Abstract

We use an analytical calculation based on the Fourier-transform method to study the transmission spectra of multilayer dielectric optical structures as a function of the relative widths of the layers that constitute the unit cell. We can select which harmonics of the fundamental design frequency are transmitted. The results of this Fourier-transform approach are compared with the exact transmission calculated by means of the transfer matrix method and provide a more intuitive understanding of the transmission spectrum. A simple phasor diagram is derived from this Fourier-transform analysis for this purpose. Inasmuch as it is difficult for us to perform experiments in the optical region, we fabricate rf analogs of these structures, using coaxial cables that have different impedances. Experimental results agree with theory.

© 2005 Optical Society of America

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References

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  1. H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics Publishing, 1986), Chap. 6, pp. 248–255.
  2. P. Yeh, Optical Waves in Layered Media (Wiley, 1988), Chaps. 5 and 6.
  3. H. Taniyama, “Waveguide structures using one-dimensional photonic crystal,” J. Appl. Phys. 91, 3511–3515 (2002).
    [CrossRef]
  4. M. Deopura, C. K. Ullal, B. Temelkuran, Y. Fink, “Dielectric omnidirectional visible reflector,” Opt. Lett. 26, 1197–1199 (2001).
    [CrossRef]
  5. A. Thelen, “Multilayer filters with wide transmittance bands,” J. Opt. Soc. Am. 53, 1266–1270 (1963).
    [CrossRef]
  6. W. E. Johnson, R. L. Crane, “Introduction to rugate filter technology,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 88–108 (1993).
    [CrossRef]
  7. W. H. Southwell, “Extended-bandwidth reflector designs by using wavelets,” Appl. Opt. 36, 314–318 (1997).
    [CrossRef] [PubMed]
  8. P. L. Swart, P. V. Bulkin, B. M. Lacquet, “Rugate filter manufacturing by electron cyclotron resonance plasma-enhanced chemical vapor deposition of SiNx,” Opt. Eng. 36, 1214–1219 (1997).
    [CrossRef]
  9. P. G. Verly, J. A. Dobrowolski, W. J. Wild, R. L. Burton, “Synthesis of high rejection filters with the Fourier transform method,” Appl. Opt. 28, 2864–2875 (1989).
    [CrossRef] [PubMed]
  10. B. E. Perilloux, Thin-Film Design: Modulated Thickness and Other Stopband Design Methods (SPIE Press, 2002), Chaps. 1 and 2.
    [CrossRef]
  11. J. A. Dobrowolski, D. Lowe, “Optical thin film synthesis program based on the use of Fourier transform,” Appl. Opt. 17, 3039–3050 (1978).
    [CrossRef] [PubMed]
  12. B. G. Bovard, “Derivation of a matrix describing a rugate dielectric thin film,” Appl. Opt. 27, 1998–2005 (1988).
    [CrossRef] [PubMed]
  13. B. G. Bovard, “Fourier transform technique applied to quarter-wave optical coatings,” Appl. Opt. 27, 3062–3063 (1988).
    [CrossRef] [PubMed]
  14. B. G. Bovard, “Rugate filter design: the modified Fourier transform technique,” Appl. Opt. 29, 24–30 (1990).
    [CrossRef] [PubMed]
  15. P. G. Verly, “Fourier transform technique with frequency filtering for optical thin-film design,” Appl. Opt. 34, 688–694 (1995).
    [CrossRef] [PubMed]
  16. M. C. Parker, R. J. Mears, S. D. Walker, “A Fourier transform theory for photon localization and evanescence in photonic bandgap structures,” J. Opt. A Pure Appl. Opt. 3, 171–183 (2001).
    [CrossRef]
  17. P. Baumeister, “Simulation of a rugate filter via a stepped-index dielectric multilayer,” Appl. Opt. 25, 2644–2645 (1986).
    [CrossRef] [PubMed]
  18. M. M. Sánchez-López, J. A. Davis, K. Crabtree, “Coaxial cable analogs of multilayer dielectric optical coatings,” Am. J. Phys. 71, 1314–1319 (2003).
    [CrossRef]
  19. G. J. Schneider, S. Hanna, J. L. Davis, G. H. Watson, “Defect modes in coaxial photonic crystals,” J. Appl. Phys. 90, 2642–2649 (2001).
    [CrossRef]
  20. J. N. Munday, W. M. Robertson, “Negative group velocity pulse tunneling through a coaxial photonic crystal,” Appl. Phys. Lett. 81, 2127–2129 (2002).
    [CrossRef]
  21. A. Haché, A. Slimani, “A model coaxial photonic crystal for studying band structures, dispersion, field localization, and superluminal effects,” Am. J. Phys. 72, 916–921 (2004).
    [CrossRef]
  22. J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).
  23. S. Zhu, N. Liu, H. Zheng, H. Chen, “Time delay of light propagation through defect modes of one-dimensional photonic band-gap structures,” Opt. Commun. 174, 139–144 (2000).
    [CrossRef]
  24. L. Carretero, M. Ulibarrena, S. Blaya, A. Fimia, “One-dimensional photonic crystals with an amplitude-modulated dielectric constant in the unit cell,” Appl. Opt. 43, 2895–2899 (2004).
    [CrossRef] [PubMed]
  25. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), Chap. 3.
  26. G. F. Miner, Lines and Electromagnetic Fields for Engineers (Oxford U. Press, 1996), Chap. 1.
  27. N. G. R. Broderick, R. T. Bratfalean, T. M. Monro, D. J. Richardson, “Temperature and wavelength tuning of second, third and fourth-harmonic generation in a two-dimensional hexagonally poled nonlinear crystal,” J. Opt. Soc. Am. B 19, 2263–2272 (2002).
    [CrossRef]
  28. J. Mes, E. J. Van Duijn, R. Zinkstok, S. Witte, W. Hogervorst, “Third-harmonic generation of a continuous-wave Ti: sapphire laser in external resonant cavities,” Appl. Phys. Lett. 82, 4423–4425 (2003).
    [CrossRef]

