Abstract

Concept, theory and simulations of a new type of waveguide device, a multiaperture Fourier-transform planar waveguide spectrometer, are presented. The spectrometer is formed by an array of Mach-Zehnder interferometers generating a wavelength dependent spatial fringe pattern at the array output. The input light spectrum is calculated using a discrete Fourier transformation of the output spatial fringes. The multiaperture input significantly increases the optical throughput (étendue) compared to conventional single input spectrometers. Design rules for the arrayed spectrometer are deduced from performance specifications such as wavelength range and spectral resolution. A design example with spectral resolution 0.025 nm and range 2.5 nm is presented, where the optical throughput is increased by a factor of 200 compared to a single input device.

© 2007 Optical Society of America

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References

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2007 (3)

2006 (2)

I. Powell and P. Cheben, “Modeling of the generic spatial heterodyne spectrometer and comparison with conventional spectrometer,” Appl. Opt. 45, 9079–9086 (2006).
[Crossref] [PubMed]

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A: Pure Appl. Opt. 8, 840–848 (2006).
[Crossref]

2005 (1)

2003 (1)

2000 (1)

K. Takada, T. Tanaka, M. Abe, T. Yanagisawa, M. Ishii, and K. Okamoto, “Beam-adjustment-free crosstalk reduction in 10GHz-spaced arrayed-waveguide grating via photosensitivity under UV laser irradiation through metal mask,” Electron. Lett. 36, 60–61 (2000).
[Crossref]

1996 (1)

H. Yamada, K. Takada, Y. Inoue, Y. Ohmori, and S. Mitachi, “Statically-phase-compensated 10GHzspaced arrayed-waveguide grating,” Electron. Lett. 32, 1580–1582-61 (1996).
[Crossref]

1994 (1)

K. Takada, Y. Inoue, H. Yamada, and M. Horiguchi, “Measurement of phase error distributions in silicabased arrayed-waveguide grating multiplexers by using Fourier transform spectroscopy,” Electron. Lett. 30, 1671–1672 (1994).
[Crossref]

1992 (1)

J. Harlander, R. J. Reynolds, and F. L. Roesler, “Spatial heterodyne spectroscopy for the exploration of diffuse interstellar emission lines at far-ultraviolet wavelengths,” Astrophys. J. 396, 730–740 (1992).
[Crossref]

1991 (1)

1981 (1)

1974 (1)

1964 (1)

1958 (1)

P. Fellgett, “A propos de la théorie du spectromètre interférentiel multiplex,” J. Phys. Radium 19, 187–191 (1958).
[Crossref]

1954 (1)

Abe, M.

K. Takada, T. Tanaka, M. Abe, T. Yanagisawa, M. Ishii, and K. Okamoto, “Beam-adjustment-free crosstalk reduction in 10GHz-spaced arrayed-waveguide grating via photosensitivity under UV laser irradiation through metal mask,” Electron. Lett. 36, 60–61 (2000).
[Crossref]

Alieva, T.

Beer, R.

R. H. Norton and R. Beer, “New apodizing functions for Fourier spectrometry,” J. Opt. Soc. Am.66, 259–264 (1976); erratum: 67, 419 (1977), http://www.opticsinfobase.org/abstract.cfm?URI=josa-67-3-419.
[Crossref]

Broquin, J.-E.

L. Labadie, P. Kern, P. Labeye, E. LeCoarer, C. Vigreux-Bercovici, A. Pradel, J.-E. Broquin, and V. Kirschner, “Technology challenges for space interferometry: the option of mid-infrared integrated optics,” Adv. Space Res., in press, available online 20 July 2007, http://dx.doi.org/10.1016/j.asr.2007.07.013.

Brown, S.

Y. Lin, G. Shepherd, B. Solheim, M. Shepherd, S. Brown, J. Harlander, and J. Whiteway, “Introduction to spatial heterodyne observations of water (SHOW) project and its instrument development,” Proc. XIV Int. TOVS Study Conf., 25–31 May 2005, Beijing, China, 835–843 (2005).

Buchwald, W. R.

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A: Pure Appl. Opt. 8, 840–848 (2006).
[Crossref]

Calvo, M. L.

Cameron, D. G.

Chamberlain, J.

