Abstract

Digital signal processing techniques are used for synthesizing tunable optical filters with variable bandwidth and centered reference frequency including the tunability of the low-pass, high-pass, bandpass, and bandstop optical filters. Potential applications of such filters are discussed, and the design techniques and properties of recursive digital filters are outlined. The basic filter structures, namely, the first-order all-pole optical filter (FOAPOF) and the first-order all-zero optical filter (FOAZOF), are described, and finally the design process of tunable optical filters and the designs of the second-order Butterworth low-pass, high-pass, bandpass, and bandstop tunable optical filters are presented. Indeed, we identify that the all-zero and all-pole networks are equivalent with well known principles of optics of interference and resonance, respectively. It is thus very straightforward to implement tunable optical filters, which is a unique feature.

© 2009 Optical Society of America

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  1. S. Suzuki, K. Oda, and Y. Hibino, “Integrated-optic double-ring resonators with a wide free spectral range of 100 GHz,” J. Lightwave Technol. 13, 1766-1771 (1995).
    [CrossRef]
  2. E. Pawlowski, K. Takiguchi, M. Okuno, K. Sasayama, A. Himeno, K. Okamato, and Y. Ohmori, “Variable bandwidth and tunable center frequency filter using transversal-form programmable optical filter,” Electron. Lett. 32, 113-114(1996).
    [CrossRef]
  3. I. P. Kaminow, P. P. Iannone, J. Stone, and L. W. Stulz, “FDMA-FSK star network with a tunable optical filter demultiplexer,” J. Lightwave Technol. 6, 1406-1414 (1988).
    [CrossRef]
  4. A. A. M. Saleh and J. Stone, “Two-stage Fabry-Perot filters as demultiplexers in optical FDMA LAN's,” J. Lightwave Technol. 7, 323-330 (1989).
    [CrossRef]
  5. M. Kuznetsov, “Cascaded coupler Mach-Zehnder channel dropping filters for wavelength-division-multiplexed optical systems,” J. Lightwave Technol. 12, 226-230 (1994).
    [CrossRef]
  6. N. Q. Ngo and L. N. Binh, “Novel realization of monotonic Butterworth-type lowpass, highpass and bandpass optical filters using phase-modulated fiber-optic interferometers and ring resonators,” J. Lightwave Technol. 12, 827-841(1994).
    [CrossRef]
  7. N. Q. Ngo, X. Dai, and L. N. Binh, “Realization of first-order monotonic Butterworth-type lowpass and highpass optical filters: experimental verification,” Microwave Opt. Technol. Lett. 8, 306-309 (1995).
    [CrossRef]
  8. L. N. Binh, Photonic Signal Processing (CRC Press, 2007).
    [CrossRef]
  9. C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley, 1999).
    [CrossRef]
  10. T. J. Cavicchi, Digital Signal Processing (Wiley, 2000).
  11. K. Takiguchi, K. Jinguji, K. Okamato, and Y. Ohmori, “Dispersion compensation using a variable group-delay dispersion equaliser,” Electron. Lett. 31, 2129-2194 (1995).
    [CrossRef]
  12. T. Nakagawa, T. Hirota, T. Ohira, M. Aikawa, K. Suto, and E. Yoneda, “New MMIC's for tuners in multichannel video distribution systems using optical fiber networks,” IEEE Trans. Microwave Theory Technol. 43, 1686-1691 (1995).
    [CrossRef]

1996 (1)

E. Pawlowski, K. Takiguchi, M. Okuno, K. Sasayama, A. Himeno, K. Okamato, and Y. Ohmori, “Variable bandwidth and tunable center frequency filter using transversal-form programmable optical filter,” Electron. Lett. 32, 113-114(1996).
[CrossRef]

1995 (4)

S. Suzuki, K. Oda, and Y. Hibino, “Integrated-optic double-ring resonators with a wide free spectral range of 100 GHz,” J. Lightwave Technol. 13, 1766-1771 (1995).
[CrossRef]

