Abstract

New forms using Dirac bra-ket notations and their transformations to express electrically filtered currents are presented for optical systems using either binary differential phase-shift keying (2-DPSK) or ON-OFF keying (OOK) with lumped first-order PMD and PDL, arbitrary optical and electrical filtering and pulse shaping. Based on these forms, the moment generating functions (MGFs) and bit-error-ratios (BERs) for different systems are obtained. Our results show that, for a given BER, 2-DPSK requires ~5dB lower input signal-to-noise ratio than OOK. By comparing BERs for different polarization systems, we also show that the PDL-induced partially polarized noise can significantly improve system performance and reduce BER variation caused by the random couplings between signal polarization, PDL and PMD vectors.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. B. Huttner, C. Geiser, and N. Gisin, "Polarization-induced distortion in optical fiber networks with polarizationmode dispersion and polarization-dependent losses," IEEE J. Select. Topics Quantum Electron. 6, 317-329 (2000).
    [CrossRef]
  2. I. T. Lima, A. O. Lima, Y. Sun, H. Jiao, J. Zweck, C. R. Menyuk, and G. M. Carter, "A receiver model for optical fiber communication systems with arbitrarily polarized noise," J. Lightwave Technol. 23, 1478-1490 (2005).
    [CrossRef]
  3. A. Mecozzi and M. Shtaif, "Signal-to-noise-ratio degradation caused by polarization-dependent loss and the effect of dynamic gain equalization," J. Lightwave Technol. 22, 1856-1871 (2004).
    [CrossRef]
  4. M. Shtaif and O. Rosenberg, "Polarization-dependent loss as a waveform-distorting mechanism and its effect on fiber-optical systems," J. Lightwave Technol. 23, 923-930 (2005).
    [CrossRef]
  5. L. Chen, Z. Zhang, and X. Bao, "Combined PMD-PDL effects on BERs in simplified optical systems: an analytical approach," Opt. Express 15, 2106-2119 (2007).
    [CrossRef] [PubMed]
  6. P. J. Winzer, S. Chandrasekhar, and H. Kim, "Impact of filtering on RZ-DPSK reception," IEEE Photon. Technol. Lett. 15, 840-842 (2003).
    [CrossRef]
  7. D. Marcuse, "Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers," J. Lightwave Technol. 8, 1816-1823 (1990).
    [CrossRef]
  8. P. A. Humblet and M. Azizoglu, "On the bit error rate of lightwave systems with optical amplifiers," J. Lightwave Technol. 9, 1576-1582 (1991).
    [CrossRef]
  9. E. Forestieri, "Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre-and postdetection filtering," J. Lightwave Technol. 18, 1493-1503 (2000).
    [CrossRef]
  10. J. L. Rebola and A. V. T. Cartaxo, "Performance evaluation of optically preamplified receivers with partially polarized noise and arbitrary optical filtering: a rigorous approach," IEE Proc. Optoelectron. 152, 251-262 (2005).
    [CrossRef]
  11. J. Wang and J. M. Kahn, "Impact of chromatic and polarization-mode dispersions on DPSK systems using interferometric demodulation and direct detection," J. Lightwave Technol. 22, 362-371 (2004).
    [CrossRef]
  12. L. Xie, L. Chen, S. Hadjifaradji, and X. Bao, "WDM high speed chirped DPSK fiber optical system transmission modeling in presence of PMD, PDL, and CD," Opt. Fiber Technol. 12, 276-281 (2006).
    [CrossRef]
  13. P. Lu, L. Chen, and X. Bao, "Polarization mode dispersion and polarization dependent loss for a pulse in singlemode fiber," J. Lightwave Technol. 19, 856-859 (2001).
    [CrossRef]
  14. H. Kogelnik, L. E. Nelson, and J. P. Gordon, "Emulation and inversion of polarization-mode dispersion," J. Lightwave Technol. 21, 482-495 (2003).
    [CrossRef]

2007 (1)

2006 (1)

L. Xie, L. Chen, S. Hadjifaradji, and X. Bao, "WDM high speed chirped DPSK fiber optical system transmission modeling in presence of PMD, PDL, and CD," Opt. Fiber Technol. 12, 276-281 (2006).
[CrossRef]

2005 (3)

2004 (2)

2003 (2)

H. Kogelnik, L. E. Nelson, and J. P. Gordon, "Emulation and inversion of polarization-mode dispersion," J. Lightwave Technol. 21, 482-495 (2003).
[CrossRef]

