Abstract

In this paper we present passive photonic device performing periodic and ultra fast spectral analysis of RF signals modulated on optical carrier. The spectral scanning is demonstrated in two approaches. First by passing the light through a couple of special bulk periscopes that split the beam into a set of parallel channels or combine a set of channels into one beam. One surface of each periscope is coated with high reflectivity coating such that the set of parallel beams travel several times through the structure due to their partial back reflection in each passage through the periscope. In each passage in the system the channel experience different delay in comparison with the original signal. This relative delay is accumulative and it is generated by placing glass bars with different length for each one of the channels. This structure realizes Finite Impulse Response (FIR) filter that performs the spectral scanning. The second approach involves similar configuration but it is realized with fibers and Y couplers rather than bulk optics. In this case the filter that performs the spectral scanning is an Infinite Impulse Response (IIR) filter having much sharper spectral sampling capability.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. Saitoh , T. Nakamura , and K. Y. Takahashi , “ High-speed (1.3 ms/scan) optical spectrum analyzer utilizing MEMS scanning mirror ” ECOC, Royal Inst. of Technol. 562 – 563 ,   3 , Kista, Sweden ( 2004 ).
  2. Y. Takahashi , “ High-speed MEMS-OSA and its application to fiber sensors ,” SPIE Proceedings   5502 , 33 – 38 ( 2004 ).
    [CrossRef]
  3. Y. Q. Lu , F. Du , Y.H. Wu , and S. T. Wu “ Liquid-crystal-based Fourier optical spectrum analyzer without moving parts ,” Jpn. J. Appl. Phys. Part 2-Letters   44 , 291 – 293 ( 2005 ).
    [CrossRef]
  4. T. Y. Itoh , Y. Aizawa , K. T. Kurokawa , and H. Tsuda , “ Optical spectrum analyzer based on arrayed waveguide grating for high-speed optical communication systems ,” IEEE Photonics Technol. Lett.   17 , 432 – 434 ( 2005 ).
    [CrossRef]
  5. K. Takada , H. Yamada , and K. Okamoto , “ Optical spectrum analyzer using cascaded AWGs with different channel spacings ,” IEEE Photonics Technol. Lett.   11 , 863 – 864 ( 1999 ).
    [CrossRef]
  6. Z. Zalevsky , A. Shemer , V. Eckhouse , D. Mendlovic , and S. Zach , “ RF-Photonic Filter for Highly Resolved and Ultra-Fast Information Extraction ,” J. Opt. Soc. Am. A   22 , 1668 – 1677 ( 2005 ).
    [CrossRef]
  7. V. Lavielle , I. Lorgere , J. L. Le Gout , S. Tonda , and D. Dolfi , “ Wideband versatile radio-frequency spectrum analyzer, ” Opt. Lett.   28 , 384 – 386 . ( 2003 ).
    [CrossRef] [PubMed]
  8. C. K. Madsen , “ General IIR optical filter design for WDM applications using all-pass filters ,” J. Lightwave Technol.   18 , 860 – 868 ( 2000 ).
    [CrossRef]
  9. H. R. Fetterman , Y. Chang , D. C. Scott , and S. R. Forrest , et al., “ Optically controlled phased array radar receiver using SLM switched real time delays ,” IEEE Microwave Guid. Wave Lett.   5 , 414 – 416 ( 1995 ).
    [CrossRef]
  10. L. Xu , R. Taylor , and S. R. Forrest , “ True time-delay phased-array antenna feed system based on optical heterodyne techniques ,” IEEE Photon. Technol. Lett.   8 , 160 – 162 ( 1996 ).
    [CrossRef]
  11. R. A. Minasian and D. B. Hunter , “ Photonic signal processing of microwave signals using fiber Bragg gratings ,” Proc. OFC ,   ThH3 339 – 340 ( 1997 ).
  12. D. Norton , S. Johns , C. Keefer , and R. Soref , “ Tunable microwave filter using high dispersion fiber time delays ,” IEEE Photonics Technol. Lett   6 , 831 – 832 ( 1994 ).
    [CrossRef]
  13. J. Campany , D. Pastor , and B. Ortega , “ New and flexible fiber-optics delay line filters using chirped fiber Bragg gratings and laser arrays ,” IEEE Trans. Microwave Theory Tech.   47 , 1321 – 1326 ( 1999 ).
    [CrossRef]
  14. M. Y. Frankel and R. D. Esman , “ Fiber optic tunable microwave transversal filter ,” IEEE Photonics Technol. Lett.   7 , 191 – 193 ( 1995 ).
    [CrossRef]
  15. F. Coppinger , S. Yegnanarayanan , P. D. Trinh , and B. Jalali , “ Continuously tunable photonic radio-frequency notch filter ,” IEEE Photonics Technol. Lett.   9 , 339 – 341 ( 1997 ).
    [CrossRef]

