Abstract

A novel method to simultaneously extract the polarization state and relative spectral phase of an ultrashort laser pulse from an angle-multiplexed spatial-spectral interferometric measurement is proposed and experimentally demonstrated. Spectral interference is produced between an arbitrary polarized signal pulse and two orthogonal linearly polarized reference pulses. The accuracy of this technique has been verified by reconstructing the known relative spectral phase arising from material dispersion and the known elliptical polarization state. Measurement of the relative spectral phase and the spatially variable polarization state of a radially polarized pulse is also demonstrated. An additional independent measurement of the spectral phase of reference pulses provides absolute spectral and temporal characteristics of the signal pulse.

© 2013 OSA

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References

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  1. D. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron.29, 571–579 (1993).
    [CrossRef]
  2. C. Iaconis and I. A. Walmsley, “Self-referencing spectral interferometry for measuring ultrashort optical pulses,” IEEE J. Quantum Electron.35, 501–509 (1999).
    [CrossRef]
  3. M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett.21, 1948–1950 (1996).
    [CrossRef] [PubMed]
  4. Y. I. Salamin, “Accurate fields of a radially polarized Gaussian laser beam,” New J. Phys.8, 133 (2006).
    [CrossRef]
  5. K. J. Moh, X.-C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett.89, 251114 (2006).
    [CrossRef]
  6. W. J. Walecki, D. N. Fittinghoff, A. L. Smirl, and R. Trebino, “Characterization of the polarization state of weak ultrashort coherent signals by dual-channel spectral interferometry,” Opt. Lett.22, 81–83 (1997).
    [CrossRef] [PubMed]
  7. C. Froehly, A. Lacourt, and J. C. Viénot, “Time impulse response and time frequency response of optical pupils: Experimental confirmations and applications,” Nouv. Rev. Opt.4, 183–196 (1973).
    [CrossRef]
  8. P. Schlup, O. Masihzadeh, L. Xu, R. Trebino, and R. Bartels, “Tomographic retrieval of the polarization state of an ultrafast laser pulse,” Opt. Lett.33, 267–269 (2008).
    [CrossRef] [PubMed]
  9. D. Meshulach, D. Yelin, and Y. Silberberg, “White light dispersion measurements by one and two-dimensional spectral Interference,” IEEE J. Quantum Electron.33, 1969–1974 (1997).
    [CrossRef]
  10. D. Meshulach, D. Yelin, and Y. Silberberg, “Real-time spatial-spectral interference measurements of ultrashort optical pulses,” J. Opt. Soc. Am. B14, 2095–2098 (1997).
    [CrossRef]
  11. I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photonics1, 308–437 (2009).
    [CrossRef]
  12. A. Börzsönyi, A. P. Kovács, and K. Osvay, “What we can learn about ultrashort pulses by linear optical methods,” Appl. Sci.3, 515–544 (2013).
    [CrossRef]
  13. P. Bowlan, P. Gabolde, A. Shreenath, K. McGresham, and R. Trebino, “Crossed-beam spectral interferometry: A simple, high-spectral-resolution method for completely characterizing complex ultrashort pulses in real time,” Opt. Express14, 11892–11900 (2006).
    [CrossRef] [PubMed]
  14. A. Börzsönyi, A. P. Kovács, M. Görbe, and K. Osvay, “Advances and limitations of the phase dispersion measurement by spectrally and spatially resolved interferometry,” Opt. Commun.281, 3051–3061 (2008).
    [CrossRef]
  15. O. Tcherbakoff, E. Mével, D. Descamps, J. Plumridge, and E. Constant, “Time-gated high-order harmonic generation,” Phys. Rev. A68, 043804 (2003).
    [CrossRef]
  16. P. Tzallas, E. Skantzakis, C. Kalpouzos, E. P. Benis, G. D. Tsakiris, and D. Charalambidis, “Generation of intense continuum extreme-ultraviolet radiation by many-cycle laser fields,” Nat. Phys.3, 846–850 (2007).
    [CrossRef]

2013 (1)

