Abstract

In this paper we realize an RF photonic chirp compression system that compresses a continuous stream of incoming RF data (modulated on top of an optical carrier) into a train of temporal short pulses. Each pulse in the train can be separated and treated individually while being sampled by low rate optical switch and without temporal loses of the incoming flow of information. Each such pulse can be filtered and analyzed differently. The main advantage of the proposed system is its capability of being able to handle, seamlessly, high rate information flow with all-optical means and with low rate optical switches.

© 2008 Optical Society of America

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  1. Z. Zalevsky, A. Shemer, D. Mendlovic, and S. Zach, "Passive and periodically ultra fast RF-photonic spectral scanner," Opt. Express 14, 8367-8381 (2006).
    [CrossRef] [PubMed]
  2. O. Raz, R. Rotman, Y. Danziger, and M. Tur, "Implementation of photonic true time delay using high-order-mode dispersion compensating fibers," IEEE Photon. Technol. Lett. 16, 1367-1369 (2004).
    [CrossRef]
  3. R. Rotman, O. Raz, and M. Tur, "Analysis of a true time delay photonic beam former for transmission of a linear frequency-modulated waveform," J. Lightwave Technol. 23, 4026-4036 (2005).
    [CrossRef]
  4. 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]
  5. O. Levinson and M. Horowitz, "Generation of Complex Microwave and Millimeter-Wave Pulses Using Dispersion and Kerr Effect in Optical Fiber Systems," J. Lightwave Technol. 21, 1179-1186 (2003).
    [CrossRef]
  6. H. Chi and J. Yao, "An approach to photonic generation of high-frequency phase-coded RF pulses," IEEE Photon. Technol. Lett. 19, 768-770 (2007).
    [CrossRef]
  7. A. Zeitouny, S. Stepanov, O. Levinson, and M. Horowitz, "Optical generation of linearly chirped microwave pulses using fiber Bragg gratings, " IEEE Photon. Technol. Lett. 17, 660-662 (2005).
    [CrossRef]
  8. 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 Photon. Technol. Lett. 17, 432-434 (2005).
    [CrossRef]
  9. 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]
  10. 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]
  11. H. R. Fetterman, Y. Chang, D. C. Scott, S. R. Forrest,  et al., "Optically controlled phased array radar receiver using SLM switched real time delays," IEEE Microwave and Guid. Wave Lett. 5, 414-416 (1995).
    [CrossRef]
  12. R. A. Minasian and D. B. Hunter, "Photonic signal processing of microwave signals using fiber Bragg gratings," Proc. OFC, ThH3, 339-340 (1997).
  13. D. Norton, S. Johns, C. Keefer, and R. Soref, "Tunable microwave filter using high dispersion fiber time delays," IEEE Photon. Technol. Lett. 6, 831-832 (1994).
    [CrossRef]
  14. M. Y. Frankel and R. D. Esman, "Fiber optic tunable microwave transversal filter," IEEE Photon. 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 Photon. Technol. Lett. 9, 339-341 (1997).
    [CrossRef]
  16. 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]
  17. A. W. Lohmann and D. Mendlovic, "Temporal filtering with time lenses," Appl. Opt. 31, 6212-6219 (1992).
    [CrossRef] [PubMed]
  18. M. Born and E. Wolf, Principles of optics: Electromagnetic theory of propagation, interference and diffraction of light, 7th ed. (Cambridge University Press, 1999), pp 468-480.
  19. A. J. Lowery, S. Wang, and M. Premaratne, "Calculation of power limit due to fiber nonlinearity in optical OFDM systems," Opt. Express 15, 13282-13287 (2007).
    [CrossRef] [PubMed]
  20. E. Shumakher, A. Hayat, A. Freimain, M. Nazarathy, and G. Eisenstein, "Timing extraction of a 10 Gbit/s NRZ signal using an electro-optic multiplication scheme," IEEE Photonic Technol. Lett. 16, 2353-2355 (2004).
    [CrossRef]

2007 (2)

H. Chi and J. Yao, "An approach to photonic generation of high-frequency phase-coded RF pulses," IEEE Photon. Technol. Lett. 19, 768-770 (2007).
[CrossRef]