2005 (3)

Y. Q. Lu , F. Du , Y.H. Wu , and S. T. Wu “ Liquid-crystal-based Fourier optical spectrum analyzer without moving parts ,” Jpn. J. Appl. Phys. Part 2-Letters   44 , 291 – 293 ( 2005 ).
[CrossRef]

T. Y. Itoh , Y. Aizawa , K. T. Kurokawa , and H. Tsuda , “ Optical spectrum analyzer based on arrayed waveguide grating for high-speed optical communication systems ,” IEEE Photonics Technol. Lett.   17 , 432 – 434 ( 2005 ).
[CrossRef]

Z. Zalevsky , A. Shemer , V. Eckhouse , D. Mendlovic , and S. Zach , “ RF-Photonic Filter for Highly Resolved and Ultra-Fast Information Extraction ,” J. Opt. Soc. Am. A   22 , 1668 – 1677 ( 2005 ).
[CrossRef]

2004 (2)

T. Saitoh , T. Nakamura , and K. Y. Takahashi , “ High-speed (1.3 ms/scan) optical spectrum analyzer utilizing MEMS scanning mirror ” ECOC, Royal Inst. of Technol. 562 – 563 ,   3 , Kista, Sweden ( 2004 ).

Y. Takahashi , “ High-speed MEMS-OSA and its application to fiber sensors ,” SPIE Proceedings   5502 , 33 – 38 ( 2004 ).
[CrossRef]

2003 (1)

2000 (1)

1999 (2)

J. Campany , D. Pastor , and B. Ortega , “ New and flexible fiber-optics delay line filters using chirped fiber Bragg gratings and laser arrays ,” IEEE Trans. Microwave Theory Tech.   47 , 1321 – 1326 ( 1999 ).
[CrossRef]

K. Takada , H. Yamada , and K. Okamoto , “ Optical spectrum analyzer using cascaded AWGs with different channel spacings ,” IEEE Photonics Technol. Lett.   11 , 863 – 864 ( 1999 ).
[CrossRef]

1997 (2)

R. A. Minasian and D. B. Hunter , “ Photonic signal processing of microwave signals using fiber Bragg gratings ,” Proc. OFC ,   ThH3 339 – 340 ( 1997 ).

F. Coppinger , S. Yegnanarayanan , P. D. Trinh , and B. Jalali , “ Continuously tunable photonic radio-frequency notch filter ,” IEEE Photonics Technol. Lett.   9 , 339 – 341 ( 1997 ).
[CrossRef]

1996 (1)

L. Xu , R. Taylor , and S. R. Forrest , “ True time-delay phased-array antenna feed system based on optical heterodyne techniques ,” IEEE Photon. Technol. Lett.   8 , 160 – 162 ( 1996 ).
[CrossRef]

1995 (2)

M. Y. Frankel and R. D. Esman , “ Fiber optic tunable microwave transversal filter ,” IEEE Photonics Technol. Lett.   7 , 191 – 193 ( 1995 ).
[CrossRef]

H. R. Fetterman , Y. Chang , D. C. Scott , and S. R. Forrest , et al., “ Optically controlled phased array radar receiver using SLM switched real time delays ,” IEEE Microwave Guid. Wave Lett.   5 , 414 – 416 ( 1995 ).
[CrossRef]

1994 (1)

D. Norton , S. Johns , C. Keefer , and R. Soref , “ Tunable microwave filter using high dispersion fiber time delays ,” IEEE Photonics Technol. Lett   6 , 831 – 832 ( 1994 ).
[CrossRef]

Aizawa, Y.

T. Y. Itoh , Y. Aizawa , K. T. Kurokawa , and H. Tsuda , “ Optical spectrum analyzer based on arrayed waveguide grating for high-speed optical communication systems ,” IEEE Photonics Technol. Lett.   17 , 432 – 434 ( 2005 ).
[CrossRef]

Campany, J.