2004 (2)

A. Haché, A. Slimani, “A model coaxial photonic crystal for studying band structures, dispersion, field localization, and superluminal effects,” Am. J. Phys. 72, 916–921 (2004).
[CrossRef]

L. Carretero, M. Ulibarrena, S. Blaya, A. Fimia, “One-dimensional photonic crystals with an amplitude-modulated dielectric constant in the unit cell,” Appl. Opt. 43, 2895–2899 (2004).
[CrossRef] [PubMed]

2003 (2)

M. M. Sánchez-López, J. A. Davis, K. Crabtree, “Coaxial cable analogs of multilayer dielectric optical coatings,” Am. J. Phys. 71, 1314–1319 (2003).
[CrossRef]

J. Mes, E. J. Van Duijn, R. Zinkstok, S. Witte, W. Hogervorst, “Third-harmonic generation of a continuous-wave Ti: sapphire laser in external resonant cavities,” Appl. Phys. Lett. 82, 4423–4425 (2003).
[CrossRef]

2002 (3)

H. Taniyama, “Waveguide structures using one-dimensional photonic crystal,” J. Appl. Phys. 91, 3511–3515 (2002).
[CrossRef]

J. N. Munday, W. M. Robertson, “Negative group velocity pulse tunneling through a coaxial photonic crystal,” Appl. Phys. Lett. 81, 2127–2129 (2002).
[CrossRef]

N. G. R. Broderick, R. T. Bratfalean, T. M. Monro, D. J. Richardson, “Temperature and wavelength tuning of second, third and fourth-harmonic generation in a two-dimensional hexagonally poled nonlinear crystal,” J. Opt. Soc. Am. B 19, 2263–2272 (2002).
[CrossRef]

2001 (3)

M. C. Parker, R. J. Mears, S. D. Walker, “A Fourier transform theory for photon localization and evanescence in photonic bandgap structures,” J. Opt. A Pure Appl. Opt. 3, 171–183 (2001).
[CrossRef]

G. J. Schneider, S. Hanna, J. L. Davis, G. H. Watson, “Defect modes in coaxial photonic crystals,” J. Appl. Phys. 90, 2642–2649 (2001).
[CrossRef]

M. Deopura, C. K. Ullal, B. Temelkuran, Y. Fink, “Dielectric omnidirectional visible reflector,” Opt. Lett. 26, 1197–1199 (2001).
[CrossRef]

2000 (1)

S. Zhu, N. Liu, H. Zheng, H. Chen, “Time delay of light propagation through defect modes of one-dimensional photonic band-gap structures,” Opt. Commun. 174, 139–144 (2000).
[CrossRef]

