Extended bandwidth wavelength swept laser source for high resolution optical frequency domain imaging

Sahar Hosseinzadeh Kassani; Martin Villiger; Néstor Uribe-Patarroyo; Changsu Jun; Reza Khazaeinezhad; Norman Lippok; Brett E. Bouma

doi:10.1364/OE.25.008255

Extended bandwidth wavelength swept laser source for high resolution optical frequency domain imaging

Sahar Hosseinzadeh Kassani, Martin Villiger, Néstor Uribe-Patarroyo, Changsu Jun, Reza Khazaeinezhad, Norman Lippok, and Brett E. Bouma

Optics Express
Vol. 25,
Issue 7,
pp. 8255-8266
(2017)
•https://doi.org/10.1364/OE.25.008255

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Sahar Hosseinzadeh Kassani, Martin Villiger, Néstor Uribe-Patarroyo, Changsu Jun, Reza Khazaeinezhad, Norman Lippok, and Brett E. Bouma, "Extended bandwidth wavelength swept laser source for high resolution optical frequency domain imaging," Opt. Express 25, 8255-8266 (2017)

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Related Topics
Table of Contents Category
- Imaging Systems and Displays
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Fourier optics and signal processing (070.0070)
- Imaging systems (110.0110)
- Optical coherence tomography (110.4500)
- Lasers, tunable (140.3600)

About this Article
History
- Original Manuscript: January 25, 2017
- Revised Manuscript: March 13, 2017
- Manuscript Accepted: March 13, 2017
- Published: March 30, 2017
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 12, Iss. 6

Accessible

Open Access

Abstract

Improving the axial resolution by providing wider bandwidth wavelength swept lasers remains an important issue for optical frequency domain imaging (OFDI). Here, we demonstrate a wide tuning range, all-fiber wavelength swept laser at a center wavelength of 1250 nm by combining two ring cavities that share a single Fabry-Perot tunable filter. The two cavities contain semiconductor optical amplifiers with central wavelengths of 1190 nm and 1292 nm, respectively. To avoid disturbing interference effects in the overlapping spectral region, we modulated the amplifiers in order to obtain consecutive wavelength sweeps in the two spectral regions. The two sweeps were fused together in post-processing to achieve a total scanning range of 223 nm, corresponding to 3.3 µm axial resolution in air. We confirm improved image quality and reduced speckle size in tomograms of swine esophagus ex vivo, and human skin and nailbed in vivo.

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References

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J. F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, “Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography,” Opt. Lett. 28(21), 2067–2069 (2003).
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L. Liu, J. A. Gardecki, S. K. Nadkarni, J. D. Toussaint, Y. Yagi, B. E. Bouma, and G. J. Tearney, “Imaging the subcellular structure of human coronary atherosclerosis using micro-optical coherence tomography,” Nat. Med. 17(8), 1010–1014 (2011).
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W.-Y. Oh, S.-H. Yun, G. J. Tearney, and B. E. Bouma, “Wide Tuning Range Wavelength-Swept Laser With Two Semiconductor Optical Amplifiers,” IEEE Photonics Technol. Lett. 17(3), 678–680 (2005).
[Crossref] [PubMed]
H. Kim, J. Lim, J. Ha, and W. Jang, “Broadband wavelength swept source combining a quantum dot and a quantum well SOA in the wavelength range of 1153–1366 nm,” Electron. Lett. 49(19), 1205–1206 (2013).
[Crossref]
M. Y. Jeon, J. Zhang, Q. Wang, and Z. Chen, “High-speed and wide bandwidth Fourier domain mode-locked wavelength swept laser with multiple SOAs,” Opt. Express 16(4), 2547–2554 (2008).
[Crossref] [PubMed]
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[Crossref]
C. Jun, M. Villiger, W.-Y. Oh, and B. E. Bouma, “All-fiber wavelength swept ring laser based on Fabry-Perot filter for optical frequency domain imaging,” Opt. Express 22(21), 25805–25814 (2014).
[Crossref] [PubMed]
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[Crossref] [PubMed]
A. Bilenca, S. H. Yun, G. J. Tearney, and B. E. Bouma, “Numerical study of wavelength-swept semiconductor ring lasers: the role of refractive-index nonlinearities in semiconductor optical amplifiers and implications for biomedical imaging applications,” Opt. Lett. 31(6), 760–762 (2006).
[Crossref] [PubMed]
S. Yun, G. Tearney, J. de Boer, N. Iftimia, and B. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express 11(22), 2953–2963 (2003).
[Crossref] [PubMed]
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2014 (1)