A. J. Lowery, S. Wang, and M. Premaratne, "Calculation of power limit due to fiber nonlinearity in optical OFDM systems," Opt. Express 15, 13282-13287 (2007).
[CrossRef] [PubMed]

2006 (1)

2005 (4)

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]

R. Rotman, O. Raz, and M. Tur, "Analysis of a true time delay photonic beam former for transmission of a linear frequency-modulated waveform," J. Lightwave Technol. 23, 4026-4036 (2005).
[CrossRef]

A. Zeitouny, S. Stepanov, O. Levinson, and M. Horowitz, "Optical generation of linearly chirped microwave pulses using fiber Bragg gratings, " IEEE Photon. Technol. Lett. 17, 660-662 (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 Photon. Technol. Lett. 17, 432-434 (2005).
[CrossRef]

2004 (2)

O. Raz, R. Rotman, Y. Danziger, and M. Tur, "Implementation of photonic true time delay using high-order-mode dispersion compensating fibers," IEEE Photon. Technol. Lett. 16, 1367-1369 (2004).
[CrossRef]

E. Shumakher, A. Hayat, A. Freimain, M. Nazarathy, and G. Eisenstein, "Timing extraction of a 10 Gbit/s NRZ signal using an electro-optic multiplication scheme," IEEE Photonic Technol. Lett. 16, 2353-2355 (2004).
[CrossRef]

2003 (2)

1999 (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]

1997 (1)

F. Coppinger, S. Yegnanarayanan, P. D. Trinh, and B. Jalali, "Continuously tunable photonic radio-frequency notch filter," IEEE Photon. 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 Photon. Technol. Lett. 7, 191-193 (1995).
[CrossRef]

H. R. Fetterman, Y. Chang, D. C. Scott, S. R. Forrest,  et al., "Optically controlled phased array radar receiver using SLM switched real time delays," IEEE Microwave and 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 Photon. Technol. Lett. 6, 831-832 (1994).
[CrossRef]

1992 (1)

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 Photon. 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, S. R. Forrest,  et al., "Optically controlled phased array radar receiver using SLM switched real time delays," IEEE Microwave and Guid. Wave Lett. 5, 414-416 (1995).
[CrossRef]

Chi, H.

H. Chi and J. Yao, "An approach to photonic generation of high-frequency phase-coded RF pulses," IEEE Photon. Technol. Lett. 19, 768-770 (2007).
[CrossRef]

Coppinger, F.

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

Danziger, Y.

O. Raz, R. Rotman, Y. Danziger, and M. Tur, "Implementation of photonic true time delay using high-order-mode dispersion compensating fibers," IEEE Photon. Technol. Lett. 16, 1367-1369 (2004).
[CrossRef]

Dolfi, D.

Eckhouse, V.

Eisenstein, G.

E. Shumakher, A. Hayat, A. Freimain, M. Nazarathy, and G. Eisenstein, "Timing extraction of a 10 Gbit/s NRZ signal using an electro-optic multiplication scheme," IEEE Photonic Technol. Lett. 16, 2353-2355 (2004).
[CrossRef]

Esman, R. D.

M. Y. Frankel and R. D. Esman, "Fiber optic tunable microwave transversal filter," IEEE Photon. Technol. Lett. 7, 191-193 (1995).
[CrossRef]

Fetterman, H. R.

H. R. Fetterman, Y. Chang, D. C. Scott, S. R. Forrest,  et al., "Optically controlled phased array radar receiver using SLM switched real time delays," IEEE Microwave and 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, S. R. Forrest,  et al., "Optically controlled phased array radar receiver using SLM switched real time delays," IEEE Microwave and 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 Photon. Technol. Lett. 7, 191-193 (1995).
[CrossRef]

Freimain, A.

E. Shumakher, A. Hayat, A. Freimain, M. Nazarathy, and G. Eisenstein, "Timing extraction of a 10 Gbit/s NRZ signal using an electro-optic multiplication scheme," IEEE Photonic Technol. Lett. 16, 2353-2355 (2004).
[CrossRef]

Hayat, A.

E. Shumakher, A. Hayat, A. Freimain, M. Nazarathy, and G. Eisenstein, "Timing extraction of a 10 Gbit/s NRZ signal using an electro-optic multiplication scheme," IEEE Photonic Technol. Lett. 16, 2353-2355 (2004).
[CrossRef]

Horowitz, M.

