Abstract

We introduce a simple delay-scanned complete spatiotemporal intensity-and-phase measurement technique based on wavelength-multiplexed holography to characterize long, complex pulses in space and time. We demonstrate it using pulses emerging from multi-mode fiber. This technique extends the temporal range and spectral resolution of the single-frame STRIPED FISH technique without using an otherwise-required expensive ultranarrow-bandpass filter. With this technique, we measured the complete intensity and phase of up to ten fiber modes from a multi-mode fiber (normalized frequency V ≈10) over a ~3ps time range. Spatiotemporal complexities such as intermodal delay, modal dispersion, and material dispersion were also intuitively displayed by the retrieved results. Agreement between the reconstructed color movies and the monitored time-averaged spatial profiles confirms the validity to this delay-scanned STRIPED FISH method.

© 2017 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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2016 (2)

D. R. Gray, M. N. Petrovich, S. R. Sandoghchi, N. V. Wheeler, N. K. Baddela, G. T. Jasion, T. Bradley, D. J. Richardson, and F. Poletti, “Real-Time Modal Analysis via Wavelength-Swept Spatial and Spectral (S-2) Imaging,” IEEE Photonics Technol. Lett. 28, 1034–1037 (2016).

Z. Zhu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Observation of multimode solitons in few-mode fiber,” Opt. Lett. 41(20), 4819–4822 (2016).
[Crossref] [PubMed]

2015 (1)

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9(5), 306–310 (2015).
[Crossref]

2014 (1)

2013 (1)

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

2012 (1)

2011 (2)

A. S. Kurkov, E. M. Sholokhov, and Y. E. Sadovnikova, “All-fiber supercontinuum source in the range of 1550-2400 nm based on telecommunication multimode fiber,” Laser Phys. Lett. 8(8), 598–600 (2011).
[Crossref]

D. N. Schimpf, R. A. Barankov, and S. Ramachandran, “Cross-correlated (C2) imaging of fiber and waveguide modes,” Opt. Express 19(14), 13008–13019 (2011).
[Crossref] [PubMed]

2010 (3)

2009 (2)

2008 (3)

2003 (1)

2001 (1)

2000 (1)

A. Liem, D. Nickel, J. Limpert, H. Zellmer, U. Griebner, S. Unger, A. Tunnermann, and G. Korn, “High average power ultra-fast fiber chirped pulse amplification system,” Appl. Phys. B-Lasers Opt. 71(6), 889–891 (2000).
[Crossref]

1985 (1)

1982 (1)

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[Crossref]

1978 (1)

1971 (2)

Akturk, S.

Baddela, N. K.

D. R. Gray, M. N. Petrovich, S. R. Sandoghchi, N. V. Wheeler, N. K. Baddela, G. T. Jasion, T. Bradley, D. J. Richardson, and F. Poletti, “Real-Time Modal Analysis via Wavelength-Swept Spatial and Spectral (S-2) Imaging,” IEEE Photonics Technol. Lett. 28, 1034–1037 (2016).

Barankov, R. A.

Bates, P. K.

Biegert, J.

Bolle, C.

Bonaretti, F.

Bowlan, P.

Bradley, T.

D. R. Gray, M. N. Petrovich, S. R. Sandoghchi, N. V. Wheeler, N. K. Baddela, G. T. Jasion, T. Bradley, D. J. Richardson, and F. Poletti, “Real-Time Modal Analysis via Wavelength-Swept Spatial and Spectral (S-2) Imaging,” IEEE Photonics Technol. Lett. 28, 1034–1037 (2016).

Burrows, E. C.

Chalus, O.

Chauhan, V.

Christodoulides, D. N.

Z. Zhu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Observation of multimode solitons in few-mode fiber,” Opt. Lett. 41(20), 4819–4822 (2016).
[Crossref] [PubMed]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9(5), 306–310 (2015).
[Crossref]

Clarkson, W. A.

Clerici, M.

Cohen, J.

Das, S.

Davis, M.

Di Trapani, P.

Englefield, C. G.

Esmaeelpour, M.

Essiambre, R.-J.

Faccio, D.

Fini, J. M.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

Fuchs, U.

Ghalmi, S.

Gloge, D.

Gnauck, A. H.

Goud, P. A.

Gray, D. R.

D. R. Gray, M. N. Petrovich, S. R. Sandoghchi, N. V. Wheeler, N. K. Baddela, G. T. Jasion, T. Bradley, D. J. Richardson, and F. Poletti, “Real-Time Modal Analysis via Wavelength-Swept Spatial and Spectral (S-2) Imaging,” IEEE Photonics Technol. Lett. 28, 1034–1037 (2016).