N. Q. Ngo, X. Dai, and L. N. Binh, “Realization of first-order monotonic Butterworth-type lowpass and highpass optical filters: experimental verification,” Microwave Opt. Technol. Lett. 8, 306-309 (1995).
[CrossRef]

K. Takiguchi, K. Jinguji, K. Okamato, and Y. Ohmori, “Dispersion compensation using a variable group-delay dispersion equaliser,” Electron. Lett. 31, 2129-2194 (1995).
[CrossRef]

T. Nakagawa, T. Hirota, T. Ohira, M. Aikawa, K. Suto, and E. Yoneda, “New MMIC's for tuners in multichannel video distribution systems using optical fiber networks,” IEEE Trans. Microwave Theory Technol. 43, 1686-1691 (1995).
[CrossRef]

1994 (2)

M. Kuznetsov, “Cascaded coupler Mach-Zehnder channel dropping filters for wavelength-division-multiplexed optical systems,” J. Lightwave Technol. 12, 226-230 (1994).
[CrossRef]

N. Q. Ngo and L. N. Binh, “Novel realization of monotonic Butterworth-type lowpass, highpass and bandpass optical filters using phase-modulated fiber-optic interferometers and ring resonators,” J. Lightwave Technol. 12, 827-841(1994).
[CrossRef]

1989 (1)

A. A. M. Saleh and J. Stone, “Two-stage Fabry-Perot filters as demultiplexers in optical FDMA LAN's,” J. Lightwave Technol. 7, 323-330 (1989).
[CrossRef]

1988 (1)

I. P. Kaminow, P. P. Iannone, J. Stone, and L. W. Stulz, “FDMA-FSK star network with a tunable optical filter demultiplexer,” J. Lightwave Technol. 6, 1406-1414 (1988).
[CrossRef]

Aikawa, M.

T. Nakagawa, T. Hirota, T. Ohira, M. Aikawa, K. Suto, and E. Yoneda, “New MMIC's for tuners in multichannel video distribution systems using optical fiber networks,” IEEE Trans. Microwave Theory Technol. 43, 1686-1691 (1995).
[CrossRef]

Binh, L. N.

N. Q. Ngo, X. Dai, and L. N. Binh, “Realization of first-order monotonic Butterworth-type lowpass and highpass optical filters: experimental verification,” Microwave Opt. Technol. Lett. 8, 306-309 (1995).
[CrossRef]

N. Q. Ngo and L. N. Binh, “Novel realization of monotonic Butterworth-type lowpass, highpass and bandpass optical filters using phase-modulated fiber-optic interferometers and ring resonators,” J. Lightwave Technol. 12, 827-841(1994).
[CrossRef]

L. N. Binh, Photonic Signal Processing (CRC Press, 2007).
[CrossRef]

Cavicchi, T. J.

T. J. Cavicchi, Digital Signal Processing (Wiley, 2000).

Dai, X.

N. Q. Ngo, X. Dai, and L. N. Binh, “Realization of first-order monotonic Butterworth-type lowpass and highpass optical filters: experimental verification,” Microwave Opt. Technol. Lett. 8, 306-309 (1995).
[CrossRef]

Hibino, Y.

S. Suzuki, K. Oda, and Y. Hibino, “Integrated-optic double-ring resonators with a wide free spectral range of 100 GHz,” J. Lightwave Technol. 13, 1766-1771 (1995).
[CrossRef]

Himeno, A.

E. Pawlowski, K. Takiguchi, M. Okuno, K. Sasayama, A. Himeno, K. Okamato, and Y. Ohmori, “Variable bandwidth and tunable center frequency filter using transversal-form programmable optical filter,” Electron. Lett. 32, 113-114(1996).
[CrossRef]

Hirota, T.

T. Nakagawa, T. Hirota, T. Ohira, M. Aikawa, K. Suto, and E. Yoneda, “New MMIC's for tuners in multichannel video distribution systems using optical fiber networks,” IEEE Trans. Microwave Theory Technol. 43, 1686-1691 (1995).
[CrossRef]

Iannone, P. P.

