Abstract

We analyze the effects of subwavelength-period resonance gratings on temporally partially coherent optical plane-wave pulse trains. The interaction of the grating with pulses is simulated with the Fourier modal method and finite-difference time-domain method whose performances are compared. Both TE and TM linearly polarized Gaussian Schell-model pulse trains are examined, and partial temporal coherence is modeled with the identical elementary-pulse method. The polarization-dependent response of the grating is seen to lead to significant variations in the average intensity, polarization properties, and degree of temporal coherence of the reflected (and transmitted) pulse trains when the coherence time and polarization state of the incident field are altered. As an important example, we demonstrate that a fully polarized incident pulse train can become partially polarized in grating reflection. The results find use in tailoring of random electromagnetic pulse trains.

© 2019 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Temporal coherence modulation of pulsed, scalar light with a Fabry–Pérot interferometer

Henri Pesonen, Kimmo Saastamoinen, Matias Koivurova, Tero Setälä, and Jari Turunen
J. Opt. Soc. Am. A 36(7) 1137-1145 (2019)

Two-time coherence of pulse trains and the integrated degree of temporal coherence

Rahul Dutta, Ari T. Friberg, Göery Genty, and Jari Turunen
J. Opt. Soc. Am. A 32(9) 1631-1637 (2015)

Polarization changes in temporal imaging with pulses of random light

Timo Voipio, Tero Setälä, and Ari T. Friberg
Opt. Express 21(7) 8987-9004 (2013)

References

  • View by:
  • |
  • |
  • |

  1. O. E. Martinez, “Design of high-power ultrashort pulse amplifiers by expansion and recompression,” IEEE J. Quantum Electron. 23, 1385–1387 (1987).
    [Crossref]
  2. A. M. Weiner, “Ultrafast optical pulse shaping: a tutorial review,” Opt. Commun. 284, 3669–3692 (2011).
    [Crossref]
  3. F. Schreier, M. Schmitz, and O. Bryngdahl, “Pulse delay at diffractive structures under resonance conditions,” Opt. Lett. 23, 1337–1339 (1998).
    [Crossref]
  4. F. Schreier and O. Bryngdahl, “Confined wave packets in the domain of Rayleigh-Wood anomalies,” J. Opt. Soc. Am. A 17, 68–73 (2000).
    [Crossref]
  5. T. Vallius, P. Vahimaa, and J. Turunen, “Pulse deformations at guided-mode resonance filters,” Opt. Express 10, 840–843 (2002).
    [Crossref]
  6. N. V. Golovastikov, D. A. Bykov, and L. L. Doskolovich, “Spatiotemporal pulse shaping using resonant diffraction gratings,” Opt. Lett. 40, 3492–3495 (2015).
    [Crossref]
  7. P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
    [Crossref]
  8. P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express 14, 5007–5012 (2006).
    [Crossref]
  9. J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011).
    [Crossref]
  10. J. Turunen and F. Wyrowski, “Time-dependent physical Stokes parameters and the degree of polarization of light,” Phys. Rev. A 99, 023824 (2019).
    [Crossref]
  11. S. S. Wang, R. Magnusson, J. S. Bagby, and M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction grating,” J. Opt. Soc. Am. A 7, 1470–1474 (1990).
    [Crossref]
  12. S. S. Wang and R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32, 2606–2613 (1993).
    [Crossref]
  13. E. Grinvald, T. Katchalski, S. Soria, S. Levit, and A. A. Friesem, “Role of photonic bandgaps in polarization-independent grating waveguide structures,” J. Opt. Soc. Am. A 25, 1435–1443 (2008).
    [Crossref]
  14. R. Petit, ed., Electromagnetic Theory of Gratings (Springer, 1980), Vol. 22 of Topics in Current Physics.
  15. M. Neviére and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2003).
  16. O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198–200 (2005).
    [Crossref]
  17. J. Tervo, T. Setälä, A. Roueff, P. Réfrégier, and A. T. Friberg, “Two-point Stokes parameters: interpretation and properties,” Opt. Lett. 34, 3074–3076 (2009).
    [Crossref]
  18. T. Voipio, T. Setälä, and A. T. Friberg, “Coherent-mode representation of partially polarized pulsed electromagnetic beams,” J. Opt. Soc. Am. A 30, 2433–2443 (2013).
    [Crossref]
  19. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11, 1137–1143 (2003).
    [Crossref]
  20. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  21. T. Voipio, T. Setälä, and A. T. Friberg, “Partial polarization theory of pulsed optical beams,” J. Opt. Soc. Am. A 30, 71–81 (2013).
    [Crossref]
  22. T. Setälä, F. Nunziata, and A. T. Friberg, “Differences between partial polarizations in the space-time and space-frequency domains,” Opt. Lett. 34, 2924–2926 (2009).
    [Crossref]
  23. D. Pietroy, M. Flury, O. Parriaux, C. Liebig, R. Stoian, and E. Quesnel, “Ultrashort laser pulse splitting upon resonant reflection on a mirror-based waveguide grating,” Opt. Express 16, 17119–17130 (2008).
    [Crossref]
  24. H. Pesonen, K. Saastamoinen, M. Koivurova, T. Setälä, and J. Turunen, “Temporal coherence modulation of pulsed, scalar light with a Fabry-Pérot interferometer,” J. Opt. Soc. Am. A 36, 1137–1145 (2019).
    [Crossref]
  25. A. T. Friberg and T. Setälä, “Electromagnetic theory of optical coherence [Invited],” J. Opt. Soc. Am. A 33, 2431–2442 (2016).
    [Crossref]

