Abstract

This paper theoretically investigates the dependence of the performance of dual-pump degenerate phase-sensitive amplification (PSA) on wavelength allocation. A fiber-based PSA under unsaturated-gain conditions is considered. Phase mismatch is formalized in terms of incident light frequencies, taking the nonlinear phase shift into account, based on which PSA performances, such as signal gain, noise figure, and phase-clamping effect, are evaluated as a function of the signal wavelength. The results quantitatively indicate that these PSA properties are degraded as the signal wavelength is detuned from the phase-matched condition.

© 2017 Optical Society of America

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References

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  1. M. Marhic, Fiber Parametric Amplifiers, Oscillators and Related Devices (Cambridge, 2008).
  2. R. Tang, J. Lasri, P. S. Devgan, V. Grigoryan, P. Kumar, and M. Vasilyev, “Gain characteristics of a frequency nondegenerate phase-sensitive fiber-optic parametric amplifier with phase self-stabilized input,” Opt. Express 13(26), 10483–10493 (2005).
    [PubMed]
  3. J. Kakande, F. Parmigiani, M. Ibesen, P. Petropoulos, and D. Richardson, “Wide bandwidth experimental study of nondegenerate phase-sensitive amplifiers in single- and dual-pump configuration,” IEEE Photonics Technol. Lett. 22(24), 1781–1783 (2010).
  4. S. Sygletos, S. K. Ibrahim, R. Weerasuriya, R. Phelan, L. G. Nielsen, A. Bogris, D. Syvridis, J. O’Gorman, and A. D. Ellis, “Phase synchronization scheme for a practical phase sensitive amplifier of ASK-NRZ signals,” Opt. Express 19(13), 12384–12391 (2011).
    [PubMed]
  5. M. Gao, T. Inoue, T. Kurosu, and S. Namiki, “Evolution of the gain extinction ratio in dual-pump phase sensitive amplification,” Opt. Lett. 37(9), 1439–1441 (2012).
    [PubMed]
  6. F. Parmitigini, G. D. Hesketh, R. Slavík, P. Horak, P. Petropoulos, and D. Richaedson, “Optical phase quantizer based on phase sensitive four wave mixing at low nonlinear phase shift,” IEEE Photonics Technol. Lett. 27(21), 2146–2149 (2014).
  7. A. Lorences-Riesgo, F. Chiarello, C. Lundström, M. Karlsson, and P. A. Andrekson, “Experimental analysis of degenerate vector phase-sensitive amplification,” Opt. Express 22(18), 21889–21902 (2014).
    [PubMed]
  8. W. Xie, I. Fsaifes, T. Labidi, and F. Bretenaker, “Investigation of degenerate dual-pump phase sensitive amplifier using multi-wave model,” Opt. Express 23(25), 31896–31907 (2015).
    [PubMed]
  9. C. J. McKinstrie and M. G. Raymer, “Four-wave-mixing cascades near the zero-dispersion frequency,” Opt. Express 14(21), 9600–9610 (2006).
    [PubMed]
  10. K. Inoue, “Four-wave mixing in an optical fiber in the zero-dispersion wavelength region,” J. Lightwave Technol. 10(11), 1553–1561 (1992).
  11. K. Inoue, “Quantum noise in parametric amplification under phase-mismatched conditions,” Opt. Commun. 366, 71–76 (2016).
  12. P. L. Voss and P. Kumar, “Raman-noise-induced noise-figure limit for χ(3) parametric amplifiers,” Opt. Lett. 29(5), 445–447 (2004).
    [PubMed]

2016 (1)

K. Inoue, “Quantum noise in parametric amplification under phase-mismatched conditions,” Opt. Commun. 366, 71–76 (2016).

2015 (1)

2014 (2)

A. Lorences-Riesgo, F. Chiarello, C. Lundström, M. Karlsson, and P. A. Andrekson, “Experimental analysis of degenerate vector phase-sensitive amplification,” Opt. Express 22(18), 21889–21902 (2014).
[PubMed]

F. Parmitigini, G. D. Hesketh, R. Slavík, P. Horak, P. Petropoulos, and D. Richaedson, “Optical phase quantizer based on phase sensitive four wave mixing at low nonlinear phase shift,” IEEE Photonics Technol. Lett. 27(21), 2146–2149 (2014).