Griebner, U.

A. Liem, D. Nickel, J. Limpert, H. Zellmer, U. Griebner, S. Unger, A. Tunnermann, and G. Korn, “High average power ultra-fast fiber chirped pulse amplification system,” Appl. Phys. B-Lasers Opt. 71(6), 889–891 (2000).
[Crossref]

Gu, X.

Guang, Z.

Horak, P.

Ikeda, M.

Ina, H.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[Crossref]

Jasion, G. T.

D. R. Gray, M. N. Petrovich, S. R. Sandoghchi, N. V. Wheeler, N. K. Baddela, G. T. Jasion, T. Bradley, D. J. Richardson, and F. Poletti, “Real-Time Modal Analysis via Wavelength-Swept Spatial and Spectral (S-2) Imaging,” IEEE Photonics Technol. Lett. 28, 1034–1037 (2016).

Kimmel, M.

Kitayama, K.

Kobayashi, S.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[Crossref]

Korn, G.

A. Liem, D. Nickel, J. Limpert, H. Zellmer, U. Griebner, S. Unger, A. Tunnermann, and G. Korn, “High average power ultra-fast fiber chirped pulse amplification system,” Appl. Phys. B-Lasers Opt. 71(6), 889–891 (2000).
[Crossref]

Kurkov, A. S.

A. S. Kurkov, E. M. Sholokhov, and Y. E. Sadovnikova, “All-fiber supercontinuum source in the range of 1550-2400 nm based on telecommunication multimode fiber,” Laser Phys. Lett. 8(8), 598–600 (2011).
[Crossref]

Liem, A.

A. Liem, D. Nickel, J. Limpert, H. Zellmer, U. Griebner, S. Unger, A. Tunnermann, and G. Korn, “High average power ultra-fast fiber chirped pulse amplification system,” Appl. Phys. B-Lasers Opt. 71(6), 889–891 (2000).
[Crossref]

Limpert, J.

A. Liem, D. Nickel, J. Limpert, H. Zellmer, U. Griebner, S. Unger, A. Tunnermann, and G. Korn, “High average power ultra-fast fiber chirped pulse amplification system,” Appl. Phys. B-Lasers Opt. 71(6), 889–891 (2000).
[Crossref]

Lingle, R.

McCurdy, A. H.

Mumtaz, S.

Nelson, L. E.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

Nicholson, J. W.

Nickel, D.

A. Liem, D. Nickel, J. Limpert, H. Zellmer, U. Griebner, S. Unger, A. Tunnermann, and G. Korn, “High average power ultra-fast fiber chirped pulse amplification system,” Appl. Phys. B-Lasers Opt. 71(6), 889–891 (2000).
[Crossref]

Nilsson, J.

O’Shea, P.

Peckham, D. W.

Petrovich, M. N.

D. R. Gray, M. N. Petrovich, S. R. Sandoghchi, N. V. Wheeler, N. K. Baddela, G. T. Jasion, T. Bradley, D. J. Richardson, and F. Poletti, “Real-Time Modal Analysis via Wavelength-Swept Spatial and Spectral (S-2) Imaging,” IEEE Photonics Technol. Lett. 28, 1034–1037 (2016).

Poletti, F.

D. R. Gray, M. N. Petrovich, S. R. Sandoghchi, N. V. Wheeler, N. K. Baddela, G. T. Jasion, T. Bradley, D. J. Richardson, and F. Poletti, “Real-Time Modal Analysis via Wavelength-Swept Spatial and Spectral (S-2) Imaging,” IEEE Photonics Technol. Lett. 28, 1034–1037 (2016).

F. Poletti and P. Horak, “Description of ultrashort pulse propagation in multimode optical fibers,” J. Opt. Soc. Am. B 25(10), 1645–1654 (2008).
[Crossref]

Ramachandran, S.

Randel, S.

Rhodes, M.

Richardson, D. J.

D. R. Gray, M. N. Petrovich, S. R. Sandoghchi, N. V. Wheeler, N. K. Baddela, G. T. Jasion, T. Bradley, D. J. Richardson, and F. Poletti, “Real-Time Modal Analysis via Wavelength-Swept Spatial and Spectral (S-2) Imaging,” IEEE Photonics Technol. Lett. 28, 1034–1037 (2016).

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” J. Opt. Soc. Am. B 27(11), B63–B92 (2010).
[Crossref]

Rubino, E.

Ryf, R.

Sadovnikova, Y. E.