J. Chamberlain, The Principles of Interferometric Spectroscopy (Wiley-Interscience, Chichester, UK, 1979).

Cheben, P.

de Haseth, J. A.

P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectrometry (Wiley-Interscience, Hoboken, New Jersey, 2007).
[Crossref]

Delâge, A.

Densmore, A.

Emelett, S. J.

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A: Pure Appl. Opt. 8, 840–848 (2006).
[Crossref]

Fellgett, P.

P. Fellgett, “A propos de la théorie du spectromètre interférentiel multiplex,” J. Phys. Radium 19, 187–191 (1958).
[Crossref]

Fellgett, P. B.

P. B. Fellgett, “The theory of infra-red sensitivities and its application to investigations of stellar radiation in the near infra-red,” Ph.D. dissertation (University of Cambridge, Cambridge, UK, 1951).

Filler, A. H.

Florjanczyk, M.

Garcia, F. C.

Griffiths, P. R.

P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectrometry (Wiley-Interscience, Hoboken, New Jersey, 2007).
[Crossref]

Harlander, J.

J. Harlander, R. J. Reynolds, and F. L. Roesler, “Spatial heterodyne spectroscopy for the exploration of diffuse interstellar emission lines at far-ultraviolet wavelengths,” Astrophys. J. 396, 730–740 (1992).
[Crossref]

Y. Lin, G. Shepherd, B. Solheim, M. Shepherd, S. Brown, J. Harlander, and J. Whiteway, “Introduction to spatial heterodyne observations of water (SHOW) project and its instrument development,” Proc. XIV Int. TOVS Study Conf., 25–31 May 2005, Beijing, China, 835–843 (2005).

Horiguchi, M.

K. Takada, Y. Inoue, H. Yamada, and M. Horiguchi, “Measurement of phase error distributions in silicabased arrayed-waveguide grating multiplexers by using Fourier transform spectroscopy,” Electron. Lett. 30, 1671–1672 (1994).
[Crossref]

Ikonen, E.

Inoue, Y.

H. Yamada, K. Takada, Y. Inoue, Y. Ohmori, and S. Mitachi, “Statically-phase-compensated 10GHzspaced arrayed-waveguide grating,” Electron. Lett. 32, 1580–1582-61 (1996).
[Crossref]

K. Takada, Y. Inoue, H. Yamada, and M. Horiguchi, “Measurement of phase error distributions in silicabased arrayed-waveguide grating multiplexers by using Fourier transform spectroscopy,” Electron. Lett. 30, 1671–1672 (1994).
[Crossref]

Ishii, M.

K. Takada, T. Tanaka, M. Abe, T. Yanagisawa, M. Ishii, and K. Okamoto, “Beam-adjustment-free crosstalk reduction in 10GHz-spaced arrayed-waveguide grating via photosensitivity under UV laser irradiation through metal mask,” Electron. Lett. 36, 60–61 (2000).
[Crossref]

Jacquinot, P.

Janz, S.

Junttila, M.-L.

Kashyap, R.

Kauppinen, J.

Kauppinen, J. K.

Kern, P.

L. Labadie, P. Kern, P. Labeye, E. LeCoarer, C. Vigreux-Bercovici, A. Pradel, J.-E. Broquin, and V. Kirschner, “Technology challenges for space interferometry: the option of mid-infrared integrated optics,” Adv. Space Res., in press, available online 20 July 2007, http://dx.doi.org/10.1016/j.asr.2007.07.013.

Kirschner, V.

L. Labadie, P. Kern, P. Labeye, E. LeCoarer, C. Vigreux-Bercovici, A. Pradel, J.-E. Broquin, and V. Kirschner, “Technology challenges for space interferometry: the option of mid-infrared integrated optics,” Adv. Space Res., in press, available online 20 July 2007, http://dx.doi.org/10.1016/j.asr.2007.07.013.

Labadie, L.

L. Labadie, P. Kern, P. Labeye, E. LeCoarer, C. Vigreux-Bercovici, A. Pradel, J.-E. Broquin, and V. Kirschner, “Technology challenges for space interferometry: the option of mid-infrared integrated optics,” Adv. Space Res., in press, available online 20 July 2007, http://dx.doi.org/10.1016/j.asr.2007.07.013.

Labeye, P.