2012 (1)

2011 (1)

2010 (1)

J. Kakande, F. Parmigiani, M. Ibesen, P. Petropoulos, and D. Richardson, “Wide bandwidth experimental study of nondegenerate phase-sensitive amplifiers in single- and dual-pump configuration,” IEEE Photonics Technol. Lett. 22(24), 1781–1783 (2010).

2006 (1)

2005 (1)

2004 (1)

1992 (1)

K. Inoue, “Four-wave mixing in an optical fiber in the zero-dispersion wavelength region,” J. Lightwave Technol. 10(11), 1553–1561 (1992).

Andrekson, P. A.

Bogris, A.

Bretenaker, F.

Chiarello, F.

Devgan, P. S.

Ellis, A. D.

Fsaifes, I.

Gao, M.

Grigoryan, V.

Hesketh, G. D.

F. Parmitigini, G. D. Hesketh, R. Slavík, P. Horak, P. Petropoulos, and D. Richaedson, “Optical phase quantizer based on phase sensitive four wave mixing at low nonlinear phase shift,” IEEE Photonics Technol. Lett. 27(21), 2146–2149 (2014).

Horak, P.

F. Parmitigini, G. D. Hesketh, R. Slavík, P. Horak, P. Petropoulos, and D. Richaedson, “Optical phase quantizer based on phase sensitive four wave mixing at low nonlinear phase shift,” IEEE Photonics Technol. Lett. 27(21), 2146–2149 (2014).

Ibesen, M.

J. Kakande, F. Parmigiani, M. Ibesen, P. Petropoulos, and D. Richardson, “Wide bandwidth experimental study of nondegenerate phase-sensitive amplifiers in single- and dual-pump configuration,” IEEE Photonics Technol. Lett. 22(24), 1781–1783 (2010).

Ibrahim, S. K.

Inoue, K.

K. Inoue, “Quantum noise in parametric amplification under phase-mismatched conditions,” Opt. Commun. 366, 71–76 (2016).

K. Inoue, “Four-wave mixing in an optical fiber in the zero-dispersion wavelength region,” J. Lightwave Technol. 10(11), 1553–1561 (1992).

Inoue, T.

Kakande, J.

J. Kakande, F. Parmigiani, M. Ibesen, P. Petropoulos, and D. Richardson, “Wide bandwidth experimental study of nondegenerate phase-sensitive amplifiers in single- and dual-pump configuration,” IEEE Photonics Technol. Lett. 22(24), 1781–1783 (2010).

Karlsson, M.

Kumar, P.

Kurosu, T.

Labidi, T.

Lasri, J.

Lorences-Riesgo, A.

Lundström, C.

McKinstrie, C. J.

Namiki, S.

Nielsen, L. G.

O’Gorman, J.

Parmigiani, F.

J. Kakande, F. Parmigiani, M. Ibesen, P. Petropoulos, and D. Richardson, “Wide bandwidth experimental study of nondegenerate phase-sensitive amplifiers in single- and dual-pump configuration,” IEEE Photonics Technol. Lett. 22(24), 1781–1783 (2010).

Parmitigini, F.

F. Parmitigini, G. D. Hesketh, R. Slavík, P. Horak, P. Petropoulos, and D. Richaedson, “Optical phase quantizer based on phase sensitive four wave mixing at low nonlinear phase shift,” IEEE Photonics Technol. Lett. 27(21), 2146–2149 (2014).

Petropoulos, P.

F. Parmitigini, G. D. Hesketh, R. Slavík, P. Horak, P. Petropoulos, and D. Richaedson, “Optical phase quantizer based on phase sensitive four wave mixing at low nonlinear phase shift,” IEEE Photonics Technol. Lett. 27(21), 2146–2149 (2014).

J. Kakande, F. Parmigiani, M. Ibesen, P. Petropoulos, and D. Richardson, “Wide bandwidth experimental study of nondegenerate phase-sensitive amplifiers in single- and dual-pump configuration,” IEEE Photonics Technol. Lett. 22(24), 1781–1783 (2010).

Phelan, R.

Raymer, M. G.

Richaedson, D.

F. Parmitigini, G. D. Hesketh, R. Slavík, P. Horak, P. Petropoulos, and D. Richaedson, “Optical phase quantizer based on phase sensitive four wave mixing at low nonlinear phase shift,” IEEE Photonics Technol. Lett. 27(21), 2146–2149 (2014).

Richardson, D.

