Abstract

We propose a monolithic large-aperture narrowband optical filter based on a moiré volume Bragg grating formed by two sequentially recorded gratings with slightly different resonant wavelengths. Such recording creates a spatial modulation of refractive index with a slowly varying sinusoidal envelope. By cutting a specimen at a small angle, to a thickness of one-period of this envelope, the longitudinal envelope profile will shift from a sine profile to a cosine profile across the face of the device. The transmission peak of the filter has a tunable bandwidth while remaining at a fixed resonant wavelength by a transversal shift of incidence position. Analytical expressions for the tunable bandwidth of such a filter are calculated and experimental data from a filter operating at 1064 nm with bandwidth range 30-90 pm is demonstrated.

© 2014 Optical Society of America

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References

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  1. C. Fricke-Begemann, M. Alpers, and J. Höffner, “Daylight rejection with a new receiver for potassium resonance temperature lidars,” Opt. Lett. 27(21), 1932–1934 (2002).
    [Crossref] [PubMed]
  2. G. A. Rakuljic and V. Leyva, “Volume holographic narrow-band optical filter,” Opt. Lett. 18(6), 459–461 (1993).
    [Crossref] [PubMed]
  3. Y. Akahane, T. Asano, H. Takano, B.-S. Song, Y. Takana, and S. Noda, “Two-dimensional photonic-crystal-slab channeldrop filter with flat-top response,” Opt. Express 13(7), 2512–2530 (2005).
    [Crossref] [PubMed]
  4. Z. Cheng, D. Liu, Y. Yang, L. Yang, and H. Huang, “Interferometric filters for spectral discrimination in high-spectral-resolution lidar: performance comparisons between Fabry-Perot interferometer and field-widened Michelson interferometer,” Appl. Opt. 52(32), 7838–7850 (2013).
    [Crossref] [PubMed]
  5. J. Bland-Hawthorn, W. van Breugel, P. R. Gillingham, I. K. Baldry, and D. H. Jones, “A tunable lyot filter at prime focus: a method for tracing supercluster scales at z ∼ 1,” Astrophys. J. 563(2), 611–628 (2001).
    [Crossref]
  6. G. B. Venus, A. Sevian, V. Smirnov, and L. B. Glebov, “High-brightness narrow-line laser diode source with volume Bragg-grating feedback,” Proc. SPIE 5711, 166–176 (2005).
    [Crossref]
  7. L. B. Glebov, J. Lumeau, S. Mokhov, V. Smirnov, and B. Y. Zeldovich, “Reflection of light by composite volume holograms: Fresnel corrections and Fabry-Perot spectral filtering,” J. Opt. Soc. Am. A 25(3), 751–764 (2008).
    [Crossref] [PubMed]
  8. S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27(21), 1945–1947 (1991).
    [Crossref]
  9. R. J. Campbell and R. Kashyap, “Spectral profile and multiplexing of Bragg gratings in photosensitive fiber,” Opt. Lett. 16(12), 898–900 (1991).
    [Crossref] [PubMed]
  10. L. A. Everall, K. Sugden, J. A. R. Williams, I. Bennion, X. Liu, J. S. Aitchison, S. Thoms, and R. M. De La Rue, “Fabrication of multipassband moiré resonators in fibers by the dual-phase-mask exposure method,” Opt. Lett. 22(19), 1473–1475 (1997).
    [Crossref] [PubMed]
  11. R. Kashyap, Fiber Bragg Gratings, 2nd ed. (Academic, 2009).
  12. V. Smirnov, J. Lumeau, S. Mokhov, B. Y. Zeldovich, and L. B. Glebov, “Ultranarrow bandwidth moiré reflecting Bragg gratings recorded in photo-thermo-refractive glass,” Opt. Lett. 35(4), 592–594 (2010).
    [Crossref] [PubMed]
  13. A. Sevian, O. Andrusyak, I. Ciapurin, V. Smirnov, G. Venus, and L. Glebov, “Efficient power scaling of laser radiation by spectral beam combining,” Opt. Lett. 33(4), 384–386 (2008).
    [Crossref] [PubMed]
  14. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).
    [Crossref]
  15. V. Mizrahi and J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11(10), 1513–1517 (1993).
    [Crossref]
  16. L. Glebov, “Volume holographic elements in a photo-thermo-refractive glass,” J. Holography and Speckle 5(1), 77–84 (2009).
    [Crossref]
  17. E. Hecht, Optics, 4th ed. (Addison-Wesley, 2001), Chap. 9.6 Multiple Beam Interference.
  18. I. Divliansky, D. Ott, B. Anderson, D. Drachenberg, V. Rotar, G. Venus, and L. Glebov, “Multiplexed volume Bragg gratings for spectral beam combining of high power fiber lasers,” Proc. SPIE 8237, 823705 (2012).
    [Crossref]
  19. M. SeGall, D. Ott, I. Divliansky, and L. B. Glebov, “Effect of aberrations in a holographic system on reflecting volume Bragg gratings,” Appl. Opt. 52(32), 7826–7831 (2013).
    [Crossref] [PubMed]