A. S. Kurkov, E. M. Sholokhov, and Y. E. Sadovnikova, “All-fiber supercontinuum source in the range of 1550-2400 nm based on telecommunication multimode fiber,” Laser Phys. Lett. 8(8), 598–600 (2011).
[Crossref]

Sandoghchi, S. R.

D. R. Gray, M. N. Petrovich, S. R. Sandoghchi, N. V. Wheeler, N. K. Baddela, G. T. Jasion, T. Bradley, D. J. Richardson, and F. Poletti, “Real-Time Modal Analysis via Wavelength-Swept Spatial and Spectral (S-2) Imaging,” IEEE Photonics Technol. Lett. 28, 1034–1037 (2016).

Schimpf, D. N.

Sholokhov, E. M.

A. S. Kurkov, E. M. Sholokhov, and Y. E. Sadovnikova, “All-fiber supercontinuum source in the range of 1550-2400 nm based on telecommunication multimode fiber,” Laser Phys. Lett. 8(8), 598–600 (2011).
[Crossref]

Sierra, A.

Takeda, M.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[Crossref]

Tartara, L.

Trebino, R.

Tunnermann, A.

A. Liem, D. Nickel, J. Limpert, H. Zellmer, U. Griebner, S. Unger, A. Tunnermann, and G. Korn, “High average power ultra-fast fiber chirped pulse amplification system,” Appl. Phys. B-Lasers Opt. 71(6), 889–891 (2000).
[Crossref]

Unger, S.

A. Liem, D. Nickel, J. Limpert, H. Zellmer, U. Griebner, S. Unger, A. Tunnermann, and G. Korn, “High average power ultra-fast fiber chirped pulse amplification system,” Appl. Phys. B-Lasers Opt. 71(6), 889–891 (2000).
[Crossref]

Vaughan, P.

Wheeler, N. V.

D. R. Gray, M. N. Petrovich, S. R. Sandoghchi, N. V. Wheeler, N. K. Baddela, G. T. Jasion, T. Bradley, D. J. Richardson, and F. Poletti, “Real-Time Modal Analysis via Wavelength-Swept Spatial and Spectral (S-2) Imaging,” IEEE Photonics Technol. Lett. 28, 1034–1037 (2016).

Winzer, P. J.

Wise, F. W.

Z. Zhu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Observation of multimode solitons in few-mode fiber,” Opt. Lett. 41(20), 4819–4822 (2016).
[Crossref] [PubMed]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9(5), 306–310 (2015).
[Crossref]

Wright, L. G.

Z. Zhu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Observation of multimode solitons in few-mode fiber,” Opt. Lett. 41(20), 4819–4822 (2016).
[Crossref] [PubMed]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9(5), 306–310 (2015).
[Crossref]

Yablon, A. D.

Zeitner, U. D.

Zellmer, H.

A. Liem, D. Nickel, J. Limpert, H. Zellmer, U. Griebner, S. Unger, A. Tunnermann, and G. Korn, “High average power ultra-fast fiber chirped pulse amplification system,” Appl. Phys. B-Lasers Opt. 71(6), 889–891 (2000).
[Crossref]

Zhu, Z.

Appl. Opt. (4)

Appl. Phys. B-Lasers Opt. (1)

A. Liem, D. Nickel, J. Limpert, H. Zellmer, U. Griebner, S. Unger, A. Tunnermann, and G. Korn, “High average power ultra-fast fiber chirped pulse amplification system,” Appl. Phys. B-Lasers Opt. 71(6), 889–891 (2000).
[Crossref]

IEEE Photonics Technol. Lett. (1)

D. R. Gray, M. N. Petrovich, S. R. Sandoghchi, N. V. Wheeler, N. K. Baddela, G. T. Jasion, T. Bradley, D. J. Richardson, and F. Poletti, “Real-Time Modal Analysis via Wavelength-Swept Spatial and Spectral (S-2) Imaging,” IEEE Photonics Technol. Lett. 28, 1034–1037 (2016).