J. Kakande, F. Parmigiani, M. Ibesen, P. Petropoulos, and D. Richardson, “Wide bandwidth experimental study of nondegenerate phase-sensitive amplifiers in single- and dual-pump configuration,” IEEE Photonics Technol. Lett. 22(24), 1781–1783 (2010).

Slavík, R.

F. Parmitigini, G. D. Hesketh, R. Slavík, P. Horak, P. Petropoulos, and D. Richaedson, “Optical phase quantizer based on phase sensitive four wave mixing at low nonlinear phase shift,” IEEE Photonics Technol. Lett. 27(21), 2146–2149 (2014).

Sygletos, S.

Syvridis, D.

Tang, R.

Vasilyev, M.

Voss, P. L.

Weerasuriya, R.

Xie, W.

IEEE Photonics Technol. Lett. (2)

J. Kakande, F. Parmigiani, M. Ibesen, P. Petropoulos, and D. Richardson, “Wide bandwidth experimental study of nondegenerate phase-sensitive amplifiers in single- and dual-pump configuration,” IEEE Photonics Technol. Lett. 22(24), 1781–1783 (2010).

F. Parmitigini, G. D. Hesketh, R. Slavík, P. Horak, P. Petropoulos, and D. Richaedson, “Optical phase quantizer based on phase sensitive four wave mixing at low nonlinear phase shift,” IEEE Photonics Technol. Lett. 27(21), 2146–2149 (2014).

J. Lightwave Technol. (1)

K. Inoue, “Four-wave mixing in an optical fiber in the zero-dispersion wavelength region,” J. Lightwave Technol. 10(11), 1553–1561 (1992).

Opt. Commun. (1)

K. Inoue, “Quantum noise in parametric amplification under phase-mismatched conditions,” Opt. Commun. 366, 71–76 (2016).

Opt. Express (5)

Opt. Lett. (2)

Other (1)

M. Marhic, Fiber Parametric Amplifiers, Oscillators and Related Devices (Cambridge, 2008).

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Figures (8)

Fig. 1
Fig. 1

Frequency allocation concerned in this paper.

Fig. 2
Fig. 2

Signal gain of parametric processes other than PSA process, as a function of signal wavelength (λs) relative to the fiber zero-dispersion wavelength (λ0). (a) The frequency difference between the pump and signal lights is fixed at 1 THz, and one pump power is 0.5, 0.4, or 0.3 W. (b) One pump power is 0.5 W, and the signal-pump frequency difference is 0.7, 1.0, or 1.3 THz. In each figure, lines in the shorter and longer wavelength sides are gains due to amplification processes of (i) pump-1 → signal + idler(-), and (ii) pump-2 → signal + idler( + ), respectively. Parameters used in calculations are λ = 1.55 μm, Dcc = 0.02 ps/km-nm-nm, γ = 12 W/km, and L = 200m.

Fig. 3
Fig. 3

Generation efficiency of FWM lights at f112 = 2fp1fp2 and f221 = 2fp2fp1, defined by P112(L)/P0 and P112(L)/P0 where P112(L) and P112(L) are the FWM output power at f112 and f221, respectively, and P0 is the pump input power. (a) The frequency difference between the pump and signal lights is fixed at 1 THz, and one pump power is 0.5, 0.4, or 0.3 W. (b) One pump power is 0.5 W, and the signal-pump frequency difference is 0.7, 1.0, or 1.3 THz. Parameters used in calculations are λ = 1.55 μm, Dcc = 0.02 ps/km-nm-nm, γ = 12 W/km, and L = 200m.

Fig. 4
Fig. 4

Signal gain as a function of signal wavelength (λs) relative to the fiber zero-dispersion wavelength (λ0). The relative incident phase is fixed at a value optimized for the phase matched condition. (a) The frequency difference between the pump and signal lights is fixed at 1 THz, and one pump power is 0.5, 0.4, or 0.4 W. (b) One pump power is 0.5 W, and the signal-pump frequency difference is 0.7, 1.0, or 1.3 THz. Parameters used in calculations are λ = 1.55 μm, Dcc = 0.02 ps/km-nm-nm, γ = 12 W/km, and L = 200m.

Fig. 5
Fig. 5

Signal gain as a function of signal wavelength (λs) relative to the fiber zero-dispersion wavelength (λ0). The relative incident phase is optimized at each wavelength. The parameters used are the same as those in Fig. 4.

