Abstract

We analyze the dispersive properties of inhomogeneous nanostructures (INSs) composed of alternating layers of different materials. Analysis of the interaction between the propagating pulse and the INS provides modified dispersion characteristics. An approximate theoretical model predicting the dispersion properties of the INS is developed and compared with more accurate numeric computation results. It is shown that the dispersion coefficient can be engineered by controlling the spatial distribution of the pulse carrier, the geometry of the INS, and the refractive indices of the materials combined to construct the INS. Specifically, the dispersion coefficient can be engineered to yield various types of dispersion, including normal dispersion, anomalous dispersion, and zero dispersion. As such, the discussed INS can be useful for applications that will benefit from engineered dispersion management and control.

© 2004 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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2003 (1)

2002 (2)

2001 (3)

2000 (1)

M. A. Porras, R. Borghi, M. Santarsiero, “Few-optical-cycle Bessel–Gauss pulsed beams in free space,” Phys. Rev. E 62, 5729–5737 (2000).
[CrossRef]

1999 (5)

1998 (2)

1997 (4)

1996 (4)

1995 (2)

1994 (2)

K. D. Paulsen, “Finite-element solution of Maxwell’s equations with Helmholtz forms,” J. Opt. Soc. Am. A 11, 1434–1444 (1994).
[CrossRef]

A. J. Antos, D. K. Smith, “Design and characterization of dispersion compensating fiber based on the LP01 mode,” J. Lightwave Technol. 12, 1739–1745 (1994).
[CrossRef]

1992 (1)

J. Lu, J. F. Greenleef, “Nondiffracting X waves—exact solutions to free space scalar wave equation and their finite aperture realization,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[CrossRef]

1987 (1)

1981 (1)

1956 (1)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Agrawal, G. P.

G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, New York, 1992).

Andreas, P.

D. Mendlovic, Z. Zalevsky, P. Andreas, “A novel device for achieving negative or positive dispersion and its applications,” Optik (Stuttgart) 110, 45–50 (1999).

Angelow, G.

Antos, A. J.

A. J. Antos, D. K. Smith, “Design and characterization of dispersion compensating fiber based on the LP01 mode,” J. Lightwave Technol. 12, 1739–1745 (1994).
[CrossRef]

Bass, M.

M. Bass, E. W. V. Stryland, D. R. Williams, W. L. Wolfe, Handbook of Optics, 2nd ed. (McGraw-Hill, New York, 1995), Chap. 33, pp. 26, 33, 63, and 69.

Bilinsky, I. P.

Birks, T. A.

T. A. Birks, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, “Dispersion compensation using singlematerial fibers,” IEEE Photon. Technol. Lett. 11, 674–676 (1999).
[CrossRef]

Borghi, R.

M. A. Porras, R. Borghi, M. Santarsiero, “Few-optical-cycle Bessel–Gauss pulsed beams in free space,” Phys. Rev. E 62, 5729–5737 (2000).
[CrossRef]

Bruce, A. J.

C. K. Madsen, G. Lenz, T. N. Nielsen, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, “Integrated optical all pass filters for dispersion compensation,” in WDM Components, D. A. Nolan, ed., Vol. 29 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), pp. 142–149.

Brundrett, D. L.

Cappuzzo, M. A.

C. K. Madsen, G. Lenz, T. N. Nielsen, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, “Integrated optical all pass filters for dispersion compensation,” in WDM Components, D. A. Nolan, ed., Vol. 29 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), pp. 142–149.

Chen, Y.

Cheng,

Cheng, C.

Cho, S. H.

Choi, S.-J.

Dapkus, P. D.

De Silvestri, S.

Deguzman, P.

Djordjev, K.

Fainman, Y.

Fan, D. Y.

Z. Y. Liu, D. Y. Fan, “Propagation of pulsed zeroth-order Bessel beams,” J. Mod. Opt. 45, 17–21 (1998).
[CrossRef]

Ferencz, K.

Fujimoto, J. G.

Gallmann, L.

Gaylord, T. K.

Genoud, F. M.

Glytsis, E. N.

Gomez, L. T.

C. K. Madsen, G. Lenz, T. N. Nielsen, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, “Integrated optical all pass filters for dispersion compensation,” in WDM Components, D. A. Nolan, ed., Vol. 29 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), pp. 142–149.

Goodman, J.

J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Greenleef, J. F.

J. Lu, J. F. Greenleef, “Nondiffracting X waves—exact solutions to free space scalar wave equation and their finite aperture realization,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[CrossRef]

Gu, C.