L. Labadie, P. Kern, P. Labeye, E. LeCoarer, C. Vigreux-Bercovici, A. Pradel, J.-E. Broquin, and V. Kirschner, “Technology challenges for space interferometry: the option of mid-infrared integrated optics,” Adv. Space Res., in press, available online 20 July 2007, http://dx.doi.org/10.1016/j.asr.2007.07.013.

Lamontagne, B.

Lapointe, J.

LeCoarer, E.

L. Labadie, P. Kern, P. Labeye, E. LeCoarer, C. Vigreux-Bercovici, A. Pradel, J.-E. Broquin, and V. Kirschner, “Technology challenges for space interferometry: the option of mid-infrared integrated optics,” Adv. Space Res., in press, available online 20 July 2007, http://dx.doi.org/10.1016/j.asr.2007.07.013.

Lin, Y.

Y. Lin, G. Shepherd, B. Solheim, M. Shepherd, S. Brown, J. Harlander, and J. Whiteway, “Introduction to spatial heterodyne observations of water (SHOW) project and its instrument development,” Proc. XIV Int. TOVS Study Conf., 25–31 May 2005, Beijing, China, 835–843 (2005).

Madsen, Ch. K.

Ch. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley-Interscience, New York, 1999).

Mantsch, H. H.

Mitachi, S.

H. Yamada, K. Takada, Y. Inoue, Y. Ohmori, and S. Mitachi, “Statically-phase-compensated 10GHzspaced arrayed-waveguide grating,” Electron. Lett. 32, 1580–1582-61 (1996).
[Crossref]

Moffatt, D. J.

Norton, R. H.

R. H. Norton and R. Beer, “New apodizing functions for Fourier spectrometry,” J. Opt. Soc. Am.66, 259–264 (1976); erratum: 67, 419 (1977), http://www.opticsinfobase.org/abstract.cfm?URI=josa-67-3-419.
[Crossref]

Ohmori, Y.

H. Yamada, K. Takada, Y. Inoue, Y. Ohmori, and S. Mitachi, “Statically-phase-compensated 10GHzspaced arrayed-waveguide grating,” Electron. Lett. 32, 1580–1582-61 (1996).
[Crossref]

Okamoto, K.

K. Takada, T. Tanaka, M. Abe, T. Yanagisawa, M. Ishii, and K. Okamoto, “Beam-adjustment-free crosstalk reduction in 10GHz-spaced arrayed-waveguide grating via photosensitivity under UV laser irradiation through metal mask,” Electron. Lett. 36, 60–61 (2000).
[Crossref]

Post, E.

Powell, I.

Pradel, A.

L. Labadie, P. Kern, P. Labeye, E. LeCoarer, C. Vigreux-Bercovici, A. Pradel, J.-E. Broquin, and V. Kirschner, “Technology challenges for space interferometry: the option of mid-infrared integrated optics,” Adv. Space Res., in press, available online 20 July 2007, http://dx.doi.org/10.1016/j.asr.2007.07.013.

Reynolds, R. J.

J. Harlander, R. J. Reynolds, and F. L. Roesler, “Spatial heterodyne spectroscopy for the exploration of diffuse interstellar emission lines at far-ultraviolet wavelengths,” Astrophys. J. 396, 730–740 (1992).
[Crossref]

Rodrigo, J. A.

Roesler, F. L.

J. Harlander, R. J. Reynolds, and F. L. Roesler, “Spatial heterodyne spectroscopy for the exploration of diffuse interstellar emission lines at far-ultraviolet wavelengths,” Astrophys. J. 396, 730–740 (1992).
[Crossref]

Schmid, J. H.

Scott, A.

Shepherd, G.

Y. Lin, G. Shepherd, B. Solheim, M. Shepherd, S. Brown, J. Harlander, and J. Whiteway, “Introduction to spatial heterodyne observations of water (SHOW) project and its instrument development,” Proc. XIV Int. TOVS Study Conf., 25–31 May 2005, Beijing, China, 835–843 (2005).

Shepherd, G. P.

G. P. Shepherd, Spectral Imaging of the Atmosphere (Academic Press, London, UK, 2002).

Shepherd, M.

Y. Lin, G. Shepherd, B. Solheim, M. Shepherd, S. Brown, J. Harlander, and J. Whiteway, “Introduction to spatial heterodyne observations of water (SHOW) project and its instrument development,” Proc. XIV Int. TOVS Study Conf., 25–31 May 2005, Beijing, China, 835–843 (2005).