A. Börzsönyi, A. P. Kovács, and K. Osvay, “What we can learn about ultrashort pulses by linear optical methods,” Appl. Sci.3, 515–544 (2013).
[CrossRef]

2009 (1)

I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photonics1, 308–437 (2009).
[CrossRef]

2008 (2)

A. Börzsönyi, A. P. Kovács, M. Görbe, and K. Osvay, “Advances and limitations of the phase dispersion measurement by spectrally and spatially resolved interferometry,” Opt. Commun.281, 3051–3061 (2008).
[CrossRef]

P. Schlup, O. Masihzadeh, L. Xu, R. Trebino, and R. Bartels, “Tomographic retrieval of the polarization state of an ultrafast laser pulse,” Opt. Lett.33, 267–269 (2008).
[CrossRef] [PubMed]

2007 (1)

P. Tzallas, E. Skantzakis, C. Kalpouzos, E. P. Benis, G. D. Tsakiris, and D. Charalambidis, “Generation of intense continuum extreme-ultraviolet radiation by many-cycle laser fields,” Nat. Phys.3, 846–850 (2007).
[CrossRef]

2006 (3)

P. Bowlan, P. Gabolde, A. Shreenath, K. McGresham, and R. Trebino, “Crossed-beam spectral interferometry: A simple, high-spectral-resolution method for completely characterizing complex ultrashort pulses in real time,” Opt. Express14, 11892–11900 (2006).
[CrossRef] [PubMed]

Y. I. Salamin, “Accurate fields of a radially polarized Gaussian laser beam,” New J. Phys.8, 133 (2006).
[CrossRef]

K. J. Moh, X.-C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett.89, 251114 (2006).
[CrossRef]

2003 (1)

O. Tcherbakoff, E. Mével, D. Descamps, J. Plumridge, and E. Constant, “Time-gated high-order harmonic generation,” Phys. Rev. A68, 043804 (2003).
[CrossRef]

1999 (1)

C. Iaconis and I. A. Walmsley, “Self-referencing spectral interferometry for measuring ultrashort optical pulses,” IEEE J. Quantum Electron.35, 501–509 (1999).
[CrossRef]

1997 (3)

1996 (1)

1993 (1)

D. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron.29, 571–579 (1993).
[CrossRef]

1973 (1)

C. Froehly, A. Lacourt, and J. C. Viénot, “Time impulse response and time frequency response of optical pupils: Experimental confirmations and applications,” Nouv. Rev. Opt.4, 183–196 (1973).
[CrossRef]

Bartels, R.

Benis, E. P.

P. Tzallas, E. Skantzakis, C. Kalpouzos, E. P. Benis, G. D. Tsakiris, and D. Charalambidis, “Generation of intense continuum extreme-ultraviolet radiation by many-cycle laser fields,” Nat. Phys.3, 846–850 (2007).
[CrossRef]

Börzsönyi, A.

A. Börzsönyi, A. P. Kovács, and K. Osvay, “What we can learn about ultrashort pulses by linear optical methods,” Appl. Sci.3, 515–544 (2013).
[CrossRef]

A. Börzsönyi, A. P. Kovács, M. Görbe, and K. Osvay, “Advances and limitations of the phase dispersion measurement by spectrally and spatially resolved interferometry,” Opt. Commun.281, 3051–3061 (2008).
[CrossRef]

Bowlan, P.

Bu, J.

K. J. Moh, X.-C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett.89, 251114 (2006).
[CrossRef]

Burge, R. E.

K. J. Moh, X.-C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett.89, 251114 (2006).
[CrossRef]

Charalambidis, D.

P. Tzallas, E. Skantzakis, C. Kalpouzos, E. P. Benis, G. D. Tsakiris, and D. Charalambidis, “Generation of intense continuum extreme-ultraviolet radiation by many-cycle laser fields,” Nat. Phys.3, 846–850 (2007).
[CrossRef]

Constant, E.

O. Tcherbakoff, E. Mével, D. Descamps, J. Plumridge, and E. Constant, “Time-gated high-order harmonic generation,” Phys. Rev. A68, 043804 (2003).
[CrossRef]

Descamps, D.