A. Zeitouny, S. Stepanov, O. Levinson, and M. Horowitz, "Optical generation of linearly chirped microwave pulses using fiber Bragg gratings, " IEEE Photon. Technol. Lett. 17, 660-662 (2005).
[CrossRef]

O. Levinson and M. Horowitz, "Generation of Complex Microwave and Millimeter-Wave Pulses Using Dispersion and Kerr Effect in Optical Fiber Systems," J. Lightwave Technol. 21, 1179-1186 (2003).
[CrossRef]

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 Photon. 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 Photon. 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 Photon. 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 Photon. 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 Photon. Technol. Lett. 17, 432-434 (2005).
[CrossRef]

Lavielle, V.

Le Gout, J. L.

Levinson, O.

A. Zeitouny, S. Stepanov, O. Levinson, and M. Horowitz, "Optical generation of linearly chirped microwave pulses using fiber Bragg gratings, " IEEE Photon. Technol. Lett. 17, 660-662 (2005).
[CrossRef]

O. Levinson and M. Horowitz, "Generation of Complex Microwave and Millimeter-Wave Pulses Using Dispersion and Kerr Effect in Optical Fiber Systems," J. Lightwave Technol. 21, 1179-1186 (2003).
[CrossRef]

Lohmann, A. W.

Lorgere, I.

Lowery, A. J.

Mendlovic, D.

Nazarathy, M.

E. Shumakher, A. Hayat, A. Freimain, M. Nazarathy, and G. Eisenstein, "Timing extraction of a 10 Gbit/s NRZ signal using an electro-optic multiplication scheme," IEEE Photonic Technol. Lett. 16, 2353-2355 (2004).
[CrossRef]

Norton, D.

D. Norton, S. Johns, C. Keefer, and R. Soref, "Tunable microwave filter using high dispersion fiber time delays," IEEE Photon. Technol. Lett. 6, 831-832 (1994).
[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]

Premaratne, M.

Raz, O.

R. Rotman, O. Raz, and M. Tur, "Analysis of a true time delay photonic beam former for transmission of a linear frequency-modulated waveform," J. Lightwave Technol. 23, 4026-4036 (2005).
[CrossRef]

O. Raz, R. Rotman, Y. Danziger, and M. Tur, "Implementation of photonic true time delay using high-order-mode dispersion compensating fibers," IEEE Photon. Technol. Lett. 16, 1367-1369 (2004).
[CrossRef]

Rotman, R.

R. Rotman, O. Raz, and M. Tur, "Analysis of a true time delay photonic beam former for transmission of a linear frequency-modulated waveform," J. Lightwave Technol. 23, 4026-4036 (2005).
[CrossRef]

O. Raz, R. Rotman, Y. Danziger, and M. Tur, "Implementation of photonic true time delay using high-order-mode dispersion compensating fibers," IEEE Photon. Technol. Lett. 16, 1367-1369 (2004).
[CrossRef]

Scott, D. C.

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

Shemer, A.

Shumakher, E.

E. Shumakher, A. Hayat, A. Freimain, M. Nazarathy, and G. Eisenstein, "Timing extraction of a 10 Gbit/s NRZ signal using an electro-optic multiplication scheme," IEEE Photonic Technol. Lett. 16, 2353-2355 (2004).
[CrossRef]

Soref, R.

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

Stepanov, S.

A. Zeitouny, S. Stepanov, O. Levinson, and M. Horowitz, "Optical generation of linearly chirped microwave pulses using fiber Bragg gratings, " IEEE Photon. Technol. Lett. 17, 660-662 (2005).
[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 Photon. 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 Photon. Technol. Lett. 17, 432-434 (2005).
[CrossRef]

Tur, M.

R. Rotman, O. Raz, and M. Tur, "Analysis of a true time delay photonic beam former for transmission of a linear frequency-modulated waveform," J. Lightwave Technol. 23, 4026-4036 (2005).
[CrossRef]

O. Raz, R. Rotman, Y. Danziger, and M. Tur, "Implementation of photonic true time delay using high-order-mode dispersion compensating fibers," IEEE Photon. Technol. Lett. 16, 1367-1369 (2004).
[CrossRef]

Wang, S.