J. Campany , D. Pastor , and B. Ortega , “ New and flexible fiber-optics delay line filters using chirped fiber Bragg gratings and laser arrays ,” IEEE Trans. Microwave Theory Tech.   47 , 1321 – 1326 ( 1999 ).
[CrossRef]

Chang, Y.

H. R. Fetterman , Y. Chang , D. C. Scott , and S. R. Forrest , et al., “ Optically controlled phased array radar receiver using SLM switched real time delays ,” IEEE Microwave Guid. Wave Lett.   5 , 414 – 416 ( 1995 ).
[CrossRef]

Coppinger, F.

F. Coppinger , S. Yegnanarayanan , P. D. Trinh , and B. Jalali , “ Continuously tunable photonic radio-frequency notch filter ,” IEEE Photonics Technol. Lett.   9 , 339 – 341 ( 1997 ).
[CrossRef]

Dolfi, D.

Du, F.

Y. Q. Lu , F. Du , Y.H. Wu , and S. T. Wu “ Liquid-crystal-based Fourier optical spectrum analyzer without moving parts ,” Jpn. J. Appl. Phys. Part 2-Letters   44 , 291 – 293 ( 2005 ).
[CrossRef]

Eckhouse, V.

Esman, R. D.

M. Y. Frankel and R. D. Esman , “ Fiber optic tunable microwave transversal filter ,” IEEE Photonics Technol. Lett.   7 , 191 – 193 ( 1995 ).
[CrossRef]

Fetterman, H. R.

H. R. Fetterman , Y. Chang , D. C. Scott , and S. R. Forrest , et al., “ Optically controlled phased array radar receiver using SLM switched real time delays ,” IEEE Microwave Guid. Wave Lett.   5 , 414 – 416 ( 1995 ).
[CrossRef]

Forrest, S. R.

L. Xu , R. Taylor , and S. R. Forrest , “ True time-delay phased-array antenna feed system based on optical heterodyne techniques ,” IEEE Photon. Technol. Lett.   8 , 160 – 162 ( 1996 ).
[CrossRef]

H. R. Fetterman , Y. Chang , D. C. Scott , and S. R. Forrest , et al., “ Optically controlled phased array radar receiver using SLM switched real time delays ,” IEEE Microwave Guid. Wave Lett.   5 , 414 – 416 ( 1995 ).
[CrossRef]

Frankel, M. Y.

M. Y. Frankel and R. D. Esman , “ Fiber optic tunable microwave transversal filter ,” IEEE Photonics Technol. Lett.   7 , 191 – 193 ( 1995 ).
[CrossRef]

Hunter, D. B.

R. A. Minasian and D. B. Hunter , “ Photonic signal processing of microwave signals using fiber Bragg gratings ,” Proc. OFC ,   ThH3 339 – 340 ( 1997 ).

Itoh, T. Y.

T. Y. Itoh , Y. Aizawa , K. T. Kurokawa , and H. Tsuda , “ Optical spectrum analyzer based on arrayed waveguide grating for high-speed optical communication systems ,” IEEE Photonics Technol. Lett.   17 , 432 – 434 ( 2005 ).
[CrossRef]

Jalali, B.

F. Coppinger , S. Yegnanarayanan , P. D. Trinh , and B. Jalali , “ Continuously tunable photonic radio-frequency notch filter ,” IEEE Photonics Technol. Lett.   9 , 339 – 341 ( 1997 ).
[CrossRef]

Johns, S.

D. Norton , S. Johns , C. Keefer , and R. Soref , “ Tunable microwave filter using high dispersion fiber time delays ,” IEEE Photonics Technol. Lett   6 , 831 – 832 ( 1994 ).
[CrossRef]

Keefer, C.

D. Norton , S. Johns , C. Keefer , and R. Soref , “ Tunable microwave filter using high dispersion fiber time delays ,” IEEE Photonics Technol. Lett   6 , 831 – 832 ( 1994 ).
[CrossRef]

Kurokawa, K. T.

T. Y. Itoh , Y. Aizawa , K. T. Kurokawa , and H. Tsuda , “ Optical spectrum analyzer based on arrayed waveguide grating for high-speed optical communication systems ,” IEEE Photonics Technol. Lett.   17 , 432 – 434 ( 2005 ).
[CrossRef]

Lavielle, V.

Le Gout, J. L.

Lorgere, I.

Lu, Y. Q.