P. J. Winzer, S. Chandrasekhar, and H. Kim, "Impact of filtering on RZ-DPSK reception," IEEE Photon. Technol. Lett. 15, 840-842 (2003).
[CrossRef]

2001 (1)

2000 (2)

E. Forestieri, "Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre-and postdetection filtering," J. Lightwave Technol. 18, 1493-1503 (2000).
[CrossRef]

B. Huttner, C. Geiser, and N. Gisin, "Polarization-induced distortion in optical fiber networks with polarizationmode dispersion and polarization-dependent losses," IEEE J. Select. Topics Quantum Electron. 6, 317-329 (2000).
[CrossRef]

1991 (1)

P. A. Humblet and M. Azizoglu, "On the bit error rate of lightwave systems with optical amplifiers," J. Lightwave Technol. 9, 1576-1582 (1991).
[CrossRef]

1990 (1)

D. Marcuse, "Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers," J. Lightwave Technol. 8, 1816-1823 (1990).
[CrossRef]

Azizoglu, M.

P. A. Humblet and M. Azizoglu, "On the bit error rate of lightwave systems with optical amplifiers," J. Lightwave Technol. 9, 1576-1582 (1991).
[CrossRef]

Bao, X.

Cartaxo, A. V. T.

J. L. Rebola and A. V. T. Cartaxo, "Performance evaluation of optically preamplified receivers with partially polarized noise and arbitrary optical filtering: a rigorous approach," IEE Proc. Optoelectron. 152, 251-262 (2005).
[CrossRef]

Carter, G. M.

Chandrasekhar, S.

P. J. Winzer, S. Chandrasekhar, and H. Kim, "Impact of filtering on RZ-DPSK reception," IEEE Photon. Technol. Lett. 15, 840-842 (2003).
[CrossRef]

Chen, L.

Forestieri, E.

Geiser, C.

B. Huttner, C. Geiser, and N. Gisin, "Polarization-induced distortion in optical fiber networks with polarizationmode dispersion and polarization-dependent losses," IEEE J. Select. Topics Quantum Electron. 6, 317-329 (2000).
[CrossRef]

Gisin, N.

B. Huttner, C. Geiser, and N. Gisin, "Polarization-induced distortion in optical fiber networks with polarizationmode dispersion and polarization-dependent losses," IEEE J. Select. Topics Quantum Electron. 6, 317-329 (2000).
[CrossRef]

Gordon, J. P.

Hadjifaradji, S.

L. Xie, L. Chen, S. Hadjifaradji, and X. Bao, "WDM high speed chirped DPSK fiber optical system transmission modeling in presence of PMD, PDL, and CD," Opt. Fiber Technol. 12, 276-281 (2006).
[CrossRef]

Humblet, P. A.

P. A. Humblet and M. Azizoglu, "On the bit error rate of lightwave systems with optical amplifiers," J. Lightwave Technol. 9, 1576-1582 (1991).
[CrossRef]

Huttner, B.

B. Huttner, C. Geiser, and N. Gisin, "Polarization-induced distortion in optical fiber networks with polarizationmode dispersion and polarization-dependent losses," IEEE J. Select. Topics Quantum Electron. 6, 317-329 (2000).
[CrossRef]

Jiao, H.

Kahn, J. M.

Kim, H.

P. J. Winzer, S. Chandrasekhar, and H. Kim, "Impact of filtering on RZ-DPSK reception," IEEE Photon. Technol. Lett. 15, 840-842 (2003).
[CrossRef]

Kogelnik, H.

Lima, A. O.

Lima, I. T.

Lu, P.

Marcuse, D.

D. Marcuse, "Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers," J. Lightwave Technol. 8, 1816-1823 (1990).
[CrossRef]

Mecozzi, A.

Menyuk, C. R.

Nelson, L. E.

Rebola, J. L.

J. L. Rebola and A. V. T. Cartaxo, "Performance evaluation of optically preamplified receivers with partially polarized noise and arbitrary optical filtering: a rigorous approach," IEE Proc. Optoelectron. 152, 251-262 (2005).
[CrossRef]

Rosenberg, O.

Shtaif, M.

Sun, Y.

Wang, J.

Winzer, P. J.

P. J. Winzer, S. Chandrasekhar, and H. Kim, "Impact of filtering on RZ-DPSK reception," IEEE Photon. Technol. Lett. 15, 840-842 (2003).
[CrossRef]

Xie, L.