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]

Yao, J.

H. Chi and J. Yao, "An approach to photonic generation of high-frequency phase-coded RF pulses," IEEE Photon. Technol. Lett. 19, 768-770 (2007).
[CrossRef]

Yegnanarayanan, S.

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

Zach, S.

Zalevsky, Z.

Zeitouny, A.

A. Zeitouny, S. Stepanov, O. Levinson, and M. Horowitz, "Optical generation of linearly chirped microwave pulses using fiber Bragg gratings, " IEEE Photon. Technol. Lett. 17, 660-662 (2005).
[CrossRef]

Appl. Opt. (1)

IEEE Microwave and Guid. Wave Lett. (1)

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

IEEE Photon. Technol. Lett. (8)

O. Raz, R. Rotman, Y. Danziger, and M. Tur, "Implementation of photonic true time delay using high-order-mode dispersion compensating fibers," IEEE Photon. Technol. Lett. 16, 1367-1369 (2004).
[CrossRef]

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. Chi and J. Yao, "An approach to photonic generation of high-frequency phase-coded RF pulses," IEEE Photon. Technol. Lett. 19, 768-770 (2007).
[CrossRef]

A. Zeitouny, S. Stepanov, O. Levinson, and M. Horowitz, "Optical generation of linearly chirped microwave pulses using fiber Bragg gratings, " IEEE Photon. Technol. Lett. 17, 660-662 (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 Photon. Technol. Lett. 17, 432-434 (2005).
[CrossRef]

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

M. Y. Frankel and R. D. Esman, "Fiber optic tunable microwave transversal filter," IEEE Photon. 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 Photon. Technol. Lett. 9, 339-341 (1997).
[CrossRef]

IEEE Photonic Technol. Lett. (1)

E. Shumakher, A. Hayat, A. Freimain, M. Nazarathy, and G. Eisenstein, "Timing extraction of a 10 Gbit/s NRZ signal using an electro-optic multiplication scheme," IEEE Photonic Technol. Lett. 16, 2353-2355 (2004).
[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. (2)

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

Opt. Express (2)

Opt. Lett. (1)

Other (2)

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

M. Born and E. Wolf, Principles of optics: Electromagnetic theory of propagation, interference and diffraction of light, 7th ed. (Cambridge University Press, 1999), pp 468-480.

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Figures (5)

Fig. 1.
Fig. 1.

Preliminary proof of concept.

Fig. 2.
Fig. 2.

Full configuration of spectral chirp coding and temporal sequence compression.

Fig. 3.
Fig. 3.

(a) Schematic sketch of the operational principle. (b) Application of information routing.

Fig. 4.
Fig. 4.

Experimental results using δL=1.5m (a). Output after chromatic delay of ≈8 ns (the two channels with their combination). (b) The same as in 4(a) but with data modulation. (c) Output after decreasing the chromatic delay in the DCF (i.e. the temporal compression). (d) The same as in 4(c) but with data modulation.

Fig. 5.
Fig. 5.

Experimental results using δL=3m (a) Output after chromatic delay of ≈16 ns (the two channels with their combination). (b) The same as in 5(a) but with data modulation. (c) Output after decreasing the chromatic delay in the DCF (i.e. after the temporal compression). (d) The same as in 5(c) but with data modulation.

Equations (9)

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

δ t = δ L c n = 3 m 3.10 8 1.5 = 15 n sec
Δ t = 2 · [ 806 p s nm ] · ( 1557.6 n m 1548.6 n m ) = 14.5 n sec
Δ t = δ τ 2 + ( β 2 z ( 2 π · δ τ ) ) 2
Δ T = N · δ τ
τ dis = β 2 z 2 · Δ μ = β 2 z · Δ λ · c 2 λ 2 = δ τ
β ( μ ) β 0 + β 1 μ + β 2 2 μ 2
β ( μ ) β 0 + β 1 μ + β 2 2 μ 2 + β 3 6 μ 3 + β 4 24 μ 4
τ dis = β 2 z · Δ μ 2 + β 3 z · Δ μ 2 6 + β 4 z · Δ μ 3 24
Δ μ min { 3 β 2 β 3 , 4 β 3 β 4 }

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