2019 (2)

2016 (1)

2015 (1)

2013 (2)

2011 (2)

A. M. Weiner, “Ultrafast optical pulse shaping: a tutorial review,” Opt. Commun. 284, 3669–3692 (2011).
[Crossref]

J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011).
[Crossref]

2009 (2)

2008 (2)

2006 (1)

2005 (1)

2003 (1)

2002 (2)

T. Vallius, P. Vahimaa, and J. Turunen, “Pulse deformations at guided-mode resonance filters,” Opt. Express 10, 840–843 (2002).
[Crossref]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[Crossref]

2000 (1)

1998 (1)

1993 (1)

1990 (1)

1987 (1)

O. E. Martinez, “Design of high-power ultrashort pulse amplifiers by expansion and recompression,” IEEE J. Quantum Electron. 23, 1385–1387 (1987).
[Crossref]

Bagby, J. S.

Bryngdahl, O.

Bykov, D. A.

Doskolovich, L. L.

Flury, M.

Friberg, A. T.

Friesem, A. A.

Golovastikov, N. V.

Grinvald, E.

Katchalski, T.

Koivurova, M.

Korotkova, O.

Levit, S.

Liebig, C.

Magnusson, R.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Martinez, O. E.

O. E. Martinez, “Design of high-power ultrashort pulse amplifiers by expansion and recompression,” IEEE J. Quantum Electron. 23, 1385–1387 (1987).
[Crossref]

Moharam, M. G.

Neviére, M.

M. Neviére and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2003).

Nunziata, F.

Pääkkönen, P.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[Crossref]

Parriaux, O.

Pesonen, H.

Pietroy, D.

Popov, E.

M. Neviére and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2003).

Quesnel, E.

Réfrégier, P.

Roueff, A.

Saastamoinen, K.

Schmitz, M.

Schreier, F.

Setälä, T.

Soria, S.

Stoian, R.

Tervo, J.

Turunen, J.

J. Turunen and F. Wyrowski, “Time-dependent physical Stokes parameters and the degree of polarization of light,” Phys. Rev. A 99, 023824 (2019).
[Crossref]

H. Pesonen, K. Saastamoinen, M. Koivurova, T. Setälä, and J. Turunen, “Temporal coherence modulation of pulsed, scalar light with a Fabry-Pérot interferometer,” J. Opt. Soc. Am. A 36, 1137–1145 (2019).
[Crossref]

J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011).
[Crossref]

P. Vahimaa and J. Turunen, “Independent-elementary-pulse representation for non-stationary fields,” Opt. Express 14, 5007–5012 (2006).
[Crossref]

T. Vallius, P. Vahimaa, and J. Turunen, “Pulse deformations at guided-mode resonance filters,” Opt. Express 10, 840–843 (2002).
[Crossref]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[Crossref]

Vahimaa, P.

Vallius, T.

Voipio, T.

Wang, S. S.

Weiner, A. M.

A. M. Weiner, “Ultrafast optical pulse shaping: a tutorial review,” Opt. Commun. 284, 3669–3692 (2011).
[Crossref]

Wolf, E.

O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198–200 (2005).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Wyrowski, F.