L. Xie, L. Chen, S. Hadjifaradji, and X. Bao, "WDM high speed chirped DPSK fiber optical system transmission modeling in presence of PMD, PDL, and CD," Opt. Fiber Technol. 12, 276-281 (2006).
[CrossRef]

Zhang, Z.

Zweck, J.

IEE Proc. Optoelectron. (1)

J. L. Rebola and A. V. T. Cartaxo, "Performance evaluation of optically preamplified receivers with partially polarized noise and arbitrary optical filtering: a rigorous approach," IEE Proc. Optoelectron. 152, 251-262 (2005).
[CrossRef]

IEEE J. Select. Topics Quantum Electron. (1)

B. Huttner, C. Geiser, and N. Gisin, "Polarization-induced distortion in optical fiber networks with polarizationmode dispersion and polarization-dependent losses," IEEE J. Select. Topics Quantum Electron. 6, 317-329 (2000).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

P. J. Winzer, S. Chandrasekhar, and H. Kim, "Impact of filtering on RZ-DPSK reception," IEEE Photon. Technol. Lett. 15, 840-842 (2003).
[CrossRef]

J. Lightwave Technol. (9)

D. Marcuse, "Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers," J. Lightwave Technol. 8, 1816-1823 (1990).
[CrossRef]

P. A. Humblet and M. Azizoglu, "On the bit error rate of lightwave systems with optical amplifiers," J. Lightwave Technol. 9, 1576-1582 (1991).
[CrossRef]

P. Lu, L. Chen, and X. Bao, "Polarization mode dispersion and polarization dependent loss for a pulse in singlemode fiber," J. Lightwave Technol. 19, 856-859 (2001).
[CrossRef]

E. Forestieri, "Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre-and postdetection filtering," J. Lightwave Technol. 18, 1493-1503 (2000).
[CrossRef]

H. Kogelnik, L. E. Nelson, and J. P. Gordon, "Emulation and inversion of polarization-mode dispersion," J. Lightwave Technol. 21, 482-495 (2003).
[CrossRef]

J. Wang and J. M. Kahn, "Impact of chromatic and polarization-mode dispersions on DPSK systems using interferometric demodulation and direct detection," J. Lightwave Technol. 22, 362-371 (2004).
[CrossRef]

A. Mecozzi and M. Shtaif, "Signal-to-noise-ratio degradation caused by polarization-dependent loss and the effect of dynamic gain equalization," J. Lightwave Technol. 22, 1856-1871 (2004).
[CrossRef]

M. Shtaif and O. Rosenberg, "Polarization-dependent loss as a waveform-distorting mechanism and its effect on fiber-optical systems," J. Lightwave Technol. 23, 923-930 (2005).
[CrossRef]

I. T. Lima, A. O. Lima, Y. Sun, H. Jiao, J. Zweck, C. R. Menyuk, and G. M. Carter, "A receiver model for optical fiber communication systems with arbitrarily polarized noise," J. Lightwave Technol. 23, 1478-1490 (2005).
[CrossRef]

Opt. Express (1)

Opt. Fiber Technol. (1)

L. Xie, L. Chen, S. Hadjifaradji, and X. Bao, "WDM high speed chirped DPSK fiber optical system transmission modeling in presence of PMD, PDL, and CD," Opt. Fiber Technol. 12, 276-281 (2006).
[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 (5)

Fig. 1.
Fig. 1.

Low-pass equivalent system with lumped PMD, PDL1, PDL2 and DPSK balanced receiver.

Fig. 2.
Fig. 2.

(a) BER versus Eb/N0 (OSNR) with α=τ=0 and (b) PMD-induced power penalty as a function of normalized DGD τ/Tb with α=0 for the OOK ({Bo,Br }={1.8/Tb ,0.65/Tb }) and the 2-DPSK ({Bo,Br }={2.2/Tb ,0.65/Tb }) systems. Bo (Br ) is the 3dB bandwidth of the Fabry-Pérot optical filter (fifth-order Bessel electrical filter), respectively [11]. BERs are evaluated using (5)–(8). Inset of (a): Time dependent filtered current caused by signal-signal beating in the OOK (solid) and the binary DPSK (dashed) systems. Crosses in (a): Monte Carlo simulation results of Ref. [11]. Stars (DPSK) and squares (OOK) in (b): numerical results of Ref. [11]. Also in (b) the power splitting ratio of PMD γ=0.5 and the required BER is 10-9.