1997 (2)

P. L. Swart, P. V. Bulkin, B. M. Lacquet, “Rugate filter manufacturing by electron cyclotron resonance plasma-enhanced chemical vapor deposition of SiNx,” Opt. Eng. 36, 1214–1219 (1997).
[CrossRef]

W. H. Southwell, “Extended-bandwidth reflector designs by using wavelets,” Appl. Opt. 36, 314–318 (1997).
[CrossRef] [PubMed]

1995 (1)

1990 (1)

1989 (1)

1988 (2)

1986 (1)

1978 (1)

1963 (1)

Baumeister, P.

Blaya, S.

Bovard, B. G.

Bratfalean, R. T.

Broderick, N. G. R.

Bulkin, P. V.

P. L. Swart, P. V. Bulkin, B. M. Lacquet, “Rugate filter manufacturing by electron cyclotron resonance plasma-enhanced chemical vapor deposition of SiNx,” Opt. Eng. 36, 1214–1219 (1997).
[CrossRef]

Burton, R. L.

Carretero, L.

Chen, H.

S. Zhu, N. Liu, H. Zheng, H. Chen, “Time delay of light propagation through defect modes of one-dimensional photonic band-gap structures,” Opt. Commun. 174, 139–144 (2000).
[CrossRef]

Crabtree, K.

M. M. Sánchez-López, J. A. Davis, K. Crabtree, “Coaxial cable analogs of multilayer dielectric optical coatings,” Am. J. Phys. 71, 1314–1319 (2003).
[CrossRef]

Crane, R. L.

W. E. Johnson, R. L. Crane, “Introduction to rugate filter technology,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 88–108 (1993).
[CrossRef]

Davis, J. A.

M. M. Sánchez-López, J. A. Davis, K. Crabtree, “Coaxial cable analogs of multilayer dielectric optical coatings,” Am. J. Phys. 71, 1314–1319 (2003).
[CrossRef]

Davis, J. L.

G. J. Schneider, S. Hanna, J. L. Davis, G. H. Watson, “Defect modes in coaxial photonic crystals,” J. Appl. Phys. 90, 2642–2649 (2001).
[CrossRef]

Deopura, M.

Dobrowolski, J. A.

Fimia, A.

Fink, Y.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), Chap. 3.

Haché, A.

A. Haché, A. Slimani, “A model coaxial photonic crystal for studying band structures, dispersion, field localization, and superluminal effects,” Am. J. Phys. 72, 916–921 (2004).
[CrossRef]

Hanna, S.

G. J. Schneider, S. Hanna, J. L. Davis, G. H. Watson, “Defect modes in coaxial photonic crystals,” J. Appl. Phys. 90, 2642–2649 (2001).
[CrossRef]

Hogervorst, W.

J. Mes, E. J. Van Duijn, R. Zinkstok, S. Witte, W. Hogervorst, “Third-harmonic generation of a continuous-wave Ti: sapphire laser in external resonant cavities,” Appl. Phys. Lett. 82, 4423–4425 (2003).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Johnson, W. E.

W. E. Johnson, R. L. Crane, “Introduction to rugate filter technology,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 88–108 (1993).
[CrossRef]

Lacquet, B. M.

P. L. Swart, P. V. Bulkin, B. M. Lacquet, “Rugate filter manufacturing by electron cyclotron resonance plasma-enhanced chemical vapor deposition of SiNx,” Opt. Eng. 36, 1214–1219 (1997).
[CrossRef]

Liu, N.

S. Zhu, N. Liu, H. Zheng, H. Chen, “Time delay of light propagation through defect modes of one-dimensional photonic band-gap structures,” Opt. Commun. 174, 139–144 (2000).
[CrossRef]

Lowe, D.

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics Publishing, 1986), Chap. 6, pp. 248–255.

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Mears, R. J.

M. C. Parker, R. J. Mears, S. D. Walker, “A Fourier transform theory for photon localization and evanescence in photonic bandgap structures,” J. Opt. A Pure Appl. Opt. 3, 171–183 (2001).
[CrossRef]

Mes, J.

J. Mes, E. J. Van Duijn, R. Zinkstok, S. Witte, W. Hogervorst, “Third-harmonic generation of a continuous-wave Ti: sapphire laser in external resonant cavities,” Appl. Phys. Lett. 82, 4423–4425 (2003).
[CrossRef]

Miner, G. F.