O. Tcherbakoff, E. Mével, D. Descamps, J. Plumridge, and E. Constant, “Time-gated high-order harmonic generation,” Phys. Rev. A68, 043804 (2003).
[CrossRef]

Dorrer, C.

I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photonics1, 308–437 (2009).
[CrossRef]

Fittinghoff, D. N.

Froehly, C.

C. Froehly, A. Lacourt, and J. C. Viénot, “Time impulse response and time frequency response of optical pupils: Experimental confirmations and applications,” Nouv. Rev. Opt.4, 183–196 (1973).
[CrossRef]

Gabolde, P.

Görbe, M.

A. Börzsönyi, A. P. Kovács, M. Görbe, and K. Osvay, “Advances and limitations of the phase dispersion measurement by spectrally and spatially resolved interferometry,” Opt. Commun.281, 3051–3061 (2008).
[CrossRef]

Iaconis, C.

C. Iaconis and I. A. Walmsley, “Self-referencing spectral interferometry for measuring ultrashort optical pulses,” IEEE J. Quantum Electron.35, 501–509 (1999).
[CrossRef]

Kalpouzos, C.

P. Tzallas, E. Skantzakis, C. Kalpouzos, E. P. Benis, G. D. Tsakiris, and D. Charalambidis, “Generation of intense continuum extreme-ultraviolet radiation by many-cycle laser fields,” Nat. Phys.3, 846–850 (2007).
[CrossRef]

Kane, D.

D. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron.29, 571–579 (1993).
[CrossRef]

Kovács, A. P.

A. Börzsönyi, A. P. Kovács, and K. Osvay, “What we can learn about ultrashort pulses by linear optical methods,” Appl. Sci.3, 515–544 (2013).
[CrossRef]

A. Börzsönyi, A. P. Kovács, M. Görbe, and K. Osvay, “Advances and limitations of the phase dispersion measurement by spectrally and spatially resolved interferometry,” Opt. Commun.281, 3051–3061 (2008).
[CrossRef]

Lacourt, A.

C. Froehly, A. Lacourt, and J. C. Viénot, “Time impulse response and time frequency response of optical pupils: Experimental confirmations and applications,” Nouv. Rev. Opt.4, 183–196 (1973).
[CrossRef]

Low, D. K. Y.

K. J. Moh, X.-C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett.89, 251114 (2006).
[CrossRef]

Masihzadeh, O.

McGresham, K.

Meshulach, D.

D. Meshulach, D. Yelin, and Y. Silberberg, “White light dispersion measurements by one and two-dimensional spectral Interference,” IEEE J. Quantum Electron.33, 1969–1974 (1997).
[CrossRef]

D. Meshulach, D. Yelin, and Y. Silberberg, “Real-time spatial-spectral interference measurements of ultrashort optical pulses,” J. Opt. Soc. Am. B14, 2095–2098 (1997).
[CrossRef]

Mével, E.

O. Tcherbakoff, E. Mével, D. Descamps, J. Plumridge, and E. Constant, “Time-gated high-order harmonic generation,” Phys. Rev. A68, 043804 (2003).
[CrossRef]

Moh, K. J.

K. J. Moh, X.-C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett.89, 251114 (2006).
[CrossRef]

Osvay, K.

A. Börzsönyi, A. P. Kovács, and K. Osvay, “What we can learn about ultrashort pulses by linear optical methods,” Appl. Sci.3, 515–544 (2013).
[CrossRef]

A. Börzsönyi, A. P. Kovács, M. Görbe, and K. Osvay, “Advances and limitations of the phase dispersion measurement by spectrally and spatially resolved interferometry,” Opt. Commun.281, 3051–3061 (2008).
[CrossRef]

Plumridge, J.

O. Tcherbakoff, E. Mével, D. Descamps, J. Plumridge, and E. Constant, “Time-gated high-order harmonic generation,” Phys. Rev. A68, 043804 (2003).
[CrossRef]

Salamin, Y. I.