2013 (2)

2012 (1)

I. Divliansky, D. Ott, B. Anderson, D. Drachenberg, V. Rotar, G. Venus, and L. Glebov, “Multiplexed volume Bragg gratings for spectral beam combining of high power fiber lasers,” Proc. SPIE 8237, 823705 (2012).
[Crossref]

2010 (1)

2009 (1)

L. Glebov, “Volume holographic elements in a photo-thermo-refractive glass,” J. Holography and Speckle 5(1), 77–84 (2009).
[Crossref]

2008 (2)

2005 (2)

Y. Akahane, T. Asano, H. Takano, B.-S. Song, Y. Takana, and S. Noda, “Two-dimensional photonic-crystal-slab channeldrop filter with flat-top response,” Opt. Express 13(7), 2512–2530 (2005).
[Crossref] [PubMed]

G. B. Venus, A. Sevian, V. Smirnov, and L. B. Glebov, “High-brightness narrow-line laser diode source with volume Bragg-grating feedback,” Proc. SPIE 5711, 166–176 (2005).
[Crossref]

2002 (1)

2001 (1)

J. Bland-Hawthorn, W. van Breugel, P. R. Gillingham, I. K. Baldry, and D. H. Jones, “A tunable lyot filter at prime focus: a method for tracing supercluster scales at z ∼ 1,” Astrophys. J. 563(2), 611–628 (2001).
[Crossref]

1997 (1)

1993 (2)

V. Mizrahi and J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11(10), 1513–1517 (1993).
[Crossref]

G. A. Rakuljic and V. Leyva, “Volume holographic narrow-band optical filter,” Opt. Lett. 18(6), 459–461 (1993).
[Crossref] [PubMed]

1991 (2)

S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27(21), 1945–1947 (1991).
[Crossref]

R. J. Campbell and R. Kashyap, “Spectral profile and multiplexing of Bragg gratings in photosensitive fiber,” Opt. Lett. 16(12), 898–900 (1991).
[Crossref] [PubMed]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).
[Crossref]

Aitchison, J. S.

Akahane, Y.

Alpers, M.

Anderson, B.

I. Divliansky, D. Ott, B. Anderson, D. Drachenberg, V. Rotar, G. Venus, and L. Glebov, “Multiplexed volume Bragg gratings for spectral beam combining of high power fiber lasers,” Proc. SPIE 8237, 823705 (2012).
[Crossref]

Andrusyak, O.

Asano, T.

Baldry, I. K.

J. Bland-Hawthorn, W. van Breugel, P. R. Gillingham, I. K. Baldry, and D. H. Jones, “A tunable lyot filter at prime focus: a method for tracing supercluster scales at z ∼ 1,” Astrophys. J. 563(2), 611–628 (2001).
[Crossref]

Bayon, F.

S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27(21), 1945–1947 (1991).
[Crossref]

Bennion, I.

Bernage, P.

S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27(21), 1945–1947 (1991).
[Crossref]

Bland-Hawthorn, J.

J. Bland-Hawthorn, W. van Breugel, P. R. Gillingham, I. K. Baldry, and D. H. Jones, “A tunable lyot filter at prime focus: a method for tracing supercluster scales at z ∼ 1,” Astrophys. J. 563(2), 611–628 (2001).
[Crossref]

Campbell, R. J.

Cheng, Z.

Ciapurin, I.

De La Rue, R. M.

Divliansky, I.