G. F. Miner, Lines and Electromagnetic Fields for Engineers (Oxford U. Press, 1996), Chap. 1.

Monro, T. M.

Munday, J. N.

J. N. Munday, W. M. Robertson, “Negative group velocity pulse tunneling through a coaxial photonic crystal,” Appl. Phys. Lett. 81, 2127–2129 (2002).
[CrossRef]

Parker, M. C.

M. C. Parker, R. J. Mears, S. D. Walker, “A Fourier transform theory for photon localization and evanescence in photonic bandgap structures,” J. Opt. A Pure Appl. Opt. 3, 171–183 (2001).
[CrossRef]

Perilloux, B. E.

B. E. Perilloux, Thin-Film Design: Modulated Thickness and Other Stopband Design Methods (SPIE Press, 2002), Chaps. 1 and 2.
[CrossRef]

Richardson, D. J.

Robertson, W. M.

J. N. Munday, W. M. Robertson, “Negative group velocity pulse tunneling through a coaxial photonic crystal,” Appl. Phys. Lett. 81, 2127–2129 (2002).
[CrossRef]

Sánchez-López, M. M.

M. M. Sánchez-López, J. A. Davis, K. Crabtree, “Coaxial cable analogs of multilayer dielectric optical coatings,” Am. J. Phys. 71, 1314–1319 (2003).
[CrossRef]

Schneider, G. J.

G. J. Schneider, S. Hanna, J. L. Davis, G. H. Watson, “Defect modes in coaxial photonic crystals,” J. Appl. Phys. 90, 2642–2649 (2001).
[CrossRef]

Slimani, A.

A. Haché, A. Slimani, “A model coaxial photonic crystal for studying band structures, dispersion, field localization, and superluminal effects,” Am. J. Phys. 72, 916–921 (2004).
[CrossRef]

Southwell, W. H.

Swart, P. L.

P. L. Swart, P. V. Bulkin, B. M. Lacquet, “Rugate filter manufacturing by electron cyclotron resonance plasma-enhanced chemical vapor deposition of SiNx,” Opt. Eng. 36, 1214–1219 (1997).
[CrossRef]

Taniyama, H.

H. Taniyama, “Waveguide structures using one-dimensional photonic crystal,” J. Appl. Phys. 91, 3511–3515 (2002).
[CrossRef]

Temelkuran, B.

Thelen, A.

Ulibarrena, M.

Ullal, C. K.

Van Duijn, E. J.

J. Mes, E. J. Van Duijn, R. Zinkstok, S. Witte, W. Hogervorst, “Third-harmonic generation of a continuous-wave Ti: sapphire laser in external resonant cavities,” Appl. Phys. Lett. 82, 4423–4425 (2003).
[CrossRef]

Verly, P. G.

Walker, S. D.

M. C. Parker, R. J. Mears, S. D. Walker, “A Fourier transform theory for photon localization and evanescence in photonic bandgap structures,” J. Opt. A Pure Appl. Opt. 3, 171–183 (2001).
[CrossRef]

Watson, G. H.

G. J. Schneider, S. Hanna, J. L. Davis, G. H. Watson, “Defect modes in coaxial photonic crystals,” J. Appl. Phys. 90, 2642–2649 (2001).
[CrossRef]

Wild, W. J.

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Witte, S.

J. Mes, E. J. Van Duijn, R. Zinkstok, S. Witte, W. Hogervorst, “Third-harmonic generation of a continuous-wave Ti: sapphire laser in external resonant cavities,” Appl. Phys. Lett. 82, 4423–4425 (2003).
[CrossRef]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, 1988), Chaps. 5 and 6.

Zheng, H.

S. Zhu, N. Liu, H. Zheng, H. Chen, “Time delay of light propagation through defect modes of one-dimensional photonic band-gap structures,” Opt. Commun. 174, 139–144 (2000).
[CrossRef]

Zhu, S.

S. Zhu, N. Liu, H. Zheng, H. Chen, “Time delay of light propagation through defect modes of one-dimensional photonic band-gap structures,” Opt. Commun. 174, 139–144 (2000).
[CrossRef]

Zinkstok, R.