C. Jun, M. Villiger, W.-Y. Oh, and B. E. Bouma, “All-fiber wavelength swept ring laser based on Fabry-Perot filter for optical frequency domain imaging,” Opt. Express 22(21), 25805–25814 (2014).
[Crossref] [PubMed]

2013 (1)

H. Kim, J. Lim, J. Ha, and W. Jang, “Broadband wavelength swept source combining a quantum dot and a quantum well SOA in the wavelength range of 1153–1366 nm,” Electron. Lett. 49(19), 1205–1206 (2013).
[Crossref]

2012 (1)

I. Grulkowski, J. J. Liu, B. Potsaid, V. Jayaraman, C. D. Lu, J. Jiang, A. E. Cable, J. S. Duker, and J. G. Fujimoto, “Retinal, anterior segment and full eye imaging using ultrahigh speed swept source OCT with vertical-cavity surface emitting lasers,” Biomed. Opt. Express 3(11), 2733–2751 (2012).
[Crossref] [PubMed]

2011 (1)

L. Liu, J. A. Gardecki, S. K. Nadkarni, J. D. Toussaint, Y. Yagi, B. E. Bouma, and G. J. Tearney, “Imaging the subcellular structure of human coronary atherosclerosis using micro-optical coherence tomography,” Nat. Med. 17(8), 1010–1014 (2011).
[Crossref] [PubMed]

2010 (2)

W. Wieser, B. R. Biedermann, T. Klein, C. M. Eigenwillig, and R. Huber, “Multi-megahertz OCT: High quality 3D imaging at 20 million A-scans and 4.5 GVoxels per second,” Opt. Express 18(14), 14685–14704 (2010).
[Crossref] [PubMed]

Y. Shimada, A. Sadr, M. F. Burrow, J. Tagami, N. Ozawa, and Y. Sumi, “Validation of swept-source optical coherence tomography (SS-OCT) for the diagnosis of occlusal caries,” J. Dent. 38(8), 655–665 (2010).
[Crossref] [PubMed]

2009 (2)

T. Xie, G. Liu, K. Kreuter, S. Mahon, H. Colt, D. Mukai, G. M. Peavy, Z. Chen, and M. Brenner, “In vivo three-dimensional imaging of normal tissue and tumors in the rabbit pleural cavity using endoscopic swept source optical coherence tomography with thoracoscopic guidance,” J. Biomed. Opt. 14(6), 064045 (2009).
[Crossref] [PubMed]

Y. Nakazaki and S. Yamashita, “Fast and wide tuning range wavelength-swept fiber laser based on dispersion tuning and its application to dynamic FBG sensing,” Opt. Express 17(10), 8310–8318 (2009).
[Crossref] [PubMed]

2008 (4)

M. Y. Jeon, J. Zhang, Q. Wang, and Z. Chen, “High-speed and wide bandwidth Fourier domain mode-locked wavelength swept laser with multiple SOAs,” Opt. Express 16(4), 2547–2554 (2008).
[Crossref] [PubMed]

M. J. Suter, B. J. Vakoc, P. S. Yachimski, M. Shishkov, G. Y. Lauwers, M. Mino-Kenudson, B. E. Bouma, N. S. Nishioka, and G. J. Tearney, “Comprehensive microscopy of the esophagus in human patients with optical frequency domain imaging,” Gastrointest. Endosc. 68(4), 745–753 (2008).
[Crossref] [PubMed]

G. Guagliumi and V. Sirbu, “Optical coherence tomography: high resolution intravascular imaging to evaluate vascular healing after coronary stenting,” Catheter. Cardiovasc. Interv. 72(2), 237–247 (2008).
[Crossref] [PubMed]