M. SeGall, D. Ott, I. Divliansky, and L. B. Glebov, “Effect of aberrations in a holographic system on reflecting volume Bragg gratings,” Appl. Opt. 52(32), 7826–7831 (2013).
[Crossref] [PubMed]

I. Divliansky, D. Ott, B. Anderson, D. Drachenberg, V. Rotar, G. Venus, and L. Glebov, “Multiplexed volume Bragg gratings for spectral beam combining of high power fiber lasers,” Proc. SPIE 8237, 823705 (2012).
[Crossref]

Douay, M.

S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27(21), 1945–1947 (1991).
[Crossref]

Drachenberg, D.

I. Divliansky, D. Ott, B. Anderson, D. Drachenberg, V. Rotar, G. Venus, and L. Glebov, “Multiplexed volume Bragg gratings for spectral beam combining of high power fiber lasers,” Proc. SPIE 8237, 823705 (2012).
[Crossref]

Everall, L. A.

Fertein, E.

S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27(21), 1945–1947 (1991).
[Crossref]

Fricke-Begemann, C.

Georges, T.

S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27(21), 1945–1947 (1991).
[Crossref]

Gillingham, P. R.

J. Bland-Hawthorn, W. van Breugel, P. R. Gillingham, I. K. Baldry, and D. H. Jones, “A tunable lyot filter at prime focus: a method for tracing supercluster scales at z ∼ 1,” Astrophys. J. 563(2), 611–628 (2001).
[Crossref]

Glebov, L.

I. Divliansky, D. Ott, B. Anderson, D. Drachenberg, V. Rotar, G. Venus, and L. Glebov, “Multiplexed volume Bragg gratings for spectral beam combining of high power fiber lasers,” Proc. SPIE 8237, 823705 (2012).
[Crossref]

L. Glebov, “Volume holographic elements in a photo-thermo-refractive glass,” J. Holography and Speckle 5(1), 77–84 (2009).
[Crossref]

A. Sevian, O. Andrusyak, I. Ciapurin, V. Smirnov, G. Venus, and L. Glebov, “Efficient power scaling of laser radiation by spectral beam combining,” Opt. Lett. 33(4), 384–386 (2008).
[Crossref] [PubMed]

Glebov, L. B.

Höffner, J.

Huang, H.

Jones, D. H.

J. Bland-Hawthorn, W. van Breugel, P. R. Gillingham, I. K. Baldry, and D. H. Jones, “A tunable lyot filter at prime focus: a method for tracing supercluster scales at z ∼ 1,” Astrophys. J. 563(2), 611–628 (2001).
[Crossref]

Kashyap, R.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).
[Crossref]

Legoubin, S.

S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27(21), 1945–1947 (1991).
[Crossref]

Leyva, V.

Liu, D.

Liu, X.

Lumeau, J.

Mizrahi, V.

V. Mizrahi and J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11(10), 1513–1517 (1993).
[Crossref]

Mokhov, S.

Niay, P.

S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27(21), 1945–1947 (1991).
[Crossref]

Noda, S.

Ott, D.

M. SeGall, D. Ott, I. Divliansky, and L. B. Glebov, “Effect of aberrations in a holographic system on reflecting volume Bragg gratings,” Appl. Opt. 52(32), 7826–7831 (2013).
[Crossref] [PubMed]

I. Divliansky, D. Ott, B. Anderson, D. Drachenberg, V. Rotar, G. Venus, and L. Glebov, “Multiplexed volume Bragg gratings for spectral beam combining of high power fiber lasers,” Proc. SPIE 8237, 823705 (2012).
[Crossref]

Rakuljic, G. A.

Rotar, V.

I. Divliansky, D. Ott, B. Anderson, D. Drachenberg, V. Rotar, G. Venus, and L. Glebov, “Multiplexed volume Bragg gratings for spectral beam combining of high power fiber lasers,” Proc. SPIE 8237, 823705 (2012).
[Crossref]

SeGall, M.

Sevian, A.

A. Sevian, O. Andrusyak, I. Ciapurin, V. Smirnov, G. Venus, and L. Glebov, “Efficient power scaling of laser radiation by spectral beam combining,” Opt. Lett. 33(4), 384–386 (2008).
[Crossref] [PubMed]

G. B. Venus, A. Sevian, V. Smirnov, and L. B. Glebov, “High-brightness narrow-line laser diode source with volume Bragg-grating feedback,” Proc. SPIE 5711, 166–176 (2005).
[Crossref]

Sipe, J. E.