Y. I. Salamin, “Accurate fields of a radially polarized Gaussian laser beam,” New J. Phys.8, 133 (2006).
[CrossRef]

Schadt, M.

Schlup, P.

Shreenath, A.

Silberberg, Y.

D. Meshulach, D. Yelin, and Y. Silberberg, “Real-time spatial-spectral interference measurements of ultrashort optical pulses,” J. Opt. Soc. Am. B14, 2095–2098 (1997).
[CrossRef]

D. Meshulach, D. Yelin, and Y. Silberberg, “White light dispersion measurements by one and two-dimensional spectral Interference,” IEEE J. Quantum Electron.33, 1969–1974 (1997).
[CrossRef]

Skantzakis, E.

P. Tzallas, E. Skantzakis, C. Kalpouzos, E. P. Benis, G. D. Tsakiris, and D. Charalambidis, “Generation of intense continuum extreme-ultraviolet radiation by many-cycle laser fields,” Nat. Phys.3, 846–850 (2007).
[CrossRef]

Smirl, A. L.

Stalder, M.

Tcherbakoff, O.

O. Tcherbakoff, E. Mével, D. Descamps, J. Plumridge, and E. Constant, “Time-gated high-order harmonic generation,” Phys. Rev. A68, 043804 (2003).
[CrossRef]

Trebino, R.

Tsakiris, G. D.

P. Tzallas, E. Skantzakis, C. Kalpouzos, E. P. Benis, G. D. Tsakiris, and D. Charalambidis, “Generation of intense continuum extreme-ultraviolet radiation by many-cycle laser fields,” Nat. Phys.3, 846–850 (2007).
[CrossRef]

Tzallas, P.

P. Tzallas, E. Skantzakis, C. Kalpouzos, E. P. Benis, G. D. Tsakiris, and D. Charalambidis, “Generation of intense continuum extreme-ultraviolet radiation by many-cycle laser fields,” Nat. Phys.3, 846–850 (2007).
[CrossRef]

Viénot, J. C.

C. Froehly, A. Lacourt, and J. C. Viénot, “Time impulse response and time frequency response of optical pupils: Experimental confirmations and applications,” Nouv. Rev. Opt.4, 183–196 (1973).
[CrossRef]

Walecki, W. J.

Walmsley, I. A.

I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photonics1, 308–437 (2009).
[CrossRef]

C. Iaconis and I. A. Walmsley, “Self-referencing spectral interferometry for measuring ultrashort optical pulses,” IEEE J. Quantum Electron.35, 501–509 (1999).
[CrossRef]

Xu, L.

Yelin, D.

D. Meshulach, D. Yelin, and Y. Silberberg, “White light dispersion measurements by one and two-dimensional spectral Interference,” IEEE J. Quantum Electron.33, 1969–1974 (1997).
[CrossRef]

D. Meshulach, D. Yelin, and Y. Silberberg, “Real-time spatial-spectral interference measurements of ultrashort optical pulses,” J. Opt. Soc. Am. B14, 2095–2098 (1997).
[CrossRef]

Yuan, X.-C.

K. J. Moh, X.-C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett.89, 251114 (2006).
[CrossRef]

Adv. Opt. Photonics (1)

I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photonics1, 308–437 (2009).
[CrossRef]

Appl. Phys. Lett. (1)

K. J. Moh, X.-C. Yuan, J. Bu, D. K. Y. Low, and R. E. Burge, “Direct noninterference cylindrical vector beam generation applied in the femtosecond regime,” Appl. Phys. Lett.89, 251114 (2006).
[CrossRef]

Appl. Sci. (1)

A. Börzsönyi, A. P. Kovács, and K. Osvay, “What we can learn about ultrashort pulses by linear optical methods,” Appl. Sci.3, 515–544 (2013).
[CrossRef]

IEEE J. Quantum Electron. (3)

D. Kane and R. Trebino, “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating,” IEEE J. Quantum Electron.29, 571–579 (1993).
[CrossRef]