W. Drexler and J. G. Fujimoto, “State-of-the-art retinal optical coherence tomography,” Prog. Retin. Eye Res. 27(1), 45–88 (2008).
[Crossref] [PubMed]

2007 (2)

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1(12), 709–716 (2007).
[Crossref]

L. A. Kranendonk, X. An, A. W. Caswell, R. E. Herold, S. T. Sanders, R. Huber, J. G. Fujimoto, Y. Okura, and Y. Urata, “High speed engine gas thermometry by Fourier-domain mode-locked laser absorption spectroscopy,” Opt. Express 15(23), 15115–15128 (2007).
[Crossref] [PubMed]

2006 (2)

A. Bilenca, S. H. Yun, G. J. Tearney, and B. E. Bouma, “Numerical study of wavelength-swept semiconductor ring lasers: the role of refractive-index nonlinearities in semiconductor optical amplifiers and implications for biomedical imaging applications,” Opt. Lett. 31(6), 760–762 (2006).
[Crossref] [PubMed]

R. Huber, M. Wojtkowski, and J. G. Fujimoto, “Fourier Domain Mode Locking (FDML): A new laser operating regime and applications for optical coherence tomography,” Opt. Express 14(8), 3225–3237 (2006).
[Crossref] [PubMed]

2005 (3)

W. Y. Oh, S. H. Yun, G. J. Tearney, and B. E. Bouma, “115 kHz tuning repetition rate ultrahigh-speed wavelength-swept semiconductor laser,” Opt. Lett. 30(23), 3159–3161 (2005).
[Crossref] [PubMed]

R. Huber, M. Wojtkowski, K. Taira, J. Fujimoto, and K. Hsu, “Amplified, frequency swept lasers for frequency domain reflectometry and OCT imaging: design and scaling principles,” Opt. Express 13(9), 3513–3528 (2005).
[Crossref] [PubMed]

W.-Y. Oh, S.-H. Yun, G. J. Tearney, and B. E. Bouma, “Wide Tuning Range Wavelength-Swept Laser With Two Semiconductor Optical Amplifiers,” IEEE Photonics Technol. Lett. 17(3), 678–680 (2005).
[Crossref] [PubMed]

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and et, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Adler, D. C.

An, X.

Bajraszewski, T.

Biedermann, B. R.

Bilenca, A.

Bouma, B.

S. H. Yun, G. Tearney, J. de Boer, and B. Bouma, “Motion artifacts in optical coherence tomography with frequency-domain ranging,” Opt. Express 12(13), 2977–2998 (2004).
[Crossref] [PubMed]

S. Yun, G. Tearney, J. de Boer, N. Iftimia, and B. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express 11(22), 2953–2963 (2003).
[Crossref] [PubMed]

Bouma, B. E.

Brenner, M.

Burrow, M. F.

Cable, A. E.

Cariou, J.

Caswell, A. W.

Cense, B.

Chang, W.

Chen, Y.

Chen, Z.

Chinn, S. R.

S. R. Chinn, E. A. Swanson, and J. G. Fujimoto, “Optical coherence tomography using a frequency-tunable optical source,” Opt. Lett. 22(5), 340–342 (1997).
[Crossref] [PubMed]

Choma, M.

M. Choma, M. Sarunic, C. Yang, and J. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express 11(18), 2183–2189 (2003).
[Crossref] [PubMed]

Colt, H.

Connolly, J.

de Boer, J.

S. H. Yun, G. Tearney, J. de Boer, and B. Bouma, “Motion artifacts in optical coherence tomography with frequency-domain ranging,” Opt. Express 12(13), 2977–2998 (2004).
[Crossref] [PubMed]

S. Yun, G. Tearney, J. de Boer, N. Iftimia, and B. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express 11(22), 2953–2963 (2003).
[Crossref] [PubMed]

de Boer, J. F.

Drexler, W.