V. Mizrahi and J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11(10), 1513–1517 (1993).
[Crossref]

Smirnov, V.

Song, B.-S.

Sugden, K.

Takana, Y.

Takano, H.

Thoms, S.

van Breugel, W.

J. Bland-Hawthorn, W. van Breugel, P. R. Gillingham, I. K. Baldry, and D. H. Jones, “A tunable lyot filter at prime focus: a method for tracing supercluster scales at z ∼ 1,” Astrophys. J. 563(2), 611–628 (2001).
[Crossref]

Venus, G.

I. Divliansky, D. Ott, B. Anderson, D. Drachenberg, V. Rotar, G. Venus, and L. Glebov, “Multiplexed volume Bragg gratings for spectral beam combining of high power fiber lasers,” Proc. SPIE 8237, 823705 (2012).
[Crossref]

A. Sevian, O. Andrusyak, I. Ciapurin, V. Smirnov, G. Venus, and L. Glebov, “Efficient power scaling of laser radiation by spectral beam combining,” Opt. Lett. 33(4), 384–386 (2008).
[Crossref] [PubMed]

Venus, G. B.

G. B. Venus, A. Sevian, V. Smirnov, and L. B. Glebov, “High-brightness narrow-line laser diode source with volume Bragg-grating feedback,” Proc. SPIE 5711, 166–176 (2005).
[Crossref]

Williams, J. A. R.

Yang, L.

Yang, Y.

Zeldovich, B. Y.

Appl. Opt. (2)

Astrophys. J. (1)

J. Bland-Hawthorn, W. van Breugel, P. R. Gillingham, I. K. Baldry, and D. H. Jones, “A tunable lyot filter at prime focus: a method for tracing supercluster scales at z ∼ 1,” Astrophys. J. 563(2), 611–628 (2001).
[Crossref]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2947 (1969).
[Crossref]

Electron. Lett. (1)

S. Legoubin, E. Fertein, M. Douay, P. Bernage, P. Niay, F. Bayon, and T. Georges, “Formation of moiré grating in core of germanosilicate fibre by transverse holographic double exposure method,” Electron. Lett. 27(21), 1945–1947 (1991).
[Crossref]

J. Holography and Speckle (1)

L. Glebov, “Volume holographic elements in a photo-thermo-refractive glass,” J. Holography and Speckle 5(1), 77–84 (2009).
[Crossref]

J. Lightwave Technol. (1)

V. Mizrahi and J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11(10), 1513–1517 (1993).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Express (1)

Opt. Lett. (6)

Proc. SPIE (2)

G. B. Venus, A. Sevian, V. Smirnov, and L. B. Glebov, “High-brightness narrow-line laser diode source with volume Bragg-grating feedback,” Proc. SPIE 5711, 166–176 (2005).
[Crossref]

I. Divliansky, D. Ott, B. Anderson, D. Drachenberg, V. Rotar, G. Venus, and L. Glebov, “Multiplexed volume Bragg gratings for spectral beam combining of high power fiber lasers,” Proc. SPIE 8237, 823705 (2012).
[Crossref]

Other (2)

R. Kashyap, Fiber Bragg Gratings, 2nd ed. (Academic, 2009).

E. Hecht, Optics, 4th ed. (Addison-Wesley, 2001), Chap. 9.6 Multiple Beam Interference.

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Figures (3)

Fig. 1
Fig. 1

(a) Scheme of a one-period moiré VBG with RIM envelope profiles of shifted phases. (b) Corresponding numerically calculated transmission peaks versus dimensionless detuning Φ = Dl for different envelope profiles.

Fig. 2
Fig. 2

Experimental (dots) and numerically modeled (line) irradiance transmission versus spectral detuning δλ = λ–λ0: (a) sine moiré envelope profile; (b) cosine profile.

Fig. 3
Fig. 3

FWHM bandwidth of moiré VBG versus the tunable envelope phase parameter 2φ/π: dots – experimental measurements, solid line – numerical modeling, dashed line – analytical expression (20) for the bandwidth of a Lorentzian peak, square dot with dotted line – analytical expression (21) for the bandwidth of a flat-top peak of a double coherent resonant cavity.