Fig. 6
Fig. 6

Noise figure NF as a function of signal wavelength (λs) relative to the fiber zero-dispersion wavelength (λ0). The relative incident phase is fixed at a value optimized for the phase matched condition. The parameters used are the same as those in Fig. 3.

Fig. 7
Fig. 7

Suppression ration of phase deviation as a function of signal wavelength (λs) relative to the fiber zero-dispersion wavelength (λ0). The relative incident phase is fixed at a value optimized for the phase matched condition. The parameters used are the same as those in Fig. 4.

Fig. 8
Fig. 8

Suppression ration of phase deviation as a function of signal wavelength (λs) relative to the fiber zero-dispersion wavelength (λ0). The relative incident phase is optimized at each wavelength. The parameters used are the same as those in Fig. 4.

Equations (61)

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d E p1 dz =iγ(| E p1 | 2 +2| E p2 | 2 +2| E s | 2 ) E p1 +iγ E s 2 E p2 * e iΔ β 0 z ,
d E p2 dz =iγ(2| E p1 | 2 +| E p2 | 2 +2| E s | 2 ) E p2 +iγ E s 2 E p1 * e iΔ β 0 z ,
d E s dz =iγ(2| E p1 | 2 +2| E p2 | 2 +| E s | 2 ) E s +2iγ E p1 E p2 E s * e iΔ β 0 z ,
d E s dz =4iγ P 0 E s +2γ P 0 e iφ e 6iγ P 0 z E s * e iΔ β 0 z ,
E out ={cosh( g 0 L)+i(Δβ/2 g 0 )sinh( g 0 L)} E in + e iφ 1+ (Δβ/2 g 0 ) 2 sinh( g 0 L) E in * ,
E out ={cos( g 1 L)+i(Δβ/2 g 1 )sin( g 1 L)} E in + e iφ (Δβ/2 g 1 ) 2 1 sin( g 0 L) E in * ,
β 0 (f)= β 0 ( f 0 )+(f f 0 ) [ d β 0 df ] f 0 + (f f 0 ) 2 π λ 4 3 c 2 [ d D c dλ ] f 0 ,
Δβ=( f s f 0 ) ( f p1 f s ) 2 2π λ 4 D cc c 2 +2γ P 0 ,
E out0 = A 0 e i ϕ 0 E in + B 0 e iφ E in * ,
A 0 = cos h 2 ( g 0 L)+ (Δβ/2 g 0 ) 2 sin h 2 ( g 0 L) ,
ϕ 0 =arctan[ (Δβ/2 g 0 )sinh( g 0 L) cosh( g 0 L) ],
B 0 = 1+ (Δβ/2 g 0 ) 2 sinh( g 0 L).
E out1 = A 1 e i ϕ 1 E in + B 1 e iφ E in * ,
A 1 = cos 2 ( g 1 L)+ (Δβ/2 g 1 ) 2 sin 2 ( g 1 L) ,
ϕ 1 =arctan[ (Δβ/2 g 1 )sin( g 1 L) cos( g 1 L) ],
B 1 = (Δβ/2 g 1 ) 2 1 sin( g 1 L).
d E p1 dz =iγ( P p1 +2 P p2 ) E 1 +2iγ E s E i E p1 * e iΔ β 0 z ,
d E s dz =2iγ( P p1 + P p2 ) E s +iγ E p1 2 E i * e iΔ β 0 z ,
d E i dz =2iγ( P p1 + P p2 ) E i +iγ E p1 2 E s * e iΔ β 0 z ,
E s (L)={cosh( g 0 L)+i(Δ β /2 g 0 )sinh( g 0 L)} E s (0)