I. P. Kaminow, P. P. Iannone, J. Stone, and L. W. Stulz, “FDMA-FSK star network with a tunable optical filter demultiplexer,” J. Lightwave Technol. 6, 1406-1414 (1988).
[CrossRef]

Jinguji, K.

K. Takiguchi, K. Jinguji, K. Okamato, and Y. Ohmori, “Dispersion compensation using a variable group-delay dispersion equaliser,” Electron. Lett. 31, 2129-2194 (1995).
[CrossRef]

Kaminow, I. P.

I. P. Kaminow, P. P. Iannone, J. Stone, and L. W. Stulz, “FDMA-FSK star network with a tunable optical filter demultiplexer,” J. Lightwave Technol. 6, 1406-1414 (1988).
[CrossRef]

Kuznetsov, M.

M. Kuznetsov, “Cascaded coupler Mach-Zehnder channel dropping filters for wavelength-division-multiplexed optical systems,” J. Lightwave Technol. 12, 226-230 (1994).
[CrossRef]

Madsen, C. K.

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley, 1999).
[CrossRef]

Nakagawa, T.

T. Nakagawa, T. Hirota, T. Ohira, M. Aikawa, K. Suto, and E. Yoneda, “New MMIC's for tuners in multichannel video distribution systems using optical fiber networks,” IEEE Trans. Microwave Theory Technol. 43, 1686-1691 (1995).
[CrossRef]

Ngo, N. Q.

N. Q. Ngo, X. Dai, and L. N. Binh, “Realization of first-order monotonic Butterworth-type lowpass and highpass optical filters: experimental verification,” Microwave Opt. Technol. Lett. 8, 306-309 (1995).
[CrossRef]

N. Q. Ngo and L. N. Binh, “Novel realization of monotonic Butterworth-type lowpass, highpass and bandpass optical filters using phase-modulated fiber-optic interferometers and ring resonators,” J. Lightwave Technol. 12, 827-841(1994).
[CrossRef]

Oda, K.

S. Suzuki, K. Oda, and Y. Hibino, “Integrated-optic double-ring resonators with a wide free spectral range of 100 GHz,” J. Lightwave Technol. 13, 1766-1771 (1995).
[CrossRef]

Ohira, T.

T. Nakagawa, T. Hirota, T. Ohira, M. Aikawa, K. Suto, and E. Yoneda, “New MMIC's for tuners in multichannel video distribution systems using optical fiber networks,” IEEE Trans. Microwave Theory Technol. 43, 1686-1691 (1995).
[CrossRef]

Ohmori, Y.

E. Pawlowski, K. Takiguchi, M. Okuno, K. Sasayama, A. Himeno, K. Okamato, and Y. Ohmori, “Variable bandwidth and tunable center frequency filter using transversal-form programmable optical filter,” Electron. Lett. 32, 113-114(1996).
[CrossRef]

K. Takiguchi, K. Jinguji, K. Okamato, and Y. Ohmori, “Dispersion compensation using a variable group-delay dispersion equaliser,” Electron. Lett. 31, 2129-2194 (1995).
[CrossRef]

Okamato, K.

E. Pawlowski, K. Takiguchi, M. Okuno, K. Sasayama, A. Himeno, K. Okamato, and Y. Ohmori, “Variable bandwidth and tunable center frequency filter using transversal-form programmable optical filter,” Electron. Lett. 32, 113-114(1996).
[CrossRef]

K. Takiguchi, K. Jinguji, K. Okamato, and Y. Ohmori, “Dispersion compensation using a variable group-delay dispersion equaliser,” Electron. Lett. 31, 2129-2194 (1995).
[CrossRef]

Okuno, M.

E. Pawlowski, K. Takiguchi, M. Okuno, K. Sasayama, A. Himeno, K. Okamato, and Y. Ohmori, “Variable bandwidth and tunable center frequency filter using transversal-form programmable optical filter,” Electron. Lett. 32, 113-114(1996).
[CrossRef]

Pawlowski, E.