Y. Q. Lu , F. Du , Y.H. Wu , and S. T. Wu “ Liquid-crystal-based Fourier optical spectrum analyzer without moving parts ,” Jpn. J. Appl. Phys. Part 2-Letters   44 , 291 – 293 ( 2005 ).
[CrossRef]

Madsen, C. K.

Mendlovic, D.

Minasian, R. A.

R. A. Minasian and D. B. Hunter , “ Photonic signal processing of microwave signals using fiber Bragg gratings ,” Proc. OFC ,   ThH3 339 – 340 ( 1997 ).

Nakamura, T.

T. Saitoh , T. Nakamura , and K. Y. Takahashi , “ High-speed (1.3 ms/scan) optical spectrum analyzer utilizing MEMS scanning mirror ” ECOC, Royal Inst. of Technol. 562 – 563 ,   3 , Kista, Sweden ( 2004 ).

Norton, D.

D. Norton , S. Johns , C. Keefer , and R. Soref , “ Tunable microwave filter using high dispersion fiber time delays ,” IEEE Photonics Technol. Lett   6 , 831 – 832 ( 1994 ).
[CrossRef]

Okamoto, K.

K. Takada , H. Yamada , and K. Okamoto , “ Optical spectrum analyzer using cascaded AWGs with different channel spacings ,” IEEE Photonics Technol. Lett.   11 , 863 – 864 ( 1999 ).
[CrossRef]

Ortega, B.

J. Campany , D. Pastor , and B. Ortega , “ New and flexible fiber-optics delay line filters using chirped fiber Bragg gratings and laser arrays ,” IEEE Trans. Microwave Theory Tech.   47 , 1321 – 1326 ( 1999 ).
[CrossRef]

Pastor, D.

J. Campany , D. Pastor , and B. Ortega , “ New and flexible fiber-optics delay line filters using chirped fiber Bragg gratings and laser arrays ,” IEEE Trans. Microwave Theory Tech.   47 , 1321 – 1326 ( 1999 ).
[CrossRef]

Saitoh, T.

T. Saitoh , T. Nakamura , and K. Y. Takahashi , “ High-speed (1.3 ms/scan) optical spectrum analyzer utilizing MEMS scanning mirror ” ECOC, Royal Inst. of Technol. 562 – 563 ,   3 , Kista, Sweden ( 2004 ).

Scott, D. C.

H. R. Fetterman , Y. Chang , D. C. Scott , and S. R. Forrest , et al., “ Optically controlled phased array radar receiver using SLM switched real time delays ,” IEEE Microwave Guid. Wave Lett.   5 , 414 – 416 ( 1995 ).
[CrossRef]

Shemer, A.

Soref, R.

D. Norton , S. Johns , C. Keefer , and R. Soref , “ Tunable microwave filter using high dispersion fiber time delays ,” IEEE Photonics Technol. Lett   6 , 831 – 832 ( 1994 ).
[CrossRef]

Takada, K.

K. Takada , H. Yamada , and K. Okamoto , “ Optical spectrum analyzer using cascaded AWGs with different channel spacings ,” IEEE Photonics Technol. Lett.   11 , 863 – 864 ( 1999 ).
[CrossRef]

Takahashi, K. Y.

T. Saitoh , T. Nakamura , and K. Y. Takahashi , “ High-speed (1.3 ms/scan) optical spectrum analyzer utilizing MEMS scanning mirror ” ECOC, Royal Inst. of Technol. 562 – 563 ,   3 , Kista, Sweden ( 2004 ).

Takahashi, Y.

Y. Takahashi , “ High-speed MEMS-OSA and its application to fiber sensors ,” SPIE Proceedings   5502 , 33 – 38 ( 2004 ).
[CrossRef]

Taylor, R.

L. Xu , R. Taylor , and S. R. Forrest , “ True time-delay phased-array antenna feed system based on optical heterodyne techniques ,” IEEE Photon. Technol. Lett.   8 , 160 – 162 ( 1996 ).
[CrossRef]

Tonda, S.

Trinh, P. D.

F. Coppinger , S. Yegnanarayanan , P. D. Trinh , and B. Jalali , “ Continuously tunable photonic radio-frequency notch filter ,” IEEE Photonics Technol. Lett.   9 , 339 – 341 ( 1997 ).
[CrossRef]

Tsuda, H.