J. Lightwave Technol. (1)

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

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72(1), 156–160 (1982).
[Crossref]

J. Opt. Soc. Am. B (3)

Laser Phys. Lett. (1)

A. S. Kurkov, E. M. Sholokhov, and Y. E. Sadovnikova, “All-fiber supercontinuum source in the range of 1550-2400 nm based on telecommunication multimode fiber,” Laser Phys. Lett. 8(8), 598–600 (2011).
[Crossref]

Nat. Photonics (2)

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Controllable spatiotemporal nonlinear effects in multimode fibres,” Nat. Photonics 9(5), 306–310 (2015).
[Crossref]

Opt. Express (7)

S. Akturk, M. Kimmel, P. O’Shea, and R. Trebino, “Measuring pulse-front tilt in ultrashort pulses using GRENOUILLE,” Opt. Express 11(5), 491–501 (2003).
[Crossref] [PubMed]

F. Bonaretti, D. Faccio, M. Clerici, J. Biegert, and P. Di Trapani, “Spatiotemporal Amplitude and Phase Retrieval of Bessel-X pulses using a Hartmann-Shack Sensor,” Opt. Express 17(12), 9804–9809 (2009).
[Crossref] [PubMed]

J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express 16(10), 7233–7243 (2008).
[Crossref] [PubMed]

P. Bowlan, U. Fuchs, R. Trebino, and U. D. Zeitner, “Measuring the spatiotemporal electric field of tightly focused ultrashort pulses with sub-micron spatial resolution,” Opt. Express 16(18), 13663–13675 (2008).
[Crossref] [PubMed]

J. Cohen, P. Bowlan, V. Chauhan, P. Vaughan, and R. Trebino, “Measuring extremely complex pulses with time-bandwidth products exceeding 65,000 using multiple-delay crossed-beam spectral interferometry,” Opt. Express 18(24), 24451–24460 (2010).
[Crossref] [PubMed]

D. N. Schimpf, R. A. Barankov, and S. Ramachandran, “Cross-correlated (C2) imaging of fiber and waveguide modes,” Opt. Express 19(14), 13008–13019 (2011).
[Crossref] [PubMed]

J. Cohen, P. Bowlan, V. Chauhan, and R. Trebino, “Measuring temporally complex ultrashort pulses using multiple-delay crossed-beam spectral interferometry,” Opt. Express 18(7), 6583–6597 (2010).
[Crossref] [PubMed]

Opt. Lett. (3)

Other (3)

R. P. Encyclopedia, “Multimode fibers,” https://www.rp-photonics.com/multimode_fibers.html .

J. A. Buck, Fundamentals of Optical Fibers (John Wiley & Sons, 2004).

Z. Guang, M. Rhodes, and R. Trebino, “Measuring spatiotemporal intensity-and-phase complexity of multimode fiber output pulses,” in Proceedings 9740, Frontiers in Ultrafast Optics: Biomedical, Scientific, and Industrial Applications XVI; 97400G (2016).

Supplementary Material (4)

NameDescription
» Visualization 1       A color movie from a human eye perspective generated to demonstrate the spatio-temporal propagation of the output of a multimode fiber in the well-centered coupling situation.Spatially in a two-dimensional movie frame and temporally between frames, t
» Visualization 2       A color movie from a human eye perspective generated to demonstrate the spatio-temporal propagation of the output of a multimode fiber in a small offsets in x- and y- directions situation. Spatially in a two-dimensional movie frame and temporally bet
» Visualization 3       A color movie from a human eye perspective generated to demonstrate the spatio-temporal propagation of the output of a multimode fiber in the small x-direction offset situation. Spatially in a two-dimensional movie frame and temporally between frames
» Visualization 4       A color movie from a human eye perspective generated to demonstrate the spatio-temporal propagation of the output of a multimode fiber in a large offset coupling situation. Spatially in a two-dimensional movie frame and temporally between frames, the

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Figures (5)

Fig. 1
Fig. 1 (a) Excluding the elements in the dashed orange rectangle, this is a schematic of a single-frame STRIPED FISH apparatus. By incorporating the variable delay stage (depicted in the dashed rectangle) into a single-frame STRIPED FISH setup, this becomes a schematic for delay-scanned STRIPED FISH. (b) A simulated STRIPED FISH trace for a simple flat-phase Gaussian pulse. False colors are used to indicate the different wavelengths of the holograms. In this work, we extend single-frame STRIPED FISH to multiple frames, one for each delay over a potentially large range of delays.
Fig. 2
Fig. 2 Experimental layout of delay-scanned STRIPED FISH setup for measuring long output pulses from MMF.
Fig. 3
Fig. 3 Three measured delay-scanned STRIPED FISH traces at −1ps, 0ps, and 0.83ps (well-centered coupling). The wavelengths of the 30 holograms are in ascending order from right to left and from top to bottom between 781nm and 813nm. Each hologram occupies 330 × 330 pixels.
Fig. 4
Fig. 4 Measured results of MMF output pulses of different fiber coupling situations. Centered coupling situation: (a) near-field spatial profile; (b) calculated spectrogram; (c-e) three selected time snapshots from the reconstructed color movie (see Visualization 1); Small offsets in both x- and y-direction situation: (f) near-field spatial profile; (g) calculated spectrogram; (h-j) three selected time snapshots from the reconstructed color movie (see Visualization 2); Small offset in only x-direction situation: (k) near-field spatial profile; (l) calculated spectrogram; (m-o) three selected time snapshots from the reconstructed color movie (see Visualization 3); Large offset situation: (p) near-field spatial profile; (q) calculated spectrogram; (r-t) three selected time snapshots from the reconstructed color movie (see Visualization 4). Spectrograms in the second column are spatially integrated and showing overall spectro-temporal energy distribution and dispersion of the existing fiber modes. The labeled dashed and dotted lines in the spectrograms are in accordance with the label of the time snapshots. The snapshots show the spatial intensities (by brightness) and frequencies present (by color) at different times. Every snapshot has one or two dominant LP modes, which are labelled in the snapshots. Temporal sums of these snapshots of these four situations agree with the monitored near-field spatial profile well, providing validity to delay-scanned STRIPED FISH.
Fig. 5
Fig. 5 Decomposition analysis of the concatenated pulses. (a) centered coupling situation; (b) small offsets in both x- and y-direction situation; (c) small offset in only x-direction situation; (d) large offset situation.