E s (L)={cos( g 1 L)+i(Δ β /2 g 1 )sin( g 1 L)} E s (0),
Δ β =( f p1 f 0 ) ( f s f p1 ) 2 2π λ 4 D cc c 2 2γ P 0 ,
d E p2 dz =iγ(2 P p1 + P p2 ) E p1 +2iγ E s E i+ E p2 * e iΔ β 0+ z ,
d E s dz =2iγ( P p1 + P p2 ) E s +iγ E p2 2 E i+ * e iΔ β 0+ z ,
d E i+ dz =2iγ( P p1 + P p2 ) E i+ +iγ E p2 2 E s * e iΔ β 0+ z
E s (L)={cosh( g +0 L)+i(Δ β + /2 g +0 )sinh( g +0 L)} E s (0)
E s (L)={cos( g +1 L)+i(Δ β + /2 g +1 )sin( g +1 L)} E s (0)
Δ β + =(2 f s f p1 f 0 ) ( f s f p1 ) 2 2π λ 4 D cc c 2 2γ P 0 ,
d E 112 dz =2iγ( P p1 + P p2 ) E 112 +iγ E p1 2 E p2 * e iΔ β 112(0) z ,
d E 221 dz =2iγ( P p1 + P p2 ) E 221 +iγ E p2 2 E p1 * e iΔ β 221(0) z ,
E 112 (L)= e iφ (Δ β 112 /2 g 112 ) 2 1 sin( g 112 L) E p2 * (0),
E 221 (L)= e iφ (Δ β 221 /2 g 221 ) 2 1 sin( g 221 L) E p1 * (0),
P 112 (L)={ (Δ β 112 /2 g 112 ) 2 1} sin 2 ( g 112 L) P 0 ,
P 221 (L)={ (Δ β 221 /2 g 221 ) 2 1} sin 2 ( g 221 L) P 0 ,
Δ β 112 =24{( f p1 f s )( f s f 0 )} ( f p1 f s ) 2 2π λ 4 D cc c 2 γ P 0 ,
Δ β 221 =24{( f p1 f s )+( f s f 0 )} ( f p1 f s ) 2 2π λ 4 D cc c 2 γ P 0 ,
E outi ={ A i e i( ϕ i + θ in ) E in + B i e i(φ θ in ) }| E in |,.......(i= 0, 1)
G i =| A i e i( ϕ i + θ in ) + B i e i(φ θ in ) | 2 = A i 2 + B i 2 +2 A i B i cos( ϕ i +2 θ in φ),
G i (Δβ=0)= A i 2 + B i 2 +2 A i B i cos(2 θ in φ).
G 0 = A 0 2 + B 0 2 +2 A 0 B 0 cos ϕ 0 = A 0 2 + B 0 2 +2 B 0 Re[ A 0 e i ϕ 0 ] = cosh 2 ( g 0 L)+ (Δβ/2 g 0 ) 2 sin h 2 ( g 0 L)+{1+ (Δβ/2 g 0 ) 2 } sinh 2 ( g 0 L) +2 1+ (Δβ/2 g 0 ) 2 sinh( g 0 L)cosh( g 0 L) =cosh(2 g 0 L)+2 (Δβ/2 g 0 ) 2 sinh 2 ( g 0 L)+ 1+ (Δβ/2 g 0 ) 2 sinh(2 g 0 L)
G 1 = A 1 2 + B 1 2 +2 B 1 Re[ A 1 e i ϕ 0 ] = cos 2 ( g 1 L)+ (Δβ/2 g 1 ) 2 sin 2 ( g 1 L)+{1+ (Δβ/2 g 1 ) 2 } sin 2 ( g 1 L) +2 (Δβ/2 g 1 ) 2 1 sin( g 1 L)cos( g 1 L) =cos(2 g 1 L)+2 (Δβ/2 g 1 ) 2 sinh 2 ( g 1 L)+ (Δβ/2 g 1 ) 2 1 sin(2 g 1 L)
G 0 = A 0 2 + B 0 2 +2 A 0 B 0 =cosh(2 g 0 L)+2 (Δβ/2 g 0 ) 2 sin h 2 ( g 0 L) +2 {cos h 2 ( g 0 L)+ (Δβ/2 g 0 ) 2 cosh(2 g 0 L)+ (Δβ/2 g 0 ) 4 sin h 2 ( g 0 L)} 1/2 sinh( g 0 L)
G 1 = A 1 2 + B 1 2 +2 A 1 B 1 =cos(2 g 1 L)+2 (Δβ/2 g 1 ) 2 sin 2 ( g 1 L) +2 [ (Δβ/2 g 1 ) 2 cos(2 g 1 L)+ (Δβ/2 g 1 ) 4 sin 2 ( g 1 L) cos 2 ( g 1 L)] 1/2 sin( g 1 L)