E. Pawlowski, K. Takiguchi, M. Okuno, K. Sasayama, A. Himeno, K. Okamato, and Y. Ohmori, “Variable bandwidth and tunable center frequency filter using transversal-form programmable optical filter,” Electron. Lett. 32, 113-114(1996).
[CrossRef]

Saleh, A. A. M.

A. A. M. Saleh and J. Stone, “Two-stage Fabry-Perot filters as demultiplexers in optical FDMA LAN's,” J. Lightwave Technol. 7, 323-330 (1989).
[CrossRef]

Sasayama, K.

E. Pawlowski, K. Takiguchi, M. Okuno, K. Sasayama, A. Himeno, K. Okamato, and Y. Ohmori, “Variable bandwidth and tunable center frequency filter using transversal-form programmable optical filter,” Electron. Lett. 32, 113-114(1996).
[CrossRef]

Stone, J.

A. A. M. Saleh and J. Stone, “Two-stage Fabry-Perot filters as demultiplexers in optical FDMA LAN's,” J. Lightwave Technol. 7, 323-330 (1989).
[CrossRef]

I. P. Kaminow, P. P. Iannone, J. Stone, and L. W. Stulz, “FDMA-FSK star network with a tunable optical filter demultiplexer,” J. Lightwave Technol. 6, 1406-1414 (1988).
[CrossRef]

Stulz, L. W.

I. P. Kaminow, P. P. Iannone, J. Stone, and L. W. Stulz, “FDMA-FSK star network with a tunable optical filter demultiplexer,” J. Lightwave Technol. 6, 1406-1414 (1988).
[CrossRef]

Suto, K.

T. Nakagawa, T. Hirota, T. Ohira, M. Aikawa, K. Suto, and E. Yoneda, “New MMIC's for tuners in multichannel video distribution systems using optical fiber networks,” IEEE Trans. Microwave Theory Technol. 43, 1686-1691 (1995).
[CrossRef]

Suzuki, S.

S. Suzuki, K. Oda, and Y. Hibino, “Integrated-optic double-ring resonators with a wide free spectral range of 100 GHz,” J. Lightwave Technol. 13, 1766-1771 (1995).
[CrossRef]

Takiguchi, K.

E. Pawlowski, K. Takiguchi, M. Okuno, K. Sasayama, A. Himeno, K. Okamato, and Y. Ohmori, “Variable bandwidth and tunable center frequency filter using transversal-form programmable optical filter,” Electron. Lett. 32, 113-114(1996).
[CrossRef]

K. Takiguchi, K. Jinguji, K. Okamato, and Y. Ohmori, “Dispersion compensation using a variable group-delay dispersion equaliser,” Electron. Lett. 31, 2129-2194 (1995).
[CrossRef]

Yoneda, E.

T. Nakagawa, T. Hirota, T. Ohira, M. Aikawa, K. Suto, and E. Yoneda, “New MMIC's for tuners in multichannel video distribution systems using optical fiber networks,” IEEE Trans. Microwave Theory Technol. 43, 1686-1691 (1995).
[CrossRef]

Zhao, J. H.

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley, 1999).
[CrossRef]

Electron. Lett. (2)

E. Pawlowski, K. Takiguchi, M. Okuno, K. Sasayama, A. Himeno, K. Okamato, and Y. Ohmori, “Variable bandwidth and tunable center frequency filter using transversal-form programmable optical filter,” Electron. Lett. 32, 113-114(1996).
[CrossRef]

K. Takiguchi, K. Jinguji, K. Okamato, and Y. Ohmori, “Dispersion compensation using a variable group-delay dispersion equaliser,” Electron. Lett. 31, 2129-2194 (1995).
[CrossRef]

IEEE Trans. Microwave Theory Technol. (1)

T. Nakagawa, T. Hirota, T. Ohira, M. Aikawa, K. Suto, and E. Yoneda, “New MMIC's for tuners in multichannel video distribution systems using optical fiber networks,” IEEE Trans. Microwave Theory Technol. 43, 1686-1691 (1995).
[CrossRef]