Equations (29)

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cos( Q 1 z)+cos( Q 2 z)=2cos(Qz)cos(qz),Q= 1 2 ( Q 1 + Q 2 ),q= 1 2 ( Q 1 Q 2 ),
n(z)= n 0 + n 1,φ (z)cos(Qz+γ),Q= 4π n 0 / λ 0 , λ 0 = 1 2 ( λ 1 + λ 2 ), n 1,φ (z)= N 1 sin( πz /l +φ),l= λ 0 2 / (2 n 0 Δλ) ,Δλ= λ 2 λ 1 ,
{ d dz A=iκ(z)B e 2iDz+iγ , d dz B=iκ(z)A e 2iDziγ , κ(z)= π n 1,ϕ (z) / λ 0 ,D=k 1 2 Q,k= 2π n 0 /λ ,
λ res = λ 0 cos θ in λ 0 (1 1 2 θ air 2 / n 0 2 )
( A(z) B(z) )= M ^ (z)( A(0) B(0) ), M ^ (z)=( M 11 (z) M 12 (z) M 21 (z) M 22 (z) ).
r= B(0) / A(0) = M 21 (L) / M 22 (L) ,R= | r | 2 = tanh 2 S, t= A(L) / A(0) =1/ M 22 (L) ,T= | t | 2 =1R=1/ cosh 2 S .
M ^ 0 (z)=( coshS(z) i e iγ sinhS(z) i e iγ sinhS(z) coshS(z) ),S(z)=π/ λ 0 0 z n 1 ( z )d z , r 0 =i e iγ tanhS(L).
s M =S(l)=π/ λ 0 0 l n 1,ϕ=0 (z)dz = 2 N 1 l / λ 0 .
( A B ) z=l+0 = M ^ ( A B ) z=0 , M ^ = K ^ (kl) m ^ (s) K ^ (kl) m ^ (s), m ^ (s)=( coshssinhs sinhscoshs ), K ^ (Φ)=( e iΦ 0 0 e iΦ ).
M 22 = e iΦ (cosΦicosh(2s)sinΦ),Φ=(k k 0 )l 2π n 0 lδλ / λ 0 2 ,δλ=λ λ 0 , T=1/ | M 22 | 2 =1/ (1+F sin 2 Φ) ,F= sinh 2 (2s).
Δ λ FWHM =2| δ λ HM |= λ 0 2 / (π n 0 l) Φ HM = λ 0 2 / (π n 0 lsinh(2s)) .
( A B ) z=l = m ^ I ( A B ) z=0 , m ^ I = K ^ (Φ+ γ 2 ) μ ^ (s,Φ) K ^ ( γ 2 ),Φ=Dl=(k k 0 )l, μ ^ =( α β β * α * ), α=coshG+iΦ/G sinhG,β=is/G sinhG,G= s 2 Φ 2 ,s= s U = π n 1 l / λ 0 .
( A B ) z=2l = m ^ II ( A B ) z=l , m ^ II = K ^ (kl) m ^ I (s,Φ, Ql+γ 2 ) K ^ (kl), M ^ = m ^ II m ^ I = K ^ (2Φ+ γ 2 ) μ ^ (s,Φ) μ ^ (s,Φ) K ^ ( γ 2 ), M 22 = e 2iΦ (1i s 1 sinh(2s)Φ2 s 2 sinh 2 s Φ 2 +O( Φ 3 )), T=1/ | M 22 | 2 1/ (1+ F U Φ 2 ) , F U =4 s U 2 sinh 4 s U .
Δ λ U,FWHM = λ 0 2 / (π n 0 l F U ) = λ 0 2 s U / (2π n 0 l sinh 2 s U )
z= ςl /π , 0ς2π, d dς ( A B )= 1 2 s M sin(ς+φ)( 0 i e 2i Φς /π +iγ i e 2i Φς /π iγ 0 )( A B ), ( A B )= K ^ ( Φ π ς+ γ 2 + π 4 )( a b ) d dς ( a b )= w ^ (ς)( a b ), w ^ (ς)= w ^ 0 (ς)+iΦ σ ^ , w ^ 0 (ς)= 1 2 s M sin(ς+φ)( 0 1 1 0 ), σ ^ = 1 π ( 1 0 0 1 ).
d dς ( a b )= w ^ (ς)( a(ς) b(ς) ),( a(ς) b(ς) )= m ^ (ς)( a(0) b(0) ) d dς m ^ = w ^ (ς) m ^ (ς), m ^ (0)= 1 ^ .