Solheim, B.

M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D.-X. Xu, “Planar waveguide spatial heterodyne spectrometer,” Proc. SPIE 6796, 67963J-1 (2007).

J. A. Rodrigo, P. Cheben, T. Alieva, M. L. Calvo, M. Florjańczyk, S. Janz, A. Scott, B. Solheim, D.-X. Xu, and A. Delâge, “Fresnel diffraction effects in Fourier-transform arrayed waveguide grating spectrometer,” Opt. Express 15, 16431–16441 (2007).
[Crossref] [PubMed]

Y. Lin, G. Shepherd, B. Solheim, M. Shepherd, S. Brown, J. Harlander, and J. Whiteway, “Introduction to spatial heterodyne observations of water (SHOW) project and its instrument development,” Proc. XIV Int. TOVS Study Conf., 25–31 May 2005, Beijing, China, 835–843 (2005).

B. Solheim, “Spatial Heterodyne Spectroscopy (SHS), Spatial Heterodyne Observations of Water (SHOW),” presented at the 12-th ASSFTS (Atmospheric Science from Space using Fourier Transform Spectrometry) workshop, Quebec City, Canada, 18 May 2005.

Soref, R. A.

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A: Pure Appl. Opt. 8, 840–848 (2006).
[Crossref]

Steel, W. H.

Takada, K.

K. Takada, T. Tanaka, M. Abe, T. Yanagisawa, M. Ishii, and K. Okamoto, “Beam-adjustment-free crosstalk reduction in 10GHz-spaced arrayed-waveguide grating via photosensitivity under UV laser irradiation through metal mask,” Electron. Lett. 36, 60–61 (2000).
[Crossref]

H. Yamada, K. Takada, Y. Inoue, Y. Ohmori, and S. Mitachi, “Statically-phase-compensated 10GHzspaced arrayed-waveguide grating,” Electron. Lett. 32, 1580–1582-61 (1996).
[Crossref]

K. Takada, Y. Inoue, H. Yamada, and M. Horiguchi, “Measurement of phase error distributions in silicabased arrayed-waveguide grating multiplexers by using Fourier transform spectroscopy,” Electron. Lett. 30, 1671–1672 (1994).
[Crossref]

Tanaka, T.

K. Takada, T. Tanaka, M. Abe, T. Yanagisawa, M. Ishii, and K. Okamoto, “Beam-adjustment-free crosstalk reduction in 10GHz-spaced arrayed-waveguide grating via photosensitivity under UV laser irradiation through metal mask,” Electron. Lett. 36, 60–61 (2000).
[Crossref]

Vigreux-Bercovici, C.

L. Labadie, P. Kern, P. Labeye, E. LeCoarer, C. Vigreux-Bercovici, A. Pradel, J.-E. Broquin, and V. Kirschner, “Technology challenges for space interferometry: the option of mid-infrared integrated optics,” Adv. Space Res., in press, available online 20 July 2007, http://dx.doi.org/10.1016/j.asr.2007.07.013.

Vogelaar, L.

Waldron, P.

Whiteway, J.

Y. Lin, G. Shepherd, B. Solheim, M. Shepherd, S. Brown, J. Harlander, and J. Whiteway, “Introduction to spatial heterodyne observations of water (SHOW) project and its instrument development,” Proc. XIV Int. TOVS Study Conf., 25–31 May 2005, Beijing, China, 835–843 (2005).

Xu, D.-X.

Yamada, H.

H. Yamada, K. Takada, Y. Inoue, Y. Ohmori, and S. Mitachi, “Statically-phase-compensated 10GHzspaced arrayed-waveguide grating,” Electron. Lett. 32, 1580–1582-61 (1996).
[Crossref]

K. Takada, Y. Inoue, H. Yamada, and M. Horiguchi, “Measurement of phase error distributions in silicabased arrayed-waveguide grating multiplexers by using Fourier transform spectroscopy,” Electron. Lett. 30, 1671–1672 (1994).
[Crossref]

Yanagisawa, T.