J. Turunen and F. Wyrowski, “Time-dependent physical Stokes parameters and the degree of polarization of light,” Phys. Rev. A 99, 023824 (2019).
[Crossref]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[Crossref]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

O. E. Martinez, “Design of high-power ultrashort pulse amplifiers by expansion and recompression,” IEEE J. Quantum Electron. 23, 1385–1387 (1987).
[Crossref]

J. Mod. Opt. (1)

J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011).
[Crossref]

J. Opt. Soc. Am. A (7)

Opt. Commun. (2)

A. M. Weiner, “Ultrafast optical pulse shaping: a tutorial review,” Opt. Commun. 284, 3669–3692 (2011).
[Crossref]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[Crossref]

Opt. Express (4)

Opt. Lett. (5)

Phys. Rev. A (1)

J. Turunen and F. Wyrowski, “Time-dependent physical Stokes parameters and the degree of polarization of light,” Phys. Rev. A 99, 023824 (2019).
[Crossref]

Other (3)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

R. Petit, ed., Electromagnetic Theory of Gratings (Springer, 1980), Vol. 22 of Topics in Current Physics.

M. Neviére and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2003).

Supplementary Material (1)

NameDescription
» Visualization 1       The evolution of the degree of polarization P (left figure) and normalized Stokes parameters of reflected time domain pulse train from a grating. Varied parameter is the coherence time of the incident pulse.

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 (13)

Fig. 1.
Fig. 1. Illustration of the analyzed subwavelength diffractive element and its effect on the spectrum. (a) A pulse train at normal incidence impinges from a medium with refractive index $ {n_{\text{i}}} $ on the grating with a period $ d $. The grating consists of a corrugation with width $ w $ and thickness $ {h_{\text{g}}} $ on a homogeneous layer whose thickness is $ {h_{\text{h}}} $. The refractive index of these two structures is $ {n_{\text{g}}} $. The grating resides on a semi-infinite substrate with refractive index $ {n_{\text{s}}} $. A virtual plane (dashed line) where the transmitted field is calculated is placed at a distance $ {h_{\text{s}}} $ from the grating. (b) In the case of purely TE- or TM-polarized illumination, the spectral elementary pulse of the incoming field (dashed line) with center angular frequency $ {\omega _{\text{c}}} $ and bandwidth $ \Omega $ splits in transmission into two parts (red solid line) exhibiting a sharp dip at $ {\omega _{\text{F}}} $. The width of the reflection peak (blue solid line) is $ {\Omega _{\text{D}}} $.
Fig. 2.
Fig. 2. Effect of grating I (see Table 1) on the spectral elementary pulse of a GSM pulse train in the cases of (a) TE polarization and (b) TM polarization. The curves correspond to the spectrum of the incident elementary pulse (black dotted line), its transmitted part calculated with FMM (blue dashed line) and FDTD (red dashed line), and the reflected part obtained with FMM (blue solid line). The incident-pulse parameters for both polarizations are $ \Omega = 0.1 \times {10^{15}}\,\, {\rm rad}/{\rm s} $ and $ {\omega _{\text{c}}} = 3.18 \times {10^{15}} \,\, {\rm rad}/{\rm s} $.
Fig. 3.
Fig. 3. Effect of grating I (see Table 1) on the intensity of the temporal elementary pulse of a GSM pulse train in the cases of (a) TE polarization and (b) TM polarization. The curves correspond to the incident elementary pulse (black dotted line), its transmitted part calculated with FMM (blue dashed line) and FDTD (red dashed line), and the reflected part obtained with FMM (blue solid line). The close-ups of the pulse tail are presented in (c) and (d) for TE and TM polarizations, respectively. The pulse parameters are as in Fig. 2.
Fig. 4.
Fig. 4. Illustration of the mean intensities of (a) TE-polarized pulses and (b) TM-polarized pulses whose temporal elementary pulses are shown in Fig. 3. The curves are related to the incident (black dotted line), reflected (solid blue line), and transmitted (FMM, blue dashed line; FDTD, red dashed line) pulse trains. The plots are for grating I, and the incident-pulse parameters are as in Fig. 2.