Fig. 3.
Fig. 3.

BER versus DOP for the 2-DPSK system with Eb/N0 =12dB and (a) τ/Tb =0 and (b) τ/Tb =0.3. The BERi PDL1 (i=pa,or) is obtained using (5)–(8) for the system with unpolarized noise, while the BERi PDL2 (i=pa,or) is calculated using (7), (8) and (11) for the case of partially polarized noise. pa (or) means the input signal polarization |p s 〉 is parallel (orthogonal) to the minimum attenuation direction |α⃗0〉 of PDL1 (or |k⃗0 of PDL2). The PDL-induced degree of polariztion is given by DOP = ( 1 e 2 x ) ( 1 + e 2 x ) , where x=a (x=ak) is the PDL value of PDL1 (PDL2), respectively. Insets: the pdf as a function of BER for systems with unpolarized noise (dashed) and partially polarized noise (solid) at DOP=0.28 or α=αk ≈2.5dB. To show clearly the two pdf curves in the inset of (a), the dashed pdf curve (unpolarized noise) is shifted up by 0.3.

Fig. 4.
Fig. 4.

BER versus DOP for the OOK format with Eb/N0 =18dB and τ/Tb =0. Calculations of BERj i (i=pa,or, j=PDL1,PDL2) are explained in the caption of Fig. 3. Also, for the OOK system, the DPSK induced factors Di (i=ss,nn,ns) detailed in (27) should be reduced to unity.

Fig. 5.
Fig. 5.

Low-pass equivalent (a) OOK system and (b) 2-DPSK system in the absence of PMD and PDL, assuming both signal and noise are aligned in the same direction.

Equations (65)