W. Drexler and J. G. Fujimoto, “State-of-the-art retinal optical coherence tomography,” Prog. Retin. Eye Res. 27(1), 45–88 (2008).
[Crossref] [PubMed]

U. Morgner, W. Drexler, F. X. Kärtner, X. D. Li, C. Pitris, E. P. Ippen, and J. G. Fujimoto, “Spectroscopic optical coherence tomography,” Opt. Lett. 25(2), 111–113 (2000).
[Crossref] [PubMed]

Duker, J.

Duker, J. S.

Eigenwillig, C. M.

et,

Fercher, A.

R. Leitgeb, C. Hitzenberger, and A. Fercher, “Performance of fourier domain vs. time domain optical coherence tomography,” Opt. Express 11(8), 889–894 (2003).
[Crossref] [PubMed]

Fercher, A. F.

F. Lexer, C. K. Hitzenberger, A. F. Fercher, and M. Kulhavy, “Wavelength-tuning interferometry of intraocular distances,” Appl. Opt. 36(25), 6548–6553 (1997).
[Crossref] [PubMed]

Flotte, T.

Fujimoto, J.

Fujimoto, J. G.

W. Drexler and J. G. Fujimoto, “State-of-the-art retinal optical coherence tomography,” Prog. Retin. Eye Res. 27(1), 45–88 (2008).
[Crossref] [PubMed]

U. Morgner, W. Drexler, F. X. Kärtner, X. D. Li, C. Pitris, E. P. Ippen, and J. G. Fujimoto, “Spectroscopic optical coherence tomography,” Opt. Lett. 25(2), 111–113 (2000).
[Crossref] [PubMed]

S. R. Chinn, E. A. Swanson, and J. G. Fujimoto, “Optical coherence tomography using a frequency-tunable optical source,” Opt. Lett. 22(5), 340–342 (1997).
[Crossref] [PubMed]

Gardecki, J. A.

Gregory, K.

Grulkowski, I.

Guagliumi, G.

Guern, Y.

Ha, J.

Hee, M. R.

Hermann, B.

Herold, R. E.

Hitzenberger, C.

R. Leitgeb, C. Hitzenberger, and A. Fercher, “Performance of fourier domain vs. time domain optical coherence tomography,” Opt. Express 11(8), 889–894 (2003).
[Crossref] [PubMed]

Hitzenberger, C. K.

F. Lexer, C. K. Hitzenberger, A. F. Fercher, and M. Kulhavy, “Wavelength-tuning interferometry of intraocular distances,” Appl. Opt. 36(25), 6548–6553 (1997).
[Crossref] [PubMed]

Hsu, K.

Huang, D.

Huber, R.

Iftimia, N.

S. Yun, G. Tearney, J. de Boer, N. Iftimia, and B. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express 11(22), 2953–2963 (2003).
[Crossref] [PubMed]

Ippen, E. P.

U. Morgner, W. Drexler, F. X. Kärtner, X. D. Li, C. Pitris, E. P. Ippen, and J. G. Fujimoto, “Spectroscopic optical coherence tomography,” Opt. Lett. 25(2), 111–113 (2000).
[Crossref] [PubMed]

Izatt, J.

M. Choma, M. Sarunic, C. Yang, and J. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express 11(18), 2183–2189 (2003).
[Crossref] [PubMed]

Jang, W.

Jayaraman, V.

Jeon, M. Y.

Jiang, J.

Jun, C.

Kärtner, F. X.

U. Morgner, W. Drexler, F. X. Kärtner, X. D. Li, C. Pitris, E. P. Ippen, and J. G. Fujimoto, “Spectroscopic optical coherence tomography,” Opt. Lett. 25(2), 111–113 (2000).
[Crossref] [PubMed]

Kim, H.

Klein, T.

Ko, T.

Kowalczyk, A.

Kranendonk, L. A.

Kreuter, K.

Kulhavy, M.

F. Lexer, C. K. Hitzenberger, A. F. Fercher, and M. Kulhavy, “Wavelength-tuning interferometry of intraocular distances,” Appl. Opt. 36(25), 6548–6553 (1997).
[Crossref] [PubMed]

Lauwers, G. Y.

Le, T.

