S + S X 2 ,
S ± = κ ± L = κ ± L + i κ ± L , X = Δ L = X + i X .
P ̂ ( L ) = [ cosh G + i X sinh G G i S + sinh G G i S sinh G G cosh G i X sinh G G ] .
M ̂ VBG ( z 2 , z 1 ) = K ̂ ( k air , z z 1 ( k air , z Q 2 ) L ) P ̂ ( L ) K ̂ ( k air , z z 1 ) .
r ( b a ) = r = M b a M b b = e 2 i k air , z z 1 P b a P b b = e 2 i k air , z z 1 i S sinh G G cosh G i X sinh G G .
X Y = α loss L 2 cos θ in .
δ Re ( X ) = 2 π n 0 L cos θ in λ vac , 0 { δ λ vac λ vac + δ Q 2 Q + 1 cos 2 θ in [ n air 2 2 n 0 2 ( sin 2 θ air sin 2 θ air , 0 ) δ n 2 n 0 ] } .
R = r ( b a ) 2 = tanh 2 S = sinh 2 G cosh 2 G X 2 S 0 2 ;
S = arcsinh ( S 0 sinh G G ) , S 0 = S + S = S + , G = S 0 2 X 2 .
S = S 0 = π n 1 L λ vac cos θ in ,
M ̂ ( z b + 0 , z b 0 ) = K ̂ ( k air , z z b ) Σ ̂ ( S ) K ̂ ( k air , z z b ) ,
Σ ̂ ( S ) = [ cosh S sinh S sinh S cosh S ] ,
S TE , TM = ln Z 1 Z 2 ± ln cos θ 2 cos θ 1 ln n 2 n 1 ± ln cos θ 2 cos θ 1 .
M ̂ ( z 2 + 0 , z 1 0 ) = K ̂ ( k air , z z 2 ) Σ ̂ ( S 2 ) K ̂ ( k air , z z 2 ) M ̂ VBG ( z 2 , z 1 ) K ̂ ( k air , z z 1 ) Σ ̂ ( S 1 ) K ̂ ( k air , z z 1 ) .
M ̂ = K ̂ ( k air , z z 2 ) Σ ̂ ( S 2 ) K ̂ ( ( γ + Q L ) 2 ) P ̂ S 0 , X K ̂ ( γ 2 ) Σ ̂ ( S 1 ) K ̂ ( k air , z z 1 ) ,
P ̂ S 0 , X = [ cosh G + i X sinh G G i S 0 sinh G G i S 0 sinh G G cosh G i X sinh G G ] ,
G = S 0 2 X 2 ,
M ̂ = K ̂ ( k air , z z 2 ) Σ ̂ ( S 2 ) K ̂ ( φ ) Σ ̂ ( S 1 ) K ̂ ( k air , z z 1 ) ,
φ = p L = ω c n 0 L cos θ in ,
R = tanh 2 S , S max = S VBG + S 1 + S 2 ,
S min = S VBG ( S 1 + S 2 ) ,
tanh 2 ( S 1 S 2 ) R tanh 2 ( S 1 + S 2 ) , S i = arctanh R i .
[ a ( z 3 + 0 ) b ( z 3 + 0 ) ] = M ̂ [ a ( 0 ) b ( 0 ) ] ,
M ̂ = K ̂ ( β 3 ) Σ ̂ ( S b ) K ̂ ( β 2 ) P ̂ S 0 , X K ̂ ( Δ γ 2 ) P ̂ S 0 , X K ̂ ( β 1 ) Σ ̂ ( S b ) ,
β 1 = γ 1 2 , β 2 = γ 2 + Q L 2 , β 3 = k air , z z 3 ,
Δ γ = Q L + γ 1 γ 2 + 2 p l .
n ( z ) = n + l = + 1 + ( c l e i l Q z + c l * e i l Q z ) .
c 1 n .
n ( z ) = n D m = + δ ( z m D ) = l = + n e i l Q z , D = 2 π Q .
n ( z ) = n 0 + n 1 cos ( Q z + γ ) + i n ( z ) ,
κ ± , TE = ω c cos θ in ( 1 2 n 1 e ± i γ + i n ) ,
κ ± , TM = κ ± , TE ρ , ρ = cos ( 2 θ in ) .
d P ̂ ( z ) d z = W ̂ P ̂ ( z ) , W ̂ = [ i Δ i κ + i κ i Δ ] = p [ 1 1 1 1 ] , p = n ω c cos θ in = α loss 2 cos θ in ,
P ̂ ( L ) = exp ( W ̂ L ) = 1 ̂ + W ̂ L = [ 1 Y Y Y 1 + Y ] ,
Y = p L = α loss L 2 cos θ in ,
R + = R = R = ( Y 1 + Y ) 2 , T = 1 ( 1 + Y ) 2 ,
1 R T = 2 Y ( 1 + Y ) 2 .
W ̂ = α loss 2 cos θ in [ 1 2 0 1 ] .
exp ( W ̂ L ) = 1 ̂ cosh Y + W ̂ L sinh Y Y = [ e Y 2 sinh Y 0 e Y ] ,
Y = α loss L 2 cos θ in ,
R + = 0 , R = ( 1 e 2 Y ) 2 , T = e 2 Y ,
1 T R + = 1 e 2 Y , 1 T R = e 2 Y ( 1 e 2 Y ) .
F ( z ) P n ( z ) = c n z n + c n 1 z n 1 + + c 0 .
F ( z ) P 1 ( z ) = F ( z 1 ) z z 2 z 1 z 2 + F ( z 2 ) z z 1 z 2 z 1 ,
F ( z ) P 2 ( z ) = F ( z 1 ) ( z z 2 ) ( z z 3 ) ( z 1 z 2 ) ( z 1 z 3 ) + F ( z 2 ) ( z z 1 ) ( z z 3 ) ( z 2 z 1 ) ( z 2 z 3 ) + F ( z 3 ) ( z z 1 ) ( z z 2 ) ( z 3 z 1 ) ( z 3 z 2 ) .
det ( Z ̂ λ 1 ̂ ) = λ 2 ( Z 11 + Z 22 ) λ + Z 11 Z 22 Z 12 Z 21 = 0 ;
λ 1 , 2 = 1 2 ( Z 11 + Z 22 ) ± 1 4 ( Z 11 Z 22 ) 2 + Z 12 Z 21 .
F ( Z ̂ ) = F ( λ 1 ) Z ̂ λ 2 1 ̂ λ 1 λ 2 + F ( λ 2 ) Z ̂ λ 1 1 ̂ λ 2 λ 1 [ F ( λ 2 ) λ 1 F ( λ 1 ) λ 2 λ 1 λ 2 ] 1 ̂ + [ F ( λ 1 ) F ( λ 2 ) λ 1 λ 2 ] Z ̂ .
Z 22 = Z 11 , λ 1 = λ 2 = λ = λ + i λ = ( Z 11 ) 2 + Z 12 Z 21 ,
exp ( Z ̂ t ) = 1 ̂ cosh ( λ t ) + Z ̂ sinh ( λ t ) λ .
exp ( Z ̂ t ) 1 ̂ + Z ̂ t , if λ 0 , arbitrary t .

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