J. Mes, E. J. Van Duijn, R. Zinkstok, S. Witte, W. Hogervorst, “Third-harmonic generation of a continuous-wave Ti: sapphire laser in external resonant cavities,” Appl. Phys. Lett. 82, 4423–4425 (2003).
[CrossRef]

Am. J. Phys. (2)

M. M. Sánchez-López, J. A. Davis, K. Crabtree, “Coaxial cable analogs of multilayer dielectric optical coatings,” Am. J. Phys. 71, 1314–1319 (2003).
[CrossRef]

A. Haché, A. Slimani, “A model coaxial photonic crystal for studying band structures, dispersion, field localization, and superluminal effects,” Am. J. Phys. 72, 916–921 (2004).
[CrossRef]

Appl. Opt. (9)

Appl. Phys. Lett. (2)

J. N. Munday, W. M. Robertson, “Negative group velocity pulse tunneling through a coaxial photonic crystal,” Appl. Phys. Lett. 81, 2127–2129 (2002).
[CrossRef]

J. Mes, E. J. Van Duijn, R. Zinkstok, S. Witte, W. Hogervorst, “Third-harmonic generation of a continuous-wave Ti: sapphire laser in external resonant cavities,” Appl. Phys. Lett. 82, 4423–4425 (2003).
[CrossRef]

J. Appl. Phys. (2)

H. Taniyama, “Waveguide structures using one-dimensional photonic crystal,” J. Appl. Phys. 91, 3511–3515 (2002).
[CrossRef]

G. J. Schneider, S. Hanna, J. L. Davis, G. H. Watson, “Defect modes in coaxial photonic crystals,” J. Appl. Phys. 90, 2642–2649 (2001).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

M. C. Parker, R. J. Mears, S. D. Walker, “A Fourier transform theory for photon localization and evanescence in photonic bandgap structures,” J. Opt. A Pure Appl. Opt. 3, 171–183 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

S. Zhu, N. Liu, H. Zheng, H. Chen, “Time delay of light propagation through defect modes of one-dimensional photonic band-gap structures,” Opt. Commun. 174, 139–144 (2000).
[CrossRef]

Opt. Eng. (1)

P. L. Swart, P. V. Bulkin, B. M. Lacquet, “Rugate filter manufacturing by electron cyclotron resonance plasma-enhanced chemical vapor deposition of SiNx,” Opt. Eng. 36, 1214–1219 (1997).
[CrossRef]

Opt. Lett. (1)

Other (7)

H. A. Macleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics Publishing, 1986), Chap. 6, pp. 248–255.

P. Yeh, Optical Waves in Layered Media (Wiley, 1988), Chaps. 5 and 6.

W. E. Johnson, R. L. Crane, “Introduction to rugate filter technology,” in Inhomogeneous and Quasi-Inhomogeneous Optical Coatings, J. A. Dobrowolski, P. G. Verly, eds., Proc. SPIE2046, 88–108 (1993).
[CrossRef]

B. E. Perilloux, Thin-Film Design: Modulated Thickness and Other Stopband Design Methods (SPIE Press, 2002), Chaps. 1 and 2.
[CrossRef]

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978), Chap. 3.

G. F. Miner, Lines and Electromagnetic Fields for Engineers (Oxford U. Press, 1996), Chap. 1.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

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Figures (7)

Fig. 1
Fig. 1

Index profile n(x) of an (HL)N structure and its logarithmic derivative function g(x).

Fig. 2
Fig. 2

Calculated transmittance for a coaxial structure (HL)4 with H = (75 Ω, DH = 3 ft) and L = [50 Ω, dL = d(1 − a) ft]: (a) a = 1/2, DL = 3 ft; (b) a = 1/3, DL = 6 ft; (c) a = 1/4, DL = 9 ft; (d) a = 1/5, DL = 12 ft. Insets, the impedance profiles.

Fig. 3
Fig. 3

Phasor diagrams for several harmonic frequencies for a = 1/2, 1/3, 1/4, 1/5.

Fig. 4
Fig. 4

Calculated transmittance for a coaxial structure (LHCHL)3 with L = (50 Ω, 1 m), H = (75 Ω, 1 m), and C = (93 Ω, 1.27 m). Inset, the impedance profile. The phase velocity in H and L is υ = 0.67c and in C is υ = 0.85c.