Fig. 5.
Fig. 5. Examples of the magnitudes of MCFs calculated with (a) FMM, (b) FDTD for TE-polarized light; and with (c) FMM, (d) FDTD for TM-polarized field. The magnitudes of the related degrees of coherence are shown in (e)–(h). The plots are for grating I, and the incident-pulse parameters are as in Fig. 2.
Fig. 6.
Fig. 6. Dependencies of the parameters characterizing the GSM pulse train, and the elementary-pulse representation when the average pulse length is fixed at $ T = 49.2 \,\, {\rm fs} $. The solid and dashed lines show the weight-function width, $ {T_{\rm p}} $, and the coherence time, $ {T_\gamma } $, respectively, as a function of the elementary-function width $ {T_{\text{e}}} $. All parameters are normalized with the average pulse length $ T $. The values used in simulations are marked with the colored circles and triangles.
Fig. 7.
Fig. 7. Illustration of the effect of the coherence time of the incident GSM pulsed beam on the reflected and transmitted average intensities in the cases of (a) TE-polarized light and (b) TM-polarized field. The curves refer to the incident (black dotted line, centered at $ t = 200 \,\, {\rm fs} $), reflected (solid lines), and transmitted (dashed lines) intensities. The blue, orange, and green lines correspond to $ {T_\gamma }/T = 0.16 $, 0.70, and 7.02, respectively.
Fig. 8.
Fig. 8. Magnitudes of the MCFs (left column) and the related degrees of coherence (right column) of the reflected fields in the case of an incident TE-polarized GSM beam. The rows from top to bottom correspond to the coherence times of $ {T_\gamma }/T = 0.16 $, 0.70, and 7.02, respectively.
Fig. 9.
Fig. 9. Magnitudes of the MCFs (left column) and the related degrees of coherence (right column) of the reflected fields in the case of an incident TM-polarized GSM beam. The rows from top to bottom correspond to the coherence times of $ {T_\gamma }/T = 0.16 $, 0.70, 7.02, respectively.
Fig. 10.
Fig. 10. Illustration of (a)–(c) spectral and (d)–(f) temporal reflected average intensities in the case of $ + {45^ \circ } $-polarized incident GSM pulse trains. Rows from top to bottom correspond to the coherence times of $ {T_\gamma }/T = 0.10 $, 0.50, and 1.33, respectively. Black dotted, red solid, and blue solid curves indicate the incident, TE-polarized, and TM-polarized pulse trains. In all cases, the incident temporal pulse shapes are the same and centered at $ {t_0} = 200 \,\, {\rm fs} $.
Fig. 11.
Fig. 11. Illustration of polarization variations when a $ + {45^ \circ } $ linearly polarized GSM pulsed beam reflects from a grating. (a)–(c) Degree of polarization, $ P(t)$, and (d)–(f) normalized Stokes parameters, $ {s_i}(t)$, $ i = (1,2,3) $, (blue, orange, and green solid lines, respectively) and $ {S_0}(t)$ normalized with its maximum value (black dotted line). Rows from top to bottom correspond to the coherence times of $ {T_\gamma }/T = 0.10 $, 0.50, and 1.33, respectively (see also Visualization 1). The grating and pulse parameters are as in Fig. 10.
Fig. 12.
Fig. 12. Distributions of the magnitudes of the two-point Stokes parameters: (a)–(d) unnormalized quantities $ {{\cal S}_j}({t_1},{t_2}) $ and (e)–(h) normalized parameters $ {s_j}({t_1},{t_2}) $, $ j = (0, \ldots ,3) $. Rows from top to bottom correspond to increasing index $ i $. The grating and pulse parameters are otherwise as in Fig. 10, but only the case $ {T_\gamma }/T = 1.33 $ is considered.
Fig. 13.
Fig. 13. Distributions of the degree of electromagnetic coherence, $ \gamma ({t_1},{t_2}) $, of the reflected field in the case of an incident $ + {45^ \circ } $ linearly polarized GSM pulse train. The graphs (a)–(c) correspond to the incident-field coherence times of $ {T_\gamma }/T = 0.10 $, 0.50, and 1.33, respectively. White squares highlight the region of high intensity. The other field and grating parameters are as in Figs. 10 and 11.