Guo, H.

Hagness, S.

A. Taflove, S. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method2nd ed. (Artech House, Norwood, Mass., 2000).

Haus, H. A.

Hu, W.

Ippen, E. P.

Iwata, K.

Kaminow, I.

I. Kaminow, T. Li, Optical Fiber Telecommunications IV, 4th ed. (Academic, San Diego, Calif., 2002), Part B, pp. 642–724.

Kartner, F. X.

Keller, U.

Kikuta, H.

Knight, J. C.

T. A. Birks, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, “Dispersion compensation using singlematerial fibers,” IEEE Photon. Technol. Lett. 11, 674–676 (1999).
[CrossRef]

Krausz, F.

Lenz, G.

C. K. Madsen, G. Lenz, T. N. Nielsen, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, “Integrated optical all pass filters for dispersion compensation,” in WDM Components, D. A. Nolan, ed., Vol. 29 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), pp. 142–149.

Li, T.

I. Kaminow, T. Li, Optical Fiber Telecommunications IV, 4th ed. (Academic, San Diego, Calif., 2002), Part B, pp. 642–724.

Liu, Z.

Liu, Z. Y.

Z. Y. Liu, D. Y. Fan, “Propagation of pulsed zeroth-order Bessel beams,” J. Mod. Opt. 45, 17–21 (1998).
[CrossRef]

Lu, B.

Lu, J.

J. Lu, J. F. Greenleef, “Nondiffracting X waves—exact solutions to free space scalar wave equation and their finite aperture realization,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[CrossRef]

Madsen, C. K.

C. K. Madsen, G. Lenz, T. N. Nielsen, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, “Integrated optical all pass filters for dispersion compensation,” in WDM Components, D. A. Nolan, ed., Vol. 29 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), pp. 142–149.

Mait, J. N.

Matuschek, N.

Mendlovic, D.

D. Mendlovic, Z. Zalevsky, P. Andreas, “A novel device for achieving negative or positive dispersion and its applications,” Optik (Stuttgart) 110, 45–50 (1999).

Mirotznik, M. S.

Missaggia, L. J.

Mogilevtsev, D.

T. A. Birks, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, “Dispersion compensation using singlematerial fibers,” IEEE Photon. Technol. Lett. 11, 674–676 (1999).
[CrossRef]

Moharam, M. G.

Morgner, U.

Nakagawa, W.

Nielsen, T. N.

C. K. Madsen, G. Lenz, T. N. Nielsen, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, “Integrated optical all pass filters for dispersion compensation,” in WDM Components, D. A. Nolan, ed., Vol. 29 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), pp. 142–149.

Nisoli, I. M.

Nordin, G.

Ohira, Y.

Parther, D. W.

Paulsen, K. D.

Porras, M. A.

M. A. Porras, “Diffraction-free and dispersion-free pulsed beam propagation in dispersive media,” Opt. Lett. 26, 1364–1366 (2001).
[CrossRef]

M. A. Porras, R. Borghi, M. Santarsiero, “Few-optical-cycle Bessel–Gauss pulsed beams in free space,” Phys. Rev. E 62, 5729–5737 (2000).
[CrossRef]

Qeullen, F.

Richter, I.

Russell, P. St. J.

T. A. Birks, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, “Dispersion compensation using singlematerial fibers,” IEEE Photon. Technol. Lett. 11, 674–676 (1999).
[CrossRef]

Rytov, S. M.

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), pp. 182–191.

Salvekar, A.

Santarsiero, M.

M. A. Porras, R. Borghi, M. Santarsiero, “Few-optical-cycle Bessel–Gauss pulsed beams in free space,” Phys. Rev. E 62, 5729–5737 (2000).
[CrossRef]

Sartania, S.

Scherer, A.

Scheuer, V.

Sheppard, C. J. R.

Smith, D. K.

A. J. Antos, D. K. Smith, “Design and characterization of dispersion compensating fiber based on the LP01 mode,” J. Lightwave Technol. 12, 1739–1745 (1994).
[CrossRef]

Spielmann, C.

Steinmeyer, G.

Stryland, E. W. V.

M. Bass, E. W. V. Stryland, D. R. Williams, W. L. Wolfe, Handbook of Optics, 2nd ed. (McGraw-Hill, New York, 1995), Chap. 33, pp. 26, 33, 63, and 69.

Sun, P. C.

Sutter, D. H.

Svelto, O.

Szipocs, R.

Taflove, A.

A. Taflove, S. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method2nd ed. (Artech House, Norwood, Mass., 2000).

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), pp. 182–191.