J. Lightwave Technol. (5)

S. Suzuki, K. Oda, and Y. Hibino, “Integrated-optic double-ring resonators with a wide free spectral range of 100 GHz,” J. Lightwave Technol. 13, 1766-1771 (1995).
[CrossRef]

I. P. Kaminow, P. P. Iannone, J. Stone, and L. W. Stulz, “FDMA-FSK star network with a tunable optical filter demultiplexer,” J. Lightwave Technol. 6, 1406-1414 (1988).
[CrossRef]

A. A. M. Saleh and J. Stone, “Two-stage Fabry-Perot filters as demultiplexers in optical FDMA LAN's,” J. Lightwave Technol. 7, 323-330 (1989).
[CrossRef]

M. Kuznetsov, “Cascaded coupler Mach-Zehnder channel dropping filters for wavelength-division-multiplexed optical systems,” J. Lightwave Technol. 12, 226-230 (1994).
[CrossRef]

N. Q. Ngo and L. N. Binh, “Novel realization of monotonic Butterworth-type lowpass, highpass and bandpass optical filters using phase-modulated fiber-optic interferometers and ring resonators,” J. Lightwave Technol. 12, 827-841(1994).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

N. Q. Ngo, X. Dai, and L. N. Binh, “Realization of first-order monotonic Butterworth-type lowpass and highpass optical filters: experimental verification,” Microwave Opt. Technol. Lett. 8, 306-309 (1995).
[CrossRef]

Other (3)

L. N. Binh, Photonic Signal Processing (CRC Press, 2007).
[CrossRef]

C. K. Madsen and J. H. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach (Wiley, 1999).
[CrossRef]

T. J. Cavicchi, Digital Signal Processing (Wiley, 2000).

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Figures (9)

Fig. 1
Fig. 1

Schematic diagram of the kth-stage first-order all-pole optical filter (FOAPOF) using the PLC technology, microring resonator. Continuous lines are optical waveguides.

Fig. 2
Fig. 2

Schematic diagram of the kth-stage first-order all-zero optical filter (FOAZOF) using the PLC technology.

Fig. 3
Fig. 3

Block diagram representation of the Mth-order tunable optical filter.

Fig. 4
Fig. 4

Characteristics of the tuning parameters versus the normalized bandwidth of the low-pass and high-pass tunable optical filters with variable bandwidth and fixed center frequency (i.e., δ 0 = 0 ) characteristics: (a) IOA amplitude gains, G; (b) intensity coupling coefficients of both low-pass and high-pass filters.

Fig. 5
Fig. 5

Characteristics of the tuning parameters versus the normalized bandwidth of the bandpass and bandstop tunable optical filters with variable bandwidth and fixed center frequency (i.e., δ 0 = 0 ) characteristics, as obtained from Table 4: (a) IOA amplitude gains, G; (b) intensity coupling coefficients of both the bandpass and bandstop filters; (c) optical phase shifts, where ϕ 1 , ϕ 2 and φ 1 = φ 2 are the phase shifts of both the bandpass and the bandstop filters.

Fig. 6
Fig. 6

Squared magnitude responses of the (a) low-pass and (b) high-pass tunable optical filters with variable bandwidth and fixed center frequency (i.e., δ 0 = 0 ) characteristics.The numbers inside the legend box represent the normalized 3 dB cutoff frequencies (i.e., ω c T / π ), which also correspond to the normalized filter bandwidths. Note that the normalized center frequency is designed at ω T / π = 0 . The filter parameters are shown in Fig. 4.

Fig. 7
Fig. 7

Squared magnitude responses of the (a) bandpass and (b) bandstop tunable optical filters with variable bandwidth and fixed center frequency (i.e., δ 0 = 0 ) characteristics. The numbers inside the legend box represent the normalized 3 dB lower and upper corner frequencies ( ω c 1 T / π , ω c 2 T / π ) . The normalized filter bandwidth is given by ( ω c 2 ω c 1 ) T / π . Note that the normalized center frequency is designed at ω T / π = 0.5 .