Tables (4)

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Table 1 Cutoff normalized frequency Vc and amplitude profile of first 14 LP modes [24]

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Table 2 The retrieved amplitude and phase of three wavelengths at four delays (well-centered coupling)

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Table 3 The decomposition results of the measured snapshots

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Table 4 The color scheme used for the Visualizations

Equations (13)

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Δ T SF = 2 π δ ω filter = λ 0 2 c δ λ filter .
δ λ eff = λ 0 2 c ( Δ T delay + Δ T SF ) = Δ T SF Δ T delay + Δ T SF δ λ SF Δ T SF Δ T delay δ λ SF
I holo ( x , y , ω i ; τ k ) = I ref ( x , y , ω i ) + I unk ( x , y , ω i ) + E ref ( x , y , ω i ; τ k ) E unk ( x , y , ω i ; τ k ) exp ( i k i y sin α ) + E ref * ( x , y , ω i ; τ k ) E unk ( x , y , ω i ; τ k ) exp ( i k i y sin α ) .
Δ φ ( x , y , ω i ; τ k ) = Arg [ E ref ( x , y , ω i ; τ k ) E unk ( x , y , ω i ; τ k ) ] .
E slice ( x , y , ω i ; τ k ) = | E ref ( x , y , ω i ; τ k ) E unk ( x , y , ω i ; τ k ) | | E ref ( x , y , ω i ) | 2 exp [ i φ slice ( x , y , ω i ; τ k ) ] , φ slice ( x , y , ω i ; τ k ) = φ ref ( x , y , ω i ) Δ φ ( x , y , ω i ; τ k ) .
E s l i c 1 e ( x , y , t ; τ k ) = { F - 1 { E s l i c 1 e ( x , y , t ; τ k ) } , 0 , i f τ k Δ T S F 2 < t < τ k + Δ T S F 2 ; o t h e r w i s e ; } = A s l i c 1 e ( x , y , t ; τ k ) exp [ i Ø s l i c 1 e ( x , y , t ; τ k ) ] .
G w e i g h t ( t ; x , y , τ k ) = exp [ ( t τ k τ G ) 2 ] .
E unk ( x , y , t ) = k = 1 N G w e i g h t ( t ; x , y , τ k ) E slice ( x , y , t ; τ k ) k = 1 N G w e i g h t ( t ; x , y , τ k ) ,
S p ( ω , T ; x , y ) = | E ( t ; x , y ) g ( t T ) exp ( i ω t ) d t | 2 ,
R ( T ; x , y ) = S p ( ω , T ; x , y ) R ( ω ) d ω ; G ( T ; x , y ) = S p ( ω , T ; x , y ) G ( ω ) d ω ; B ( T ; x , y ) = S p ( ω , T ; x , y ) B ( ω ) d ω .
E l m i = A l m i { J l ( u l m r a ) Φ i , r a ; J l ( u l m ) K l ( w l m ) K l ( w l m r a ) Φ i , r a ; , Φ i = { cos ( l φ ) , i = e ; sin ( l φ ) , i = o ;
J l 1 ( u ) J l ( u ) = w u K l 1 ( w ) K l ( w )
| w l m i ( t ) | 2 = | E m e a s u r e d ( x , y , t ) E l m i ( x , y ) d x d y | 2 | E m e a s u r e d ( x , y , t ) | 2 d x d y | E l m i ( x , y ) | 2 d x d y .

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