T. Y. Itoh , Y. Aizawa , K. T. Kurokawa , and H. Tsuda , “ Optical spectrum analyzer based on arrayed waveguide grating for high-speed optical communication systems ,” IEEE Photonics Technol. Lett.   17 , 432 – 434 ( 2005 ).
[CrossRef]

Wu, S. T.

Y. Q. Lu , F. Du , Y.H. Wu , and S. T. Wu “ Liquid-crystal-based Fourier optical spectrum analyzer without moving parts ,” Jpn. J. Appl. Phys. Part 2-Letters   44 , 291 – 293 ( 2005 ).
[CrossRef]

Wu, Y.H.

Y. Q. Lu , F. Du , Y.H. Wu , and S. T. Wu “ Liquid-crystal-based Fourier optical spectrum analyzer without moving parts ,” Jpn. J. Appl. Phys. Part 2-Letters   44 , 291 – 293 ( 2005 ).
[CrossRef]

Xu, L.

L. Xu , R. Taylor , and S. R. Forrest , “ True time-delay phased-array antenna feed system based on optical heterodyne techniques ,” IEEE Photon. Technol. Lett.   8 , 160 – 162 ( 1996 ).
[CrossRef]

Yamada, H.

K. Takada , H. Yamada , and K. Okamoto , “ Optical spectrum analyzer using cascaded AWGs with different channel spacings ,” IEEE Photonics Technol. Lett.   11 , 863 – 864 ( 1999 ).
[CrossRef]

Yegnanarayanan, S.

F. Coppinger , S. Yegnanarayanan , P. D. Trinh , and B. Jalali , “ Continuously tunable photonic radio-frequency notch filter ,” IEEE Photonics Technol. Lett.   9 , 339 – 341 ( 1997 ).
[CrossRef]

Zach, S.

Zalevsky, Z.

ECOC, Royal Inst. of Technol. (1)

T. Saitoh , T. Nakamura , and K. Y. Takahashi , “ High-speed (1.3 ms/scan) optical spectrum analyzer utilizing MEMS scanning mirror ” ECOC, Royal Inst. of Technol. 562 – 563 ,   3 , Kista, Sweden ( 2004 ).

IEEE Microwave Guid. Wave Lett. (1)

H. R. Fetterman , Y. Chang , D. C. Scott , and S. R. Forrest , et al., “ Optically controlled phased array radar receiver using SLM switched real time delays ,” IEEE Microwave Guid. Wave Lett.   5 , 414 – 416 ( 1995 ).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

L. Xu , R. Taylor , and S. R. Forrest , “ True time-delay phased-array antenna feed system based on optical heterodyne techniques ,” IEEE Photon. Technol. Lett.   8 , 160 – 162 ( 1996 ).
[CrossRef]

IEEE Photonics Technol. Lett (1)

D. Norton , S. Johns , C. Keefer , and R. Soref , “ Tunable microwave filter using high dispersion fiber time delays ,” IEEE Photonics Technol. Lett   6 , 831 – 832 ( 1994 ).
[CrossRef]

IEEE Photonics Technol. Lett. (4)

T. Y. Itoh , Y. Aizawa , K. T. Kurokawa , and H. Tsuda , “ Optical spectrum analyzer based on arrayed waveguide grating for high-speed optical communication systems ,” IEEE Photonics Technol. Lett.   17 , 432 – 434 ( 2005 ).
[CrossRef]

K. Takada , H. Yamada , and K. Okamoto , “ Optical spectrum analyzer using cascaded AWGs with different channel spacings ,” IEEE Photonics Technol. Lett.   11 , 863 – 864 ( 1999 ).
[CrossRef]

M. Y. Frankel and R. D. Esman , “ Fiber optic tunable microwave transversal filter ,” IEEE Photonics Technol. Lett.   7 , 191 – 193 ( 1995 ).
[CrossRef]

F. Coppinger , S. Yegnanarayanan , P. D. Trinh , and B. Jalali , “ Continuously tunable photonic radio-frequency notch filter ,” IEEE Photonics Technol. Lett.   9 , 339 – 341 ( 1997 ).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

J. Campany , D. Pastor , and B. Ortega , “ New and flexible fiber-optics delay line filters using chirped fiber Bragg gratings and laser arrays ,” IEEE Trans. Microwave Theory Tech.   47 , 1321 – 1326 ( 1999 ).
[CrossRef]

J. Lightwave Technol. (1)

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

Jpn. J. Appl. Phys. Part 2-Letters (1)

Y. Q. Lu , F. Du , Y.H. Wu , and S. T. Wu “ Liquid-crystal-based Fourier optical spectrum analyzer without moving parts ,” Jpn. J. Appl. Phys. Part 2-Letters   44 , 291 – 293 ( 2005 ).
[CrossRef]

Opt. Lett. (1)

Proc. OFC (1)

R. A. Minasian and D. B. Hunter , “ Photonic signal processing of microwave signals using fiber Bragg gratings ,” Proc. OFC ,   ThH3 339 – 340 ( 1997 ).