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

i ( t k ) = 1 2 [ s o ( t k + T b ) + n o ( t k + T b ) ] · [ s o ( t k ) + n o ( t k ) ] * + c . c . ,
s in ( t ) = s in ( t ) p s = l = ( S in ) l e j 2 π l t N T b p s .
n ( t ) = n x ( t ) e x + n y ( t ) e y = m = [ ( N in ) x , m e x + ( N in ) y , m e y ] e j 2 π m ( t t k + T 0 ) T 0 .
< e s ( c 2 + 2 c a ) > = d c 2 π σ 2 e c 2 2 σ 2 e s ( c 2 + 2 c a ) = [ 1 2 σ 2 s ] 1 2 e 2 σ 2 s 2 a 2 1 2 σ 2 s
Ψ t k ( s ) = e sy ( t k ) = e sy ss m = M M e s 2 2 σ 2 b ˜ m D 2 1 s β m ( 1 s β m ) 2 ,
E [ y ( t k ) ] = d Ψ t k ( s ) d s s = 0 = y ss + m = M M 2 β m
Δ y 2 = d 2 Ψ t k ( s ) d s 2 s = 0 E 2 [ y ( t k ) ] = m = M M ( 2 β m 2 + 4 σ 2 b ˜ m D 2 ) .
BER y th ( t k ) = ± 1 2 π j C ± Ψ t k ( s ) s e sy th d s ,
BER = k = 0 N 1 BER y th ( t k ) N .
K = ( k 11 k 12 k 21 k 22 ) .
T PDL 2 = e α k 2 e α k · σ 2 = K k 0
Ψ t k ( s ) = e s ( y ss + y nn + y ns ) = e sy ss m = M M e s 2 2 σ 2 ( b ˜ k 0 D ) m 2 1 s β m 1 s β m e s 2 2 σ 2 ( b ˜ k D ) m 2 ( k k 0 ) 2 1 s β m ( k k 0 ) 2 1 s β m ( k k 0 ) 2 .
Ψ t k ( s ) = e sy ss m = M M e s 2 2 σ 2 ( b ˜ k 0 D ) m 2 k 0 2 1 s β m k 0 2 1 s β m k 0 2 e s 2 2 σ 2 ( b ˜ k D ) m 2 k 2 1 s β m k 2 1 s β m k 2 .
( b ˜ i ) m 2 = l , l ' = L L s l o * B ml * B ml ' s l ' o 1 ± p s · k 0 2 k i 2 = b m 2 1 ± p s · k 0 2 ( 1 ± D O P )
y ss = s o ( t k ) R ss s o ( t k ) [ 1 + p s · k 0 2 k 0 2 + 1 p s · k 0 2 k 2 ] d k [ 1 + DOP p s · k 0 ]
Ψ t k ( s ) = e sy ss m = M M e s 2 β m ( 1 + D O P ) 2 ( 1 + p s · k 0 ) b m 2 ( 2 λ m ) 1 s β m ( 1 + D O P ) 1 s β m ( 1 + D O P ) e s 2 β m ( 1 D O P ) 2 ( 1 p s · k 0 ) b m 2 ( 2 λ m ) 1 s β m ( 1 D O P ) 1 s β m ( 1 D O P )
= e s d k [ 1 + DOP ( p s · k 0 ) ] m = M M 1 [ 1 s β m ( 1 DOP ) ] [ 1 s β m ( 1 + DOP ) ] exp [ β m s 2 λ m [ b m ] 2
1 + DOP 2 β m s + β m s DOP 2 + ( p s · k 0 ) DOP ( 2 β m s + DOP 2 β m s ) [ 1 s β m ( 1 DOP ) ] [ 1 s β m ( 1 + DOP ) ] ] ,
s in ( t ) = l = [ s in ( t ) ] l = l ( S in ) l e j 2 π l t N T b , n in ( t ) = m = [ n in ( t ) ] m = m ( N in ) m e j 2 π m ( t t k + T 0 ) T 0 ,
s in ( t k ) = [ ( S in ) L e j 2 π L t k N T b , , ( S in ) L e j 2 π L t k N T b ] T , s in ( t k ) = [ ( S in ) L * e j 2 π L t k N T b , , ( S in ) L * e j 2 π L t k N T b ]
n in ( t k ) = N in = [ ( N in ) M , , ( N in ) M ] T , n in ( t k ) = [ ( N in ) M * , , ( N in ) M * ]
L = η B o N T b , M = η B o T 0 , T 0 = μ ( 1 B o + 1 B r ) .
s o ( t k ) = O ss Φ CD s in ( t k ) , s o ( t k ) = s in ( t k ) Φ CD O ss
n o ( t k ) = O nn N in , n o ( t k ) = N in O nn
y ( t k ) = y ss + y nn + y ns ,
y ss = s o ( t k ) R ss s o ( t k ) ,
y nn = n o ( t k ) R nn n o ( t k ) = N in O nn R nn O nn N in = Z Λ Z ,
y ns = n o ( t k ) R ns s o ( t k ) + c . c . = N in O nn R ns O ss Φ CD s in ( t k ) + c . c . = Z b ( t k ) + c . c . ,
( R ss ) l l ' H r ( l ' l N T b ) , ( R nn ) m m ' H r ( m ' m T 0 ) , ( R ns ) ml H r ( l N T b m T 0 ) ,
Λ U O nn R nn O nn U ,
Z = U N in ,
b ( t k ) = U O nn R ns O ss Φ CD s in ( t k ) = U O nn R ns s o ( t k ) B s o ( t k ) ,