m ^ (ς)= m ^ 0 (ς)[ 1 ^ +iΦ u ^ (1) (ς)+ (iΦ) 2 u ^ (2) (ς)]: m ^ (0)= 1 ^ m ^ 0 (0)= 1 ^ = m ^ 0 (2π), u ^ (1,2) (0)=0; d dς m ^ = w ^ m ^ d dς m ^ 0 = w ^ 0 m ^ 0 , d dς u ^ (1) = m ^ 0 1 σ ^ m ^ 0 , d dς u ^ (2) = m ^ 0 1 σ ^ m ^ 0 u ^ (1) =( d dς u ^ (1) ) u ^ (1) .
m ^ 0 (ς)=( coshS(ς) sinhS(ς) sinhS(ς) coshS(ς) ),S(ς)= 1 2 s(cosφcos(ς+φ)), d dς u ^ (1) = m ^ 0 1 σ ^ m ^ 0 = 1 π ( cosh(2S(ς)) sinh(2S(ς)) sinh(2S(ς)) cosh(2S(ς)) ) u ^ (1) (2π)=2 I 0 (s)( cosh(scosφ sinh(scosφ) sinh(scosφ) cosh(scosφ) ).
d dς u ^ (2) = 1 2 d dς ( u ^ (1) u ^ (1) )+ 1 2 [( d dς u ^ (1) ) u ^ (1) u ^ (1) ( d dς u ^ (1) )],
u 22 (2) (2π)= 1 2 ( u ^ (1) (2π) u ^ (1) (2π)) 22 =2 I 0 2 (s), m 22 (2π)=12i I 0 (s)cosh(scosφ)Φ2 I 0 2 (s) Φ 2 +iO( Φ 3 ),T= 1 | m 22 | 2 = 1 1+ F M Φ 2 +O( Φ 4 ) , F M =4 I 0 2 (s) sinh 2 (scosφ),s= s M ,Δ λ M,FWHM = λ 0 2 π n 0 l F M = λ 0 2 2π n 0 l I 0 ( s M )sinh( s M cosφ) .
T= 1 1+ H M Φ 4 , H M 1/2 =2 I 0 ( s M ) L 0 ( s M ),Δ λ DCM,FWHM = λ 0 2 π n 0 l H M 1/4 = λ 0 2 π n 0 l 2 I 0 ( s M ) L 0 ( s M ) .
I 0 (s)= k=0 s 2k / ( 2 k k!) 2 , L 0 (s)=2/π k=0 s 2k+1 / [(2k+1)!!] 2 .
T=1/ (1+ H R sin 4 Φ) , H R = sinh 2 (2s),Δ λ DCR,FWHM = λ 0 2 / (π n 0 l H R 1/4 ) ,
m ^ = μ ^ ( s 2 , Φ 2 ) μ ^ (s,Φ) μ ^ ( s 2 , Φ 2 ),T= | m 22 | 2 =1/ (1+ H U Φ 4 ) , H U =4 s 4 (coshs1) 2 sinh 2 s,Δ λ DCU,FWHM = λ 0 2 / (π n 0 l H U 1/4 ) ,
m ^ (2π)= 1 ^ 1 2 αl u ^ (1) (2π)+O( α 2 ),T=1/ m 22 2 12 I 0 ( s M )cosh( s M cosφ)αl.
m ^ (2π)=( m 22 * m 12 m 12 m 22 ), m 12 =1 Φ 2 U 12 (2) + Φ 4 U 12 (4) , U pq (j) = u pq (j) (2π), m 22 =12i I 0 (s)Φ2 I 0 2 (s) Φ 2 i Φ 3 U 22 (3) + Φ 4 U 22 (4) , U 12 (2) = 1 π 2 0 2π dξ 0 ξ dη sinh(s(sinηsinξ))=2 I 0 (s) L 0 (s).
| m 22 | 2 =1+H Φ 4 +...,H=4 I 0 4 (s)4 I 0 (s) U 22 (3) +2 U 22 (4) .
det m ^ =1+ [4 I 0 4 (s)4 I 0 (s) U 22 (3) +2 U 22 (4) ( U 12 (2) ) 2 ] =0 Φ 4 +...=1,
| m 22 | 2 =1+H Φ 4 ,H= ( U 12 (2) ) 2 , U 12 (2) =2 I 0 (s) L 0 (s).

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