Le Brun, G.

Le Jeune, B.

Leitgeb, R.

R. Leitgeb, C. Hitzenberger, and A. Fercher, “Performance of fourier domain vs. time domain optical coherence tomography,” Opt. Express 11(8), 889–894 (2003).
[Crossref] [PubMed]

Lexer, F.

F. Lexer, C. K. Hitzenberger, A. F. Fercher, and M. Kulhavy, “Wavelength-tuning interferometry of intraocular distances,” Appl. Opt. 36(25), 6548–6553 (1997).
[Crossref] [PubMed]

Li, X. D.

U. Morgner, W. Drexler, F. X. Kärtner, X. D. Li, C. Pitris, E. P. Ippen, and J. G. Fujimoto, “Spectroscopic optical coherence tomography,” Opt. Lett. 25(2), 111–113 (2000).
[Crossref] [PubMed]

Lim, J.

Lin, C. P.

Liu, G.

Liu, J. J.

Liu, L.

Lortrian, J.

Lu, C. D.

Mahon, S.

Mino-Kenudson, M.

Morgner, U.

U. Morgner, W. Drexler, F. X. Kärtner, X. D. Li, C. Pitris, E. P. Ippen, and J. G. Fujimoto, “Spectroscopic optical coherence tomography,” Opt. Lett. 25(2), 111–113 (2000).
[Crossref] [PubMed]

Mukai, D.

Nadkarni, S. K.

Nakazaki, Y.

Nelson, J. S.

Nishioka, N. S.

Oh, W. Y.

Oh, W.-Y.

Okura, Y.

Ozawa, N.

Park, B. H.

Peavy, G. M.

Piederrière, Y.

Pierce, M. C.

Pitris, C.

U. Morgner, W. Drexler, F. X. Kärtner, X. D. Li, C. Pitris, E. P. Ippen, and J. G. Fujimoto, “Spectroscopic optical coherence tomography,” Opt. Lett. 25(2), 111–113 (2000).
[Crossref] [PubMed]

Potsaid, B.

Puliafito, C. A.

Sadr, A.

Sanders, S. T.

Sarunic, M.

M. Choma, M. Sarunic, C. Yang, and J. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express 11(18), 2183–2189 (2003).
[Crossref] [PubMed]

Schmitt, J.

Schuman, J. S.

Shimada, Y.

Shishkov, M.

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Fig. 1 (a) Experimental setup of the extended bandwidth wavelength swept laser. (b) Amplified spontaneous emission (ASE) spectra of the two semiconductor optical amplifiers (SOAs). (c) Timing of filter driving and laser modulation signals. PC: polarization controller, WDM: Wavelength division multiplexer.

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Fig. 2 (a) Optical spectra of the extended bandwidth wavelength swept laser, and each individual single band laser. (b) Estimated axial resolution by inverse Fourier transforming the source spectra.

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Fig. 3 (a) Schematic of the OCT system for evaluation of the extended bandwidth swept source. (b) Laser time trace and interference fringe signal. SMF: single mode fiber, AO FS: acousto-optic frequency shifter, PC: Polarization controller, FBG: fiber Bragg grating, PD: photo detector

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Fig. 4 Mapped and dispersion compensated depth-scans for band-1 (a), and band-2 (b), respectively. Same depth-scans, after additional calibration and zero-padding to N₁ + N₂ points resulting in equal depth positions for band-1 (c) and band-2 (d). (e) Standard deviation of the phase difference between band-1 and band-2 across all peak positions as a function of the relative offset in wavenumber indices between the two bands. (f) Synthesized fringe signals of an individual peak after applying correct wavenumber offset and an additional scalar phase value.

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Fig. 5 (a) Axial point spread functions (PSFs) for various axial peak positions for the two individual and the combined spectra, (b) Zoomed-in version of one of the peaks (c) full-width at half-maximum (FWHM) of the axial PSFs for the individual and the combined spectra

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Fig. 6 (a),(b) and (c) OCT tomograms of swine esophagus ex vivo using Band-1, Band-2 and extended bandwidth laser. (d) Zoomed-in versions of dashed regions of interest in each of panels (a)-(c). (e) Lateral mean of the axial autocorrelation of the selected regions of interest.