Fig. 8
Fig. 8

Squared magnitude responses of the (a) low-pass and (b) high-pass tunable optical filters with fixed bandwidth and variable center frequency (i.e., δ 0 = 0.1 π ) characteristics. The numbers inside the legend box represent the new normalized 3 dB cutoff frequencies (i.e., ω c T / π = ω c T / π + δ 0 / π ). The normalized filter bandwidths are still the same as those in Fig. 6 (i.e., ω c T / π = ω c T / π δ 0 / π ). Note that the new normalized center frequency is at ω T / π = 0.1 .

Fig. 9
Fig. 9

Squared magnitude responses of the (a) bandpass and (b) bandstop tunable optical filters with fixed bandwidth and variable center frequency (i.e., δ 0 = 0.1 π ) characteristics. The numbers inside the legend box represent the new normalized 3 dB lower and upper corner frequencies ( ω c 1 T / π , ω c 2 T / π ) , where ω c 1 T / π = ω c 1 T / π + δ 0 / π and ω c 2 T / π = ω c 2 T / π + δ 0 / π . The normalized filter bandwidths are still the same as those in Fig. 7 [i.e., ( ω c 2 ω c 1 ) T / π = ( ω c 2 ω c 1 ) T / π ]. Note that the new normalized center frequency is at ω T / π = 0.6 .

Tables (1)

Tables Icon

Table 1 Filtering Characteristics at the Output Ports of the Second-Order Butterworth Tunable Optical Filter

Equations (61)