Fig. 5
Fig. 5

Phasor diagrams of the filter (LHCHL)3 for the harmonic frequencies m = 1, 2 … 6.

Fig. 6
Fig. 6

Experimentally measured transmittance for a coaxial structure (HL)4 with H = (75 Ω, DH = 3 ft) and L = [50 Ω, dL = d(1 − a) ft]: (a) a = 1/2, DL = 3 ft; (b) a = 1/3, DL = 6 ft; (c) a = 1/4, DL = 9 ft; (d) a = 1/5, DL = 12 ft.

Fig. 7
Fig. 7

Experimentally measured transmittance for a coaxial structure (LHCHL)3 with L = (50 Ω, 1 m), H = (75 Ω, 1 m), and C = (93 Ω, 1.27 m).

Tables (1)

Tables Icon

Table 1 Values of |t| at the Gap Central Frequencies of the (HL)4 Structures of Fig. 2

Equations (31)

Equations on this page are rendered with MathJax. Learn more.

g ( x ) = n ( x ) 2 n ( x ) ,
Q = 1 2 ln ( 1 + R 1 R ) R = tanh 2 ( Q ) .
g ( x ) = r [ m = δ ( x m d ) m = δ ( x x 0 m d ) ] ,
r = 1 2 ln ( n H n L ) ,
Q ( σ ) = FT [ g ( x ) ] = r [ comb ( d σ ) exp ( i 2 π x 0 σ ) comb ( d σ ) ] ,
comb ( d σ ) = 1 d m = + δ ( σ m d ) .
Q ( σ ) = 1 d m = + ξ δ ( σ m d ) ,
ξ = r [ 1 exp ( i 2 π m a ) ] .
f m = m c d = m ( 2 D L υ L + 2 D H υ H ) 1 ,
f 1 = υ 2 D ,
g N ( x ) = g ( x ) rect ( x N d / 2 N d ) ,
rect ( x ) = { 1 | x | 1 / 2 0 else
Q N ( σ ) = FT [ g N ( x ) ] = Q ( σ ) N d exp ( i π σ ) sinc ( N d σ ) ,
Q N ( σ ) = N m = + ξ exp [ i π ( σ m d ) ] = sinc [ N d ( σ m d ) ] .
Q N ( σ = n d ) = N ξ = N r [ 1 exp ( i 2 π n a ) ] ,
| t | = 1 R = 1 | cosh ( Q N ) | = 1 | cosh ( N ξ ) | ,
f 1 = ( 4 D L υ L + 4 D H υ H + 2 D C υ C ) 1 20.1 MHz .
g ( x ) = r 1 m = δ ( x m d ) + r 2 m = δ ( x d 5 m d ) r 2 m = δ ( x 2 d 5 m d ) r 1 m = δ ( x 3 d 5 m d ) ,
ξ = [ r 1 + r 2 exp ( i 2 π m 5 ) r 2 exp ( i 4 π m 5 ) r 1 exp ( i 6 π m 5 ) ] .
3 r
2 r
P ( φ ) = [ exp ( + i φ ) 0 0 exp ( i φ ) ] ,
M 12 = 1 t 12 [ 1 r 12 r 12 1 ] ,
M 0 = 1 t H L [ 1 r H L r H L 1 ] [ + i 0 0 i ] 1 t H L [ 1 r H L r H L 1 ] [ + i 0 0 i ] .
M 0 = 1 1 r H L 2 [ 1 + r H L 2 2 r H L 2 r H L 1 + r H L 2 ] .
M 0 N = 1 sin ( θ ) [ M 11 sin ( N θ ) sin [ ( N 1 ) θ ] M 12 sin ( N θ ) M 21 sin ( N θ ) M 22 sin ( N θ ) sin [ ( N 1 ) θ ] ] ,
cos ( θ ) = ( 1 / 2 ) ( M 11 + M 22 ) .
M 0 N = [ cos ( N θ ) M 12 sin ( N θ ) sin ( θ ) M 21 sin ( N θ ) sin ( θ ) cos ( N θ ) ] .
| t | = 1 | cos ( N θ ) | = 1 | cosh ( N ϕ ) | ,
cos ( θ ) = cosh ( ϕ ) = 1 + r HL 2 1 r HL 2 1 .
cosh ( ϕ ) = n H 2 + n L 2 2 n H n L = cosh ( 2 r ) ,

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