K. Takada, T. Tanaka, M. Abe, T. Yanagisawa, M. Ishii, and K. Okamoto, “Beam-adjustment-free crosstalk reduction in 10GHz-spaced arrayed-waveguide grating via photosensitivity under UV laser irradiation through metal mask,” Electron. Lett. 36, 60–61 (2000).
[Crossref]

Zhao, J. H.

Ch. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley-Interscience, New York, 1999).

Appl. Opt. (3)

Astrophys. J. (1)

J. Harlander, R. J. Reynolds, and F. L. Roesler, “Spatial heterodyne spectroscopy for the exploration of diffuse interstellar emission lines at far-ultraviolet wavelengths,” Astrophys. J. 396, 730–740 (1992).
[Crossref]

Electron. Lett. (3)

K. Takada, Y. Inoue, H. Yamada, and M. Horiguchi, “Measurement of phase error distributions in silicabased arrayed-waveguide grating multiplexers by using Fourier transform spectroscopy,” Electron. Lett. 30, 1671–1672 (1994).
[Crossref]

K. Takada, T. Tanaka, M. Abe, T. Yanagisawa, M. Ishii, and K. Okamoto, “Beam-adjustment-free crosstalk reduction in 10GHz-spaced arrayed-waveguide grating via photosensitivity under UV laser irradiation through metal mask,” Electron. Lett. 36, 60–61 (2000).
[Crossref]

H. Yamada, K. Takada, Y. Inoue, Y. Ohmori, and S. Mitachi, “Statically-phase-compensated 10GHzspaced arrayed-waveguide grating,” Electron. Lett. 32, 1580–1582-61 (1996).
[Crossref]

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

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A: Pure Appl. Opt. 8, 840–848 (2006).
[Crossref]

J. Opt. Soc. Am. (2)

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

J. Phys. Radium (1)

P. Fellgett, “A propos de la théorie du spectromètre interférentiel multiplex,” J. Phys. Radium 19, 187–191 (1958).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Proc. SPIE (1)

M. Florjańczyk, P. Cheben, S. Janz, A. Scott, B. Solheim, and D.-X. Xu, “Planar waveguide spatial heterodyne spectrometer,” Proc. SPIE 6796, 67963J-1 (2007).

Other (11)

Ch. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley-Interscience, New York, 1999).

J. Chamberlain, The Principles of Interferometric Spectroscopy (Wiley-Interscience, Chichester, UK, 1979).

P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectrometry (Wiley-Interscience, Hoboken, New Jersey, 2007).
[Crossref]

R. H. Norton and R. Beer, “New apodizing functions for Fourier spectrometry,” J. Opt. Soc. Am.66, 259–264 (1976); erratum: 67, 419 (1977), http://www.opticsinfobase.org/abstract.cfm?URI=josa-67-3-419.
[Crossref]

P. Cheben, “Wavelength dispersive planar waveguide devices: Echelle gratings and arrayed waveguide gratings,” in Optical Waveguides: From Theory to Applied Technologies, M. L. Calvo and V. Laksminarayanan, eds. (CRC Press, London, 2007).

P. B. Fellgett, “The theory of infra-red sensitivities and its application to investigations of stellar radiation in the near infra-red,” Ph.D. dissertation (University of Cambridge, Cambridge, UK, 1951).

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Y. Lin, G. Shepherd, B. Solheim, M. Shepherd, S. Brown, J. Harlander, and J. Whiteway, “Introduction to spatial heterodyne observations of water (SHOW) project and its instrument development,” Proc. XIV Int. TOVS Study Conf., 25–31 May 2005, Beijing, China, 835–843 (2005).

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[Crossref]

L. Labadie, P. Kern, P. Labeye, E. LeCoarer, C. Vigreux-Bercovici, A. Pradel, J.-E. Broquin, and V. Kirschner, “Technology challenges for space interferometry: the option of mid-infrared integrated optics,” Adv. Space Res., in press, available online 20 July 2007, http://dx.doi.org/10.1016/j.asr.2007.07.013.

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Figures (6)

Fig. 1.
Fig. 1.

The schematics of the waveguide spectrometer formed by arrayed Mach-Zehnder interferometers.

Fig. 2.
Fig. 2.

Spatial fringe formation at the arrayed MZI outputs. Monochromatic inputs at a) the Littrow wavenumber σL, b) σL+δσ, and c) σL+2δσ, and the corresponding spatial fringes; d) Superposition of monochromatic inputs and the corresponding spatial fringe pattern.