a ^ out ={cosh( g 0 L)+i(Δβ/2 g 0 )sinh( g 0 L)} a ^ in + e iφ 1+ (Δβ/2 g 0 ) 2 sinh( g 0 L) a ^ in = A 0 e i ϕ 0 a ^ in + B 0 e iφ a ^ in ,
a ^ out ={cos( g 1 L)+i(Δβ/2 g 1 )sin( g 1 L)} a ^ in + e iφ (Δβ/2 g 1 ) 2 1 sin( g 1 L) a ^ in = A 1 e i ϕ 1 a ^ in + B 1 e iφ a ^ in .
< n ^ out >=<α| a ^ out a ^ out |α> =( A i 2 + B i 2 )< a ^ in a ^ in >+ A i B i { e i(φ ϕ i ) < a ^ in 2 >+ e i(φ ϕ i ) < a ^ in 2 >}+ B i 2 =[ A i 2 + B i 2 + A i B i { e i(φ2 θ in ϕ i ) + e i(φ2 θ in ϕ i ) }] n in + B i 2 = G i n in + B i 2 ,
< n ^ out 2 >=<α| ( a ^ out a ^ out ) 2 |α> =<α| [( A i 2 + B i 2 ) a ^ in a ^ in + A i B i { e i(φ ϕ i ) a ^ in 2 + e i(φ ϕ i ) a ^ in 2 }+ B i 2 ] 2 |α> = n in 2 G i 2 + n in [ G i 2 +2 G i B i 2 +4 ( A i B i ) 2 ( A i B i ) 2 { e i(φ2 θ in ϕ i ) + e i(φ2 θ in ϕ i ) } 2 ] + B i 4 +2 ( A i B i ) 2 .
σ out 2 =< n ^ out 2 >< n ^ out > 2 = n in [ G i 2 +4 ( A i B i ) 2 ( A i B i ) 2 { e i(φ2 θ in ϕ i ) + e i(φ2 θ in ϕ i ) } 2 ]+2 ( A i B i ) 2 ,
(SNR) out = < n ^ out > 2 σ out 2 = n in 1+[4 ( A i B i ) 2 ( A i B i ) 2 { e i(φ2 θ in ϕ i ) + e i(φ2 θ in ϕ i ) } 2 ]/ G i 2 ,
NF= (SNR) in (SNR) out =1+ ( A i B i ) 2 G i 2 [4 { e i(φ2 θ in ϕ i ) + e i(φ2 θ in ϕ i ) } 2 ].
NF=1+ B 0 2 G 0 2 [4 A 0 2 { A 0 e i ϕ 0 + A 0 e i ϕ 0 } 2 ] =1+ 4 G 0 2 ( Δβ 2 g 0 ) 2 {1+ (Δβ/2 g 0 ) 2 }sin h 4 ( g 0 L)
NF=1+ B 1 2 G 1 2 [4 A 1 2 { A 1 e i ϕ 1 + A 1 e i ϕ 1 } 2 ] =1+ 4 G 1 2 ( Δβ 2 g 1 ) 2 { (Δβ/2 g 1 ) 2 1} sin 4 ( g 1 L)
NF=1+ ( A i B i ) 2 G i 2 [4 (1+1) 2 ]=1.
E outi ={ A i e i{ θ in +( ϕ i φ)/2} + B i e i{ θ in +( ϕ i φ)/2} }| E in | e i( ϕ i +φ)/2 ,
E outi ={ A i e i{ θ in0 +Δθ+( ϕ i φ)/2} + B i e i{ θ in0 +Δθ+( ϕ i φ)/2} }| E in | e i( ϕ i +φ)/2 .
E outi ={ A i e i{Δθ+ ϕ i /2} + B i e i{Δθ+ ϕ i /2} }| E in | e i( ϕ i +φ)/2 ={( A i + B i )cos(Δθ+ ϕ i /2)+i( A i B i )sin(Δθ+ ϕ i /2)}| E in | e i( ϕ i +φ)/2 .
arg[ E outi ]= ϕ i +φ 2 +arctan[ A i B i A i + B i tan(Δθ+ ϕ i /2) ].
R= 1 Δθ { arctan[ A i B i A i + B i tan( ϕ i /2+Δθ) ]arctan[ A i B i A i + B i tan( ϕ i /2) ] }.
E outi ={ A i e iΔθ + B i e iΔθ }| E in | e i( ϕ i +φ)/2 ,
arg[ E outi ]= ϕ i +φ 2 +arctan[ A i B i A i + B i tan(Δθ) ],
R= 1 Δθ arctan[ A i B i A i + B i tan(Δθ) ].