C. Iaconis and I. A. Walmsley, “Self-referencing spectral interferometry for measuring ultrashort optical pulses,” IEEE J. Quantum Electron.35, 501–509 (1999).
[CrossRef]

D. Meshulach, D. Yelin, and Y. Silberberg, “White light dispersion measurements by one and two-dimensional spectral Interference,” IEEE J. Quantum Electron.33, 1969–1974 (1997).
[CrossRef]

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

Nat. Phys. (1)

P. Tzallas, E. Skantzakis, C. Kalpouzos, E. P. Benis, G. D. Tsakiris, and D. Charalambidis, “Generation of intense continuum extreme-ultraviolet radiation by many-cycle laser fields,” Nat. Phys.3, 846–850 (2007).
[CrossRef]

New J. Phys. (1)

Y. I. Salamin, “Accurate fields of a radially polarized Gaussian laser beam,” New J. Phys.8, 133 (2006).
[CrossRef]

Nouv. Rev. Opt. (1)

C. Froehly, A. Lacourt, and J. C. Viénot, “Time impulse response and time frequency response of optical pupils: Experimental confirmations and applications,” Nouv. Rev. Opt.4, 183–196 (1973).
[CrossRef]

Opt. Commun. (1)

A. Börzsönyi, A. P. Kovács, M. Görbe, and K. Osvay, “Advances and limitations of the phase dispersion measurement by spectrally and spatially resolved interferometry,” Opt. Commun.281, 3051–3061 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. A (1)

O. Tcherbakoff, E. Mével, D. Descamps, J. Plumridge, and E. Constant, “Time-gated high-order harmonic generation,” Phys. Rev. A68, 043804 (2003).
[CrossRef]

Supplementary Material (4)

» Media 1: AVI (3768 KB)     
» Media 2: AVI (1297 KB)     
» Media 3: AVI (1221 KB)     
» Media 4: AVI (2429 KB)     

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Figures (9)

Fig. 1
Fig. 1

(a) Principle of the angle-multiplexed spatial-spectral interferometric technique. After being dispersed by a grating, two reference pulses and the signal pulse are vertically multiplexed on a cylindrical mirror and reflected at small angles θ1 and θ2. A CCD camera is used to record the 2D interference pattern produced at the focus of the cylindrical mirror. (b) The orientation angle ψ(ω) and the ellipticity angle χ(ω) of the polarization ellipse

Fig. 2
Fig. 2

The spectral phase and polarization ellipse retrieval algorithm using simulated data. (a) Simulated interferogram corresponding to two orthogonal reference pulses and a signal pulse. The reference pulses have a group delay dispersion (GDD) of −100 fs2 and −30 fs2 and a third-order dispersion (TOD) of −250 fs3 and −50 fs3, respectively. The signal pulse has a GDD of 2000 fs2 and a TOD of 1200 fs3. The phase shift δ between the two polarization components of the signal pulse varies between −π to π ( Media 1). (b) The 1D Fourier transform of this interferogram along x-dimension ( Media 2). (c) In the kx-domain, either the top or bottom sidebands were filtered out. (d) and (e) The extracted sidebands were inverse-Fourier transformed back to the x-domain. (f) and (g) The resulting product of the interfering fields is divided by the spectral field of the reference pulse to obtain the spectral field and phase difference ϕ i s ( ω ) ϕ i r ( ω ) of the corresponding polarization components of the signal and reference pulses. (h) The spectral phase difference δ(ω) between the two polarization components of the signal pulse was calculated by taking the sum of the extracted value of Δϕ(ω) and Δϕr(ω) ( Media 3). (i) The polarization ellipse parameters at central wavelength (λ = 800 nm) was retrieved by using the extracted spectral fields E 1 s ( ω ), E 2 s ( ω ) and spectral phase difference δ(ω) ( Media 4).