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H ^ ( z ) = A ^ k = 1 M ( z z ^ k ) ( z p ^ k ) = A ^ ( z z ^ 1 ) ( z p ^ 1 ) ( z z ^ 2 ) ( z p ^ 2 ) ( z z ^ M ) ( z p ^ M ) ,
p ^ k = | p ^ k | exp [ j arg ( p ^ k ) ] , ( 0 | p ^ k | < 1 ) ,
z ^ k = | z ^ k | exp [ j arg ( z ^ k ) ] , | z ^ k | = 1 ,
H ^ ap , k ( z ) = 1 ( z p ^ k ) ,
H ^ az , k ( z ) = ( z z ^ k ) ,
H ^ ap ( z ) = k = 1 M H ^ ap , k ( z ) .
H ^ az ( z ) = k = 1 M H ^ az , k ( z ) .
H ^ ( z ) = A ^ · H ^ ap ( z ) · H ^ az ( z ) .
complex cross coupled coefficient = γ w a k exp ( j θ k ) ,
complex direct coupled coefficient = γ w ( 1 a k ) exp ( j θ k ) ,
a k = ( 1 + cos ϕ k ) / 2 ( 0 a k 1 ) ,
φ k = cos 1 ( 2 a k 1 ) ( 0 φ k 2 π ) ,
θ k = tan 1 [ sin ϕ k cos ϕ k 1 ] ( π / 2 θ k π / 2 ) .
Λ 1 k = exp ( α w L 1 k ) exp ( j ω T 1 k ) ,
Λ 2 k = exp ( α w L 2 k ) exp ( j ω T 2 k ) exp ( j ϕ k ) ,
H ap , k ( ω ) = E ap , k out E ap , k in = γ w a k exp ( j 2 θ k ) Λ 2 k 1 γ w ( 1 a k ) exp ( j 2 θ k ) Λ 1 k Λ 2 k ,
L = L 1 k + L 2 k ,
T = T 1 k + T 2 k ,
z = exp ( j ω T ) .
H ap , k ( z ) = A ap , k exp [ j ( 2 θ k + ϕ k ) ] exp ( j ω T 2 k ) z z p k ,
A ap , k = γ w a k exp ( α w L 2 k ) ,
p k = γ w ( 1 a k ) exp ( α w L ) exp [ j ( 2 θ k + ϕ k ) ] .
p k = | p k | exp [ j arg ( p k ) ] ( 0 | p k | < 1 ) ,
| p k | = γ w ( 1 a k ) exp ( α w L ) ,
arg ( p k ) = 2 θ k + ϕ k .
Λ 3 k = exp ( α w L 3 k ) exp ( j ω T 3 k ) ,
Λ 4 k = exp ( α w L 4 k ) exp ( j ω T 4 k ) exp ( j ψ k ) ,
H az , k ( ω ) = E az , k out 1 E az , k in = γ az ( 1 b 1 k ) ( 1 b 2 k ) Λ 3 k γ az b 1 k b 2 k Λ 4 k ,
L = L 4 k L 3 k ,
T = T 4 k T 3 k .
H az , k ( z ) = A az , k exp ( j ω T 3 k ) z 1 ( z z k ) ,
A az , k = γ az ( 1 b 1 k ) ( 1 b 2 k ) exp ( α w L 3 k ) ,
z k = b 1 k b 2 k ( 1 b 1 k ) ( 1 b 2 k ) exp ( α w L ) exp ( j ψ k ) .
z k = | z k | exp [ j arg ( z k ) ] ,
| z k | = b 1 k b 2 k ( 1 b 1 k ) ( 1 b 2 k ) exp ( α w L ) ,
arg ( z k ) = ψ k .
H az , k * ( z ) = E az , k out 2 E az , k in = A az , k * exp [ j ( ω T 3 k + π / 2 ) ] z 1 ( z z k * ) ,
A az , k * = γ az b 2 k ( 1 b 1 k ) exp ( α w L 3 k ) ,
z k * = b 1 k ( 1 b 2 k ) b 2 k ( 1 b 1 k ) exp ( α w L ) exp [ j ( ψ k + π ) ] .
z k * = | z k * | exp [ j arg ( z k * ) ] ,
| z k * | = b 1 k ( 1 b 2 k ) b 2 k ( 1 b 1 k ) exp ( α w L ) ,
arg ( z k * ) = ψ k + π .
| H az , k ( z ) | 2 + | H az , k * ( z ) | 2 = 1.
H az ( z ) = k = 1 M H az , k ( z ) .
H ( z ) = G · H ap ( z ) · H az ( z ) ,
H ( z ) = exp [ j ( k = 1 M ( 2 θ k + ϕ k ) ω k = 1 M ( T 2 k + T 3 k ) ) ] · [ A k = 1 M ( z z k ) ( z p k ) ] ,
A = G k = 1 M A ap , k A az , k .
A = A ^ ,
p k = p ^ k ,
z k = z ^ k .
G = ( γ w γ az 1 / 2 ) M A ^ k = 1 M a k [ ( 1 b 1 k ) ( 1 b 2 k ) ] 1 / 2 exp [ α w ( L 2 k + L 3 k ) ] .
a k = 1 | p ^ k | γ w exp ( α w L ) , | p ^ k | γ w exp ( α w L ) ,
ϕ k = { arg ( p ^ k ) 2 θ k , arg ( p ^ k ) 2 θ k 0 , arg ( p ^ k ) 2 θ k + 2 π , arg ( p ^ k ) 2 θ k < 0.
b 1 k = 1 1 + exp ( 2 α w L ) ,
b 2 k = 1 / 2 ,
ψ k = { arg ( z ^ k ) , arg ( z ^ k ) 0 , arg ( z ^ k ) + 2 π , arg ( z ^ k ) < 0.
ϕ k = { arg ( p ^ k ) 2 θ k + δ 0 , arg ( p ^ k ) 2 θ k + δ 0 0 , arg ( p ^ k ) 2 θ k + δ 0 + 2 π , arg ( p ^ k ) 2 θ k + δ 0 < 0 ,
ψ k = { arg ( z ^ k ) + δ 0 , arg ( z ^ k ) + δ 0 0 , arg ( z ^ k ) + δ 0 + 2 π , arg ( z ^ k ) + δ 0 < 0.
a k = 1 1.18 | p ^ k | , | p ^ k | 0.85 ,
b 1 k = 0.523.
G = ( 0.392 ) M A ^ k = 1 M a k ,

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