Tschudi, T.

Tyan, R.

Tyan, R. C.

Walpole, J. N.

Williams, D. R.

M. Bass, E. W. V. Stryland, D. R. Williams, W. L. Wolfe, Handbook of Optics, 2nd ed. (McGraw-Hill, New York, 1995), Chap. 33, pp. 26, 33, 63, and 69.

Wolfe, W. L.

M. Bass, E. W. V. Stryland, D. R. Williams, W. L. Wolfe, Handbook of Optics, 2nd ed. (McGraw-Hill, New York, 1995), Chap. 33, pp. 26, 33, 63, and 69.

Xu, F.

Yeh, P.

Zalevsky, Z.

D. Mendlovic, Z. Zalevsky, P. Andreas, “A novel device for achieving negative or positive dispersion and its applications,” Optik (Stuttgart) 110, 45–50 (1999).

Appl. Opt. (3)

IEEE Photon. Technol. Lett. (1)

T. A. Birks, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, “Dispersion compensation using singlematerial fibers,” IEEE Photon. Technol. Lett. 11, 674–676 (1999).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

J. Lu, J. F. Greenleef, “Nondiffracting X waves—exact solutions to free space scalar wave equation and their finite aperture realization,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, 19–31 (1992).
[CrossRef]

J. Lightwave Technol. (2)

A. J. Antos, D. K. Smith, “Design and characterization of dispersion compensating fiber based on the LP01 mode,” J. Lightwave Technol. 12, 1739–1745 (1994).
[CrossRef]

K. Djordjev, S.-J. Choi, S.-J. Choi, P. D. Dapkus, “Study of the effects of the geometry on the performance of vertically coupled InP microdisk resonators,” J. Lightwave Technol. 20, 1485–1492 (2002).
[CrossRef]

J. Mod. Opt. (1)

Z. Y. Liu, D. Y. Fan, “Propagation of pulsed zeroth-order Bessel beams,” J. Mod. Opt. 45, 17–21 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Opt. Express (1)

Opt. Lett. (10)

M. A. Porras, “Diffraction-free and dispersion-free pulsed beam propagation in dispersive media,” Opt. Lett. 26, 1364–1366 (2001).
[CrossRef]

F. Qeullen, “Dispersion cancellation using linearly chirped Bragg grating filters in optical waveguides,” Opt. Lett. 12, 847–849 (1987).
[CrossRef]

C. Gu, P. Yeh, “Form birefringence dispersion in periodic layered media,” Opt. Lett. 21, 504–506 (1996).
[CrossRef] [PubMed]

R. Tyan, P. C. Sun, A. Scherer, Y. Fainman, “Polarizing beam splitter based on the anisotropic spectral reflectivity characteristic of form-birefringent multilayer gratings,” Opt. Lett. 21, 761–763 (1996).
[CrossRef] [PubMed]

F. Xu, R. Tyan, P. C. Sun, Y. Fainman, C. Cheng, A. Scherer, “Form-birefringent computer-generated holograms,” Opt. Lett. 21, 1513–1515 (1996).
[CrossRef] [PubMed]

F. Xu, R. Tyan, P. C. Sun, C. Cheng, A. Scherer, Y. Fainman, “Fabrication, modeling, and characterization of form-birefringent nanostructures,” Opt. Lett. 20, 2457–2459 (1995).
[CrossRef] [PubMed]

I. M. Nisoli, S. De Silvestri, O. Svelto, R. Szipocs, K. Ferencz, C. Spielmann, S. Sartania, F. Krausz, “Compression of high-energy laser pulses below 5 fs,” Opt. Lett. 22, 522–524 (1997).
[CrossRef] [PubMed]

I. P. Bilinsky, J. G. Fujimoto, J. N. Walpole, L. J. Missaggia, “Semiconductor-doped-silica saturable-absorber films for solid-state laser mode locking,” Opt. Lett. 23, 1766–1768 (1998).
[CrossRef]

U. Morgner, F. X. Kartner, S. H. Cho, Y. Chen, H. A. Haus, J. G. Fujimoto, E. P. Ippen, V. Scheuer, G. Angelow, T. Tschudi, “Sub-two-cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser,” Opt. Lett. 24, 411–413 (1999).
[CrossRef]

D. H. Sutter, G. Steinmeyer, L. Gallmann, N. Matuschek, F. M. Genoud, U. Keller, V. Scheuer, G. Angelow, T. Tschudi, “Semiconductor saturable-absorber mirror-assisted Kerr-lens mode-locked Ti:sapphire laser producing pulses in the two-cycle regime,” Opt. Lett. 24, 631–633 (1999).
[CrossRef]

Optik (Stuttgart) (1)

D. Mendlovic, Z. Zalevsky, P. Andreas, “A novel device for achieving negative or positive dispersion and its applications,” Optik (Stuttgart) 110, 45–50 (1999).