SPIE Proceedings (1)

Y. Takahashi , “ High-speed MEMS-OSA and its application to fiber sensors ,” SPIE Proceedings   5502 , 33 – 38 ( 2004 ).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1.
Fig. 1.

The RF optical spectrum analyzer. (a). The integrated optical system. (b). The folding concept.

Fig. 2.
Fig. 2.

Numerical simulation for the optical spectrum analyzer. (a) Zoom on one of the BPF. (b) The spectral scanning. (c) Kaiser weighting of the 32 optical tapes.

Fig. 3.
Fig. 3.

The schematic structure of the two concepts of optical benches. (a). The two periscopes concept. (b) The optical beam splitter (periscope) structure.(c). The mirror- periscope concept.

Fig. 4.
Fig. 4.

The experimental results for the third replica. (a). Several samples of captured results (only 3 are shown). Each plot contains in its upper part the temporal signal and in the lower part its FFT Sample of captured results. (b). Computing of the overall BPF for the third replica using points based on six charts similar to those shown in Fig. 4(a).

Fig. 5.
Fig. 5.

The experimental results for the fourth replica. a). Several samples of captured results (only 3 are shown). Each plot contains in its upper part the temporal signal and in the lower part its FFT. b). Computing of the overall BPF for the forth replica using points based on six charts similar to those shown in Fig. 5(a).

Fig. 6.
Fig. 6.

The experimental setup and the measurements.

Fig. 7.
Fig. 7.

Optical IIR filter.

Fig. 8.
Fig. 8.

Spectral response of an IIR filter.

Fig. 9.
Fig. 9.

All-optical spectral scanning with IIR filters.

Fig. 10.
Fig. 10.

(a). The experimental schematic sketch. (b). The experimental setup.

Fig. 11.
Fig. 11.

IIR Experimental results: Real time signal - blue line, FFT of signal- red line. (a) Stop band at 48 MHz (-11.8 dB). (b) Stop band at 56 MHz (-11.8 dB). (c) Pass band at 56 MHz.

Fig. 12.
Fig. 12.

IIR Experimental results: Real time signal - blue line, FFT of signal- red line. Gray line- FFT signal. (a) Pass band at 6.995 MHz. (b) Pass band at 9.935 MHz. (c) Stop band at 9.916 MHz. (d) Stop band at 10.53 MHz.

Tables (1)

Tables Icon

Table 1. Parameters used for the numerical simulation.

Equations (15)

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

Δt AT = 2 Δy c n
S T ( t ) = n = 0 N 1 a n s ( t nδt )
S T ( μ ) = s T ( t ) exp ( 2 πiμt ) dt = n = 0 N 1 a n s ( t nδt ) exp ( 2 πiμt ) dt = S ( μ ) F ( μ )
S ( μ ) = s ( t ) exp ( 2 πiμt ) dt
F ( μ ) = n = 0 N 1 a n exp ( 2 πiμ n δt )
w [ m ] = I 0 [ β 1 ( m α α ) ] I 0 ( β )
δt = 2 Δx ( n c 1 c ) = 2 Δx ( 1 2 10 8 1 3 10 8 ) = 0.8 n sec
r 1 r 2 · y ( t Δt ) + t 2 x ( t ) = y ( t )
r 1 r 2 · y ( t Δt ) + t 1 t 2 · x ( t ) = y ( t )
Y ( ν ) = t 1 t 2 1 r 1 r 2 exp ( 2 πiνΔt ) X ( ν )
Y ( ν ) X ( ν ) = t 1 t 2 1 + r 1 2 r 2 2 2 r 1 r 2 cos ( 2 πνΔt )
2 πν m ( min ) Δt = π + 2 πm ; m = ± 1 , ± 2 . . .
ν m ( min ) = 1 + 2 m 2 Δt
ν m ( max ) = m Δt
ΔL = cΔt / n = 2 10 8 × 0.5 10 9 = 10 cm

Metrics