Fig. 3.
Fig. 3.

Transfer matrix notation used in the model.

Fig. 4.
Fig. 4.

Transmission spectra of water vapor at 15 km altitude. The bandpass input filter spectrum is also shown.

Fig. 5.
Fig. 5.

Ideal input and apodized calculated spectra for the arrayed MZI spectrometer with 0.1 nm resolution.

Fig. 6.
Fig. 6.

Ideal input and apodized calculated spectra for the arrayed MZI spectrometer with 0.025 nm resolution.

Equations (26)

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E ( x , y , z , t ) = a e ( x , y ) e i ( ω t β z ) H ( x , y , z , t ) = a h ( x , y ) e i ( ω t β z )
P = 1 2 a 2 e × h * · z ̂ dx dy
[ a 1 , i out a 2 , i out ] = S i · [ a 1 , i in a 2 , i in ] = S c · S d , i · S s · [ a 1 , i in a 2 , i in ]
S s = γ s [ 1 κ s i κ s i κ s 1 κ s ] S c = γ c [ 1 κ c i κ c i κ c 1 κ c ]
S d , i = γ d , i e i β L 2 , i [ e α Δ L i e i β Δ L i 0 0 1 ]
P 1 out ( x i ) = P 1 , i out = 1 2 P in [ A 1 , i B i cos β Δ L i ]
P 2 out ( x i ) = P 2 , i out = 1 2 P in [ A 2 , i + B i cos β Δ L i ]
A 1 , i = 2 γ s 2 γ d , i 2 γ c 2 [ κ s κ c + ( 1 κ s ) ( 1 κ c ) 2 α Δ L i ]
A 2 , i = 2 γ s 2 γ d , i 2 γ c 2 [ κ s ( 1 κ c ) + κ c ( 1 κ c ) 2 α Δ L i ]
B i = 4 γ s 2 γ d , i 2 γ c 2 [ κ s κ c ( 1 κ s ) ( 1 κ c ) ] 1 2 e α Δ L i
0 p 1 , i out ( σ ) d σ = 1 2 0 p in ( σ ) [ A 1 , i B i cos ( 2 π σ n eff Δ L i ) ] d σ
0 p 2 , i out ( σ ) d σ = 1 2 0 p in ( σ ) [ A 2 , i + B i cos ( 2 π σ n eff Δ L i ) ] d σ
P 1 , i out = 1 2 A 1 , i P in 1 2 B i 0 p in ( σ ) cos ( 2 π σ n eff Δ L i ) d σ
P 2 , i out = 1 2 A 2 , i P in + 1 2 B i 0 p in ( σ ) cos ( 2 π σ n eff Δ L i ) d σ
F i = 1 B i ( 2 P 1 , i out A 1 , i P in ) = 1 B i ( 2 P 2 , i out A 2 , i P in )
F ( x ) = 0 p in ( σ ) cos 2 π σ x d σ = p in ( σ ¯ ) cos 2 π σ ¯ x d σ ¯
p in ( σ ¯ ) = F ( x ) cos 2 π σ ¯ x dx = 2 0 F ( x ) cos 2 π σ ¯ x dx
p in ( σ ¯ ) = Δ x N P in + 2 Δ x N i = 1 N F ( x i ) cos 2 π σ ¯ x i
P i in = 2 ( P 1 , i out + P 2 , i out ) ( A 1 , i + A 2 , i )
p in ( σ ¯ ) = Δ x N P in + 2 Δ x N i = 1 N W ( x i ) F ( x i ) cos 2 π σ ¯ x i
Δ φ = 2 π ( σ + δ σ ) Δ x 2 π σ Δ x = 2 π
δ σ = 1 λ 0 1 λ 0 + δ λ δ λ λ 0 2 = 1 R 1 λ 0
Δ L max = 1 δ σ n eff = R λ 0 n eff
N min = 2 Δ x Δ σ = 2 Δ σ δ σ = 2 Δ σ δ λ
δ σ pol σ 0 δ λ pol λ 0 = Δ n neff n neff
W ( x i ) = [ 1 ( x i x max ) 2 ] 2 = [ 1 ( Δ L i Δ L max ) 2 ] 2

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