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Fig. 7 (a), (b) OCT tomograms of human skin (back of a hand) in vivo for a single band (a) and extended bandwidth laser (b), respectively. (c), (d) OCT images of swine esophagus ex vivo for a single band (c) and extended bandwidth laser (d). (e), (f) Images of a human finger (nailbed) in vivo for a single band (e) and extended bandwidth laser (f). Insets: 2x magnified view of the dashed regions.

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Tables (1)

Table 1 Calibration and fusion process

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Equations (1)

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δ z = \frac{2 \ln 2}{π} \frac{λ_{\circ}^{2}}{n Δ λ},

	Input: S₁(m, z_p) and S₂(n, z_p)	m ∈ [0, N₁-1], n ∈ [0, N₂-1]N_1,2: Number of samples z_p: Mirror position
	Output: α, β, n_os, φ_os	α = ∆k₁/∆k₂, with ∆k₂ ≥ ∆k₁ β: Group delay offset φ_os: Phase offset k_os = n_os∆k₂
1.	Compute tomograms: $\begin{array}{l} t_{1} (p, z_{p}) = \sum_{m = 0}^{N_{1} - 1} S_{1} (m, z_{p}) \exp (- i 2 π \frac{m p}{N_{F F T}}) \\ t_{2} (q, z_{p}) = \sum_{n = 0}^{N_{2} - 1} S_{2} (n, z_{p}) \exp (- i 2 π \frac{n q}{N_{F F T}}) \end{array}$	N_FFT ≥ N₁ + N₂: Total number of sampling points after zero-padding
2.	Find axial scaling α and offset β: $p_{p} = α (q_{p} + β), α \leq 1$	p_p, q_p: Axial pixel positions of the reflector peak signal
3.	Compute adjusted t’₁ after interpolating S₁ on coarser ∆k’₁ = ∆k₁/α and adding a phase ramp to compensate the axial offset: $\begin{array}{l} {t^{'}}_{1} (p, z_{p}) = \sum_{m = 0}^{N_{1} - 1} {S^{'}}_{1} (m, z_{p}) \exp (- i 2 π \frac{m p}{N_{F F T}}) where \\ {S^{'}}_{1} ({k^{'}}_{1}) = S_{1} ({k^{'}}_{1}) \exp (i 2 π \frac{β m}{N_{F F T}}), {k^{'}}_{1} = m \frac{Δ k_{1}}{α} \end{array}$
4.	Compute candidate t’₂ for all possible offsets n_os: ${t^{'}}_{2} (q, z_{p}, n_{o s}) = \sum_{n = 0}^{N_{2} - 1} S_{2} (n, z_{p}) \exp (- i 2 π \frac{(n + n_{o s}) q}{N_{F F T}})$	n_os: Relative offset, in integer sampling points, between the two spectra
5.	Calculate phase variance across all peak positions and find best n_os; ${\tilde{n}}_{o s} = \min_{n_{o s}} [\arg ({t^{'}}_{1} (p_{p}, z_{p}) c o n j ({t^{'}}_{2} (q_{p}, z_{p}, n_{o s})))]$	ñ_os: Optimum offsett₁(q_p,z_p), t₂(p_p,z_p): Tomograms at the reflector peak signal
6.	Calculate constant scalar phase offset: $φ_{o s} = 〈 \arg ({t^{'}}_{1} (p_{p}, z_{p}) c o n j ({t^{'}}_{2} (q_{p}, z_{p}, {\tilde{n}}_{o s}))) 〉$	< >: Mean over all z_p.
7.	Compute fused spectrum: $S_{t o t} (v) = {S^{'}}_{1} (v) + S_{2} (v - {\tilde{n}}_{o s}) e^{i φ_{o s}}$	v ∈ [0, N_FFT], v = m for S’₁ and v = n for S₂, and S’₁(w) = 0 for w ≥ N₁ and S₂(w) = 0 for w < 0 and w ≥ N₂

Abstract

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