Fig. 3
Fig. 3

A schematic illustrates the experimental geometry used for angle-multiplexed spatial-spectral interferometry technique. M, mirror; SM, spherical mirror. Polarization of the signal beam is rotated to an arbitrary polarization state by a zero-order wave plate (λ/4 or λ/2) after reflection from the beamsplitter BS1. Two zero-order half-wave plates (λ/2) after two Glan-Taylor polarizers (LP1 and LP2) with an extinction ratio of ∼ 1 × 10−6 keep the polarization of two reference pulses highly orthogonal. Two delay lines (Delay1, Delay2) equipped with two high precision translation stages provide correct timing between signal pulse and two reference pulses. Three beams are vertically multiplexed at zero delay and collimated before being spectrally dispersed on the CCD using the grating (GR) and the cylindrical mirror (CM).

Fig. 4
Fig. 4

Experimental interferograms with three different temporal delays τ between the two reference pulses: (a) τ < 0, (b) τ = 0, (c) τ > 0. The extracted spectral phase shift δ(ω) corresponding to respective temporal delays of: (d) τ < 0, (e) τ = 0, (f) τ > 0.

Fig. 5
Fig. 5

Effect of measured fast phase jitter: (a) phase drift corresponding to the relative phase difference between the two reference arms; drifts in corresponding polarization ellipse parameters, such as (b) ellipticity angle Δ χ r and (c) orientation angle Δ ψ r. The corresponding drifts for those quantities is also measured over longer time scales and shown in (d)–(f).

Fig. 6
Fig. 6

(a) Extracted relative phase difference between the two components of the polarization state created by the quarter-wave plate; (b) polarization ellipse parameters δ, χ and ψ at the central wavelength (λ), corresponding to various polarization states set by the quarter-wave plate. Solid lines represent the calculated values for these parameters.

Fig. 7
Fig. 7

(a) Experimental spectral interferogram of a signal pulse linearly polarized at 45° passing through SF11 glass rod and interfering with two orthogonally polarized reference pulses. (b) The retrieved spectrum (red) is compared to the measured spectrum (blue). The extracted net spectral phase (green) introduced by the SF11 glass rod is also compared to the calculated phase (magenta) due to material dispersion. (c) Extracted polarization angles for several tested polarization states are plotted against the set linear polarization angle of the signal.

Fig. 8
Fig. 8

(a) Measured far-field image of a radially polarized beam. Also shown is the far-field image of the same beam after passing through a rotating linear analyzer set at (b) 0°, (c) 45°, (d) 90°, and (e) 135°.

Fig. 9
Fig. 9

Two interferograms were taken with two reference beams interfere with the sampled signal beam taken at point 1 (a) and point 2 (b), respectively. (c) Extracted relative spectral phases corresponding to two sampling points (1 and 2) were compared. The inset indicates the sampling position in the beam. (d) The ellipse orientation angles ψ that represent sampled points (1 and 2) were calculated and compared.

Equations (8)

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

I ( ω , x ) = i = 1 2 [ I i s ( ω ) + I i r ( ω ) + 2 I i s ( ω ) I i r ( ω ) cos ( 2 k x sin θ i + ϕ i s ( ω ) ϕ i r ( ω ) ) ] ,
Δ ϕ ( ω ) = [ ϕ 2 s ( ω ) ϕ 2 r ( ω ) ] [ ϕ 1 s ( ω ) ϕ 1 r ( ω ) ] .
Δ ϕ r ( ω ) = ϕ 2 r ( ω ) ϕ 1 r ( ω ) .
δ ( ω ) = Δ ϕ ( ω ) + Δ ϕ r ( ω ) ,
δ ( ω ) = ϕ 2 s ( ω ) ϕ 1 s ( ω ) .
sin 2 χ ( ω ) = 2 E 1 s ( ω ) E 2 s ( ω ) E 1 s ( ω ) 2 + E 2 s ( ω ) 2 sin δ ( ω ) π 4 χ ( ω ) π 4 ,
tan 2 ψ ( ω ) = 2 E 1 s ( ω ) E 2 s ( ω ) E 1 s ( ω ) 2 E 2 s ( ω ) 2 cos δ ( ω ) 0 ψ ( ω ) π .
tan α ( ω ) = E 2 s ( ω ) E 1 s ( ω ) 0 α ( ω ) π 2 .

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