Phys. Rev. E (1)

M. A. Porras, R. Borghi, M. Santarsiero, “Few-optical-cycle Bessel–Gauss pulsed beams in free space,” Phys. Rev. E 62, 5729–5737 (2000).
[CrossRef]

Sov. Phys. JETP (1)

S. M. Rytov, “Electromagnetic properties of a finely stratified medium,” Sov. Phys. JETP 2, 466–475 (1956).

Other (7)

A. Taflove, S. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method2nd ed. (Artech House, Norwood, Mass., 2000).

C. K. Madsen, G. Lenz, T. N. Nielsen, A. J. Bruce, M. A. Cappuzzo, L. T. Gomez, “Integrated optical all pass filters for dispersion compensation,” in WDM Components, D. A. Nolan, ed., Vol. 29 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1999), pp. 142–149.

I. Kaminow, T. Li, Optical Fiber Telecommunications IV, 4th ed. (Academic, San Diego, Calif., 2002), Part B, pp. 642–724.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), pp. 182–191.

G. P. Agrawal, Fiber-Optic Communication Systems (Wiley, New York, 1992).

J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

M. Bass, E. W. V. Stryland, D. R. Williams, W. L. Wolfe, Handbook of Optics, 2nd ed. (McGraw-Hill, New York, 1995), Chap. 33, pp. 26, 33, 63, and 69.

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of the INS geometry.

Fig. 2
Fig. 2

Structural linear dispersion coefficient versus d / λ in a GaAs–air periodic structure (fill factor, f = 0.5 ) for (a) TE and (b) TM polarization states. Solid curve, EMT model; dashed–dotted curve, accurate results from use of RCWA; horizontal line, inherent material dispersion. [ps/km/nm], [picoseconds per (kilometer times nanometer)].

Fig. 3
Fig. 3

Total linear dispersion coefficient versus d / λ in a GaAs–air periodic structure (fill factor f = 0.5 ) for TE (solid curve) and TM (dashed–dotted curve) polarization states.

Fig. 4
Fig. 4

Total linear dispersion coefficient versus d / λ in a fused-silica–air periodic structure (fill factor f = 0.5 ) for TE (solid curve) and TM (dashed–dotted curve) polarization states.

Fig. 5
Fig. 5

Total linear dispersion coefficient versus angular frequency α in a GaAs–air periodic structure (fill factor f = 0.5 ): (a) TE polarization and (b) TM polarization.

Fig. 6
Fig. 6

Total linear dispersion coefficient versus angular frequency α in a fused-silica–air periodic structure (fill factor f = 0.5 ): (a) TE polarization and (b) TM polarization.

Equations (40)