y ss ( t k ) = [ s o ( t k + T b ) R ss s o ( t k ) + c . c . ] 2 = s o ( t k ) R ss D s o ( t k )
y nn ( t k ) = [ n o ( t k + T B ) R nn n o ( t k ) + c . c . ] 2 = N o R nn D N o = Z Λ D Z
y ns ( t k ) = [ n o ( t k + T b ) R ns s o ( t k ) + n o ( t k ) R ns s o ( t k + T b ) + c . c . ] 2
= [ N in O nn R ns D s o ( t k ) + c . c . ] = [ Z b D ( t k ) + c . c . ]
b D ( t k ) = U O nn R ns D s o ( t k ) B D s o ( t k ) ,
( R ss D ) l l ' = ( R ss ) l l ' D l l ' s s , ( R nn D ) m m ' = ( R nn ) m m ' D m m ' n n , ( R ns D ) m l = ( R ns ) m l D m l n s
D ll ' ss = e j 2 π l N + e j 2 π l ' N 2 , D m m ' n n = e j 2 π m T b T 0 + e j 2 π m ' T b T 0 2 , D ml ns = e j 2 π m T b T 0 + e j 2 π l N 2 .
T PMD ( ω l ) = exp ( j ω l τ · σ 2 ) ,
τ 0 T PMD ( ω l ) τ 0 = e j ω l τ 2 , τ T PMD ( ω l ) τ = e j ω l τ 2 , τ 0 T PMD ( ω l ) τ = 0 .
α 0 T PDL 1 α 0 = 1 , α T PDL 1 α = e α , α 0 T PDL 1 α = 0 .
s o ( t k ) = s o ( t k ) p s = O ss P ( τ , α ) Φ CD s in ( t k ) p s ,
T PMD = ( τ 0 τ 0 + τ τ ) T PMD ( τ 0 τ 0 + τ τ ) = e j ω l τ 2 τ 0 τ 0 + e j ω l τ 2 τ τ
T PDL 1 = ( α 0 α 0 + α α ) T PDL 1 ( α 0 α 0 + α α ) = α 0 α 0 + e α α α ,
( P α 0 ) l l ' p s = δ l , l ' α 0 [ α 0 τ 0 τ 0 p s e j ω l τ 2 + α 0 τ τ p s e j ω l τ 2 ] δ l , l ' c l α 0 α 0
( P α ) l l ' p s = δ l , l ' e α α [ α τ 0 τ 0 p s e j ω l τ 2 + α τ τ p s e j ω l τ 2 ] δ l , l ' e α c l α α .
s o ( t k ) = O ss [ ( P α 0 + P α ) P s ] Φ CD s in ( t k ) = [ ( P α 0 + P α ) P s ] s 0 ( t k ) ,
y ss = s o ( t k ) R ss D s o ( t k ) = s o ( t k ) R ˜ ss D s o ( t k ) ,
Θ l l ' = e α [ c l α 0 * c l ' α 0 + e 2 α c l α * c l ' α ] Θ ll ' α 0 + Θ ll ' α ,
Θ ll ' i = e ± α [ e j ( ω l ' ω l ) τ 2 A ˜ ( τ 0 , ± α 0 ) + e j ( ω l ' ω l ) τ 2 A ˜ ( τ 0 , ± α 0 ) ± e j ( ω l ω l ' ) τ 2 C ˜ + j D ˜ 4 ± e j ( ω l + ω l ' ) τ 2 C ˜ j D ˜ 4 ]
= e ± α 2 [ cos ( ω l ' ω l ) τ 2 [ 1 ± ( τ 0 · p s ) ( τ 0 · α 0 ) ] + j sin ( ω l ' ω l ) τ 2 ( τ 0 · p s ± τ 0 · α 0 )
± ( cos ( ω l + ω l ' ) τ 2 C ˜ sin ( ω l + ω l ' ) τ 2 D ˜ ) ] ( + : i = α 0 ; : i = α )
Θ l l ' ( τ , α ; p s ) = s h α sin θ α τ sin θ s τ cos ( φ α τ φ s τ + ( ω l + ω l ' ) τ 2 )
+ cos ( ω l ' ω l ) τ 2 ( ch α + sh α cos θ α τ cos θ s τ ) + j sin ( ω l ' ω l ) τ 2 ( ch α cos θ s τ + sh α cos θ α τ )
Z = U N in = Z α 0 α 0 + Z α α .
y nn = N in O nn R nn D O nn N in = Z Λ D Z = m = M M ( ( Z α 0 ) m 2 + ( Z α ) m 2 ) λ m D ,
b D ( t k ) = B D [ P ( τ , α ) p s ] s o ( t k ) = B D [ P α 0 p s ] s o ( t k ) + B D [ P α p s ] s o ( t k )
b ˜ α 0 D α 0 + b ˜ α D α
y ns = Z b D + c . c . = Z α 0 b ˜ α 0 D + Z α b ˜ α D + c . c .
( b ˜ i D ) m 2 = l , l ' s l o * ( B D ) ml * ( B D ) ml ' s l ' o e α Θ ll ' i ( τ , α ; p s )
b ˜ m D 2 = ( b ˜ α 0 D ) m 2 + ( b ˜ α D ) m 2 = l , l ' s l o * ( B D ) ml * ( B D ) ml ' s l ' o e α Θ ll ' ( τ , α ; p s ) .
y nn = N in P PDL 2 O nn R nn D O nn P PDL 2 N in
= Z k P PDL 2 Λ D P PDL 2 Z k = m = M M [ Z k 0 2 + ( k k 0 ) 0 Z k 2 ] λ m D .
y ns = Z k P PDL 2 b D + c . c . = Z k 0 b ˜ k 0 D + k k 0 Z k b ~ k D + c . c . ,

Metrics