Tables (1)

Tables Icon

Table 1. Specifications of the Two Gratings Analyzed in this Worka,b

Equations (44)

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

Γ 0 ( t 1 , t 2 ) = [ Γ 0 x x ( t 1 , t 2 ) Γ 0 x y ( t 1 , t 2 ) Γ 0 y x ( t 1 , t 2 ) Γ 0 y y ( t 1 , t 2 ) ] ,
Γ 0 ( t 1 , t 2 ) = p ( t ) e 0 ( t 1 t ) e 0 T ( t 2 t ) d t .
Γ ( t 1 , t 2 ) = p ( t ) e ( t 1 t ) e T ( t 2 t ) d t ,
e 0 ( ω ) = 1 2 π e 0 ( t ) exp ( i ω t ) d t .
e ( ω ) = R ( ω ) e 0 ( ω ) ,
e ( t ) = 0 e ( ω ) exp ( i ω t ) d ω .
R ( ω ) = [ R x ( ω ) 0 0 R y ( ω ) ] ,
S 0 ( t 1 , t 2 ) = Γ x x ( t 1 , t 2 ) + Γ y y ( t 1 , t 2 ) ,
S 1 ( t 1 , t 2 ) = Γ x x ( t 1 , t 2 ) Γ y y ( t 1 , t 2 ) ,
S 2 ( t 1 , t 2 ) = Γ x y ( t 1 , t 2 ) + Γ y x ( t 1 , t 2 ) ,
S 3 ( t 1 , t 2 ) = i [ Γ y x ( t 1 , t 2 ) Γ x y ( t 1 , t 2 ) ] .
S 0 ( t ) = J x x ( t ) + J y y ( t ) ,
S 1 ( t ) = J x x ( t ) J y y ( t ) ,
S 2 ( t ) = J x y ( t ) + J y x ( t ) ,
S 3 ( t ) = i [ J y x ( t ) J x y ( t ) ] ,
γ i j ( t 1 , t 2 ) = Γ i j ( t 1 , t 2 ) Γ i i ( t 1 , t 1 ) Γ j j ( t 2 , t 2 ) ,
s j ( t 1 , t 2 ) = S j ( t 1 , t 2 ) S 0 ( t 1 ) S 0 ( t 2 ) ,
γ 2 ( t 1 , t 2 ) = t r [ Γ ( t 1 , t 2 ) Γ ( t 2 , t 1 ) ] S 0 ( t 1 ) S 0 ( t 2 ) ,
W ( ω 1 , ω 2 ) = 1 ( 2 π ) 2 Γ ( t 1 , t 2 ) × exp [ i ( ω 1 t 1 ω 2 t 2 ) ] d t 1 d t 2 .
W 0 ( ω 1 , ω 2 ) = 2 π e 0 ( ω 1 ) e 0 T ( ω 2 ) p ( ω 2 ω 1 ) ,
p ( ω ) = 1 2 π p ( t ) exp ( i ω t ) d t .
W ( ω 1 , ω 2 ) = 2 π e ( ω 1 ) e T ( ω 2 ) p ( ω 2 ω 1 ) ,
μ i j ( ω 1 , ω 2 ) = W i j ( ω 1 , ω 2 ) W i i ( ω 1 , ω 1 ) W j j ( ω 2 , ω 2 ) ,
S 0 ( ω ) = 2 π p ( 0 ) [ | e x ( ω ) | 2 + | e y ( ω ) | 2 ] ,
P ( ω ) = 1 4 det J ( ω ) [ t r J ( ω ) ] 2 = S 1 2 ( ω ) + S 2 2 ( ω ) + S 3 2 ( ω ) S 0 ( ω ) ,
P ( t ) = 1 4 det J ( t ) [ t r J ( t ) ] 2 = S 1 2 ( t ) + S 2 2 ( t ) + S 3 2 ( t ) S 0 ( t ) ,
I f ( t ) = p ( t ) | f ( t t ) | 2 d t ,
J i j ( t ) = 2 π 0 e i ( ω 1 ) e j ( ω 2 ) p ( ω 2 ω 1 ) × exp [ i t ( ω 2 ω 1 ) ] d ω 1 d ω 2 .
P = S 1 2 + S 2 2 + S 3 2 S 0 .
e 0 ( t ) = e 0 ( 2 π ) 1 / 4 1 T e exp ( t 2 T e 2 ) exp ( i ω c t )
e 0 ( ω ) = e 0 ( 2 π ) 3 / 4 T e exp [ ( ω ω c ) 2 Ω 2 ] ,
Ω = 2 / T e
p ( t ) = p 0 exp ( 2 t 2 T p 2 )
p ( ω ) = 1 8 π T p p 0 exp ( T p 2 8 ω 2 ) ,
W 0 ( ω 1 , ω 2 ) = e 0 e 0 T [ C 0 ( ω 1 ) C 0 ( ω 2 ) ] 1 / 2 μ 0 ( ω 1 , ω 2 ) ,
C 0 ( ω ) = C 0 exp [ 2 Ω 2 ( ω ω c ) 2 ] ,
μ 0 ( ω 1 , ω 2 ) = exp [ ( ω 2 ω 1 ) 2 2 Ω μ 2 ] ,
Ω μ = 2 / T p
Γ 0 ( t 1 , t 2 ) = e 0 e 0 T [ I 0 ( t 1 ) I 0 ( t 2 ) ] 1 / 2 γ 0 ( t 1 , t 2 ) ,
I 0 ( t ) = I 0 exp ( 2 t 2 T 2 ) ,
γ 0 ( t 1 , t 2 ) = exp [ ( t 1 t 2 ) 2 2 T γ 2 ] exp [ i ω c ( t 2 t 1 ) ] ,
T = T e 2 + T p 2 ,
T γ = T e 1 + T e 2 / T p 2 ,
S 0 ( t ) = t r ( e 0 e 0 T ) I 0 ( t ) ,

Metrics