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

E ( x ,   z = 0 ,   t ) = - E ¯ ( x ,   z = 0 ,   ω ) exp [ j ( ω - ω 0 ) t ] d ω ,
E ¯ ( x ,   z = 0 ,   ω ) = - A ( f x ,   z = 0 ) exp ( j 2 π f x x ) H ( ω ) d f x = - U ( f x ,   z = 0 ,   ω ) exp ( j 2 π f x x ) d f x ,
E ¯ ( x ,   z ,   ω ) = - A ( f x ,   z ) exp ( j 2 π f x x ) H ( ω ) d f x = - U ( f x ,   z ,   ω ) exp ( j 2 π f x x ) d f x .
2 x 2 + 2 z 2 + K ( ω ) 2 E ¯ ( x ,   z ,   ω ) = 0 .
U ( f x ,   z ,   ω ) = U ( f x ,   z = 0 ,   ω ) exp ( j β z ) ,
β = [ K 2 ( ω ) - 4 π 2 f x 2 ] 1 / 2 .
2 π f x = K ( ω ) sin ( θ ) = α ,
β ( ω ) = [ K 2 ( ω ) - K 2 ( ω ) sin 2 ( θ ) ] 1 / 2 = [ K 2 ( ω ) - α 2 ] 1 / 2 = K ( ω ) cos ( θ ) = ω C   n   cos ( θ ) = 2 π ν C   n   cos ( θ ) ,
E ¯ ( x ,   z ,   ω ) 2 0 + U ( α ,   0 ,   ω ) exp [ j β ( ω ) z ] × cos ( α x ) d α + U ( 0 ,   0 ,   ω ) exp [ jK ( ω ) z ] .
U ( α ,   0 ,   ω ) = n = 1 N U ( α n ,   0 ,   ω ) δ ( α - α n ) ,
E ¯ ( x ,   z ,   ω ) n = 1 N U ( α n ,   0 ,   ω ) exp [ j β n ( ω ) z ] cos ( α n x ) ,
E ¯ ( x ,   z ,   ω ) U ( α ,   0 ,   ω ) exp [ j β ( ω ) z ] cos ( α x ) .
β ( ω ) β ( ω 0 ) + β ( ω 0 ) ( ω - ω 0 ) + 1 2   β ( ω 0 ) ( ω - ω 0 ) 2 + = β ( ν 0 ) + 2 π V g   ( ν - ν 0 ) + π D ν ( ν - ν 0 ) 2 ,
1 V g = 1 2 π d β d ν ν 0 = d β d ω ω 0
D ν = 1 2 π d 2 β d ν 2 ν = ν 0 = 2 π d 2 β d ω 2 ω = ω 0 = d d ν 1 V g ν = ν 0 ,
E ¯ ( x ,   z ,   ν ) H 0 ( ν ) exp [ j β ( ν 0 ) z ] exp [ j 2 π z ( ν - ν 0 ) / V g ] × exp [ j π D ν z ( ν - ν 0 ) 2 ] .
E ¯ ( x ,   z ,   ν ) = H 0 ( ν ) exp [ j β ( ν ) z ] = H 0 ( ν ) exp   jK ( ν ) z ,
U ( α ,   0 ,   ν ) = H ( ν ) α = 0 0 otherwise
n TE ( 2 ) = [ n TE ( 0 ) ] 2 + 1 3 d λ   π f ( 1 - f   ) ( n 1 2 - n 2 2 ) 2 1 / 2 ,
n TM ( 2 ) = [ n TM ( 0 ) ] 2 + 1 3 d λ   π f ( 1 - f   ) 1 n 1 2 - 1 n 2 2 n TE ( 0 ) × [ n TM ( 0 ) ] 3 2 1 / 2 ,
n TE ( 0 ) = [ fn 1 2 + ( 1 - f   ) n 2 2 ] 1 / 2 ,
n TM ( 0 ) = n 1 n 2 [ fn 2 2 + ( 1 - f   ) n 1 2 ] 1 / 2 .
η TE = d π f ( 1 - f   ) ( n 1 2 - n 2 2 ) c 3 ,
η TM = d π f ( 1 - f   ) 1 n 1 2 - 1 n 2 2 n TE ( 0 ) [ n TM ( 0 ) ] 3 c 3 ,
n j ( 2 ) = { [ n j ( 0 ) ] 2 + [ η j ν ] 2 } 1 / 2 ,
n j ( 2 ) n j ( 0 ) + η j 2 ν 2 2 n j ( 0 ) .
V g ( j ) = 2 n j ( 0 ) c 2 [ n j ( 0 ) ] 2 + 3 ( η i ν 0 ) 2 ,
D ν j = 3 η j 2 cn j ( 0 ) ν 0 ,
D λ = - ν 2 c   D ν = - 3 η j 2 n j ( 0 ) c λ 3 .
exp [ j π z β 3 ( ν - ν 0 ) 2 / 3 ] ,
β 3 = 1 2 π 3 β ν 3 ν = ν 0 = 3 η j 2 cn j ( 0 ) .
L 2 = ( 2 π ) 2 T 2 D ν ,
L 3 = 3 ( 2 π ) 3 T 3 β 3 ,
L 3 / L 2 = 6 π T / ν 0 .
n 2 - 1 = i = 1 A i λ 2 λ 2 - λ i 2 .
d n TE d n = n [ 2 ( n 2 + 1 ) ] 1 / 2 = 0.678 ,
d n TM d n = 1 2 ( n 2 + 1 ) 3 / 2 = 0.0163 .
A 1 = 0.6961663 , A 2 = 0.4079426 , A 3 = 0.8974794 ,
λ 1 = 0.0684043 , λ 2 = 0.1162414 , λ 3 = 9.896161 .
D ν = 1 2 π d 2 β d ν 2 = 1 2 π   cos ( θ ) K + ( K ) 2 K 1 - 1 cos 2 ( θ ) = 1 2 π   cos ( θ ) K - ( K ) 2 K tan 2 ( θ ) ,

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