Abstract

Accurately characterizing the width of a narrow (high Q) resonance can be quite challenging, requiring either a high-resolution grating spectrometer or a Fourier transform spectrometer with a long delay range. Here, we describe a new technique to determine the Q of a resonance based on spatial measurements that is not subject to the same limitations as conventional methods. We theoretically show that this technique can give an accurate measurement of Q for essentially any set of input beam parameters, even if the spectral resolution of the measurement is inadequate to resolve the linewidth of the resonance. We confirm this striking result with both numerical simulations and experimental measurements.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

2017 (4)

P. Genevet, F. Capasso, F. Aieta, M. Khorasaninejad, and R. Devlin, Optica 4, 139 (2017).
[Crossref]

P. Ma and L. Gao, Opt. Express 25, 9676 (2017).
[Crossref]

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, Nat. Photonics 11, 543 (2017).
[Crossref]

R. Mendis, M. Nagai, W. Zhang, and D. M. Mittleman, Sci. Rep. 7, 5909 (2017).
[Crossref]

2016 (2)

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, Nat. Rev. Mater. 1, 16048 (2016).
[Crossref]

Y. Fan, N.-H. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, Adv. Opt. Mater. 4, 1824 (2016).
[Crossref]

2015 (1)

M. J. Theisen and T. G. Brown, J. Mod. Opt. 62, 244 (2015).
[Crossref]

2013 (2)

Y. Wan, Z. Zheng, W. Kong, X. Zhao, and J. Liu, IEEE Photon. J. 5, 7200107 (2013).
[Crossref]

K. Y. Bliokh and A. Aiello, J. Opt. 15, 014001 (2013).
[Crossref]

2012 (1)

I. V. Soboleva, V. V. Moskalenko, and A. A. Fedyanin, Phys. Rev. Lett. 108, 123901 (2012).
[Crossref]

2011 (2)

P. Xiao, J. Opt. Soc. Am. B 28, 1895 (2011).
[Crossref]

P. U. Jepsen, D. G. Cooke, and M. Koch, Laser Photon. Rev. 5, 124 (2011).
[Crossref]

2010 (3)

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, Nat. Mater. 9, 707 (2010).
[Crossref]

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, Rev. Mod. Phys. 82, 2257 (2010).
[Crossref]

J. Hao, H. Li, C. Yin, and Z. Cao, J. Opt. Soc. Am. B 27, 1305 (2010).
[Crossref]

2006 (1)

2003 (2)

I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, Appl. Phys. Lett. 83, 2713 (2003).
[Crossref]

S. Fan, W. Suh, and J. D. Joannopoulos, J. Opt. Soc. Am. A 20, 569 (2003).
[Crossref]

2002 (1)

S. Fan and J. D. Joannopoulos, Phys. Rev. B 65, 235112 (2002).
[Crossref]

1998 (1)

1993 (2)

1988 (1)

J.-P. Connerade and A. M. Lane, Rep. Prog. Phys. 51, 1439 (1988).
[Crossref]

1948 (1)

K. Artmann, Ann. Phys. 437, 87 (1948).
[Crossref]

1947 (1)

F. Goos and H. Hänchen, Ann. Phys. 436, 333 (1947).
[Crossref]

Aiello, A.

K. Y. Bliokh and A. Aiello, J. Opt. 15, 014001 (2013).
[Crossref]

Aieta, F.

Artmann, K.

K. Artmann, Ann. Phys. 437, 87 (1948).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh and A. Aiello, J. Opt. 15, 014001 (2013).
[Crossref]

Brown, T. G.

M. J. Theisen and T. G. Brown, J. Mod. Opt. 62, 244 (2015).
[Crossref]

Cao, Z.

Capasso, F.

Charous, A.

Chong, C. T.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, Nat. Mater. 9, 707 (2010).
[Crossref]

Connerade, J.-P.

J.-P. Connerade and A. M. Lane, Rep. Prog. Phys. 51, 1439 (1988).
[Crossref]

Cooke, D. G.

P. U. Jepsen, D. G. Cooke, and M. Koch, Laser Photon. Rev. 5, 124 (2011).
[Crossref]

Depine, R.

Devlin, R.

Erdogan, T.

Fan, S.

S. Fan, W. Suh, and J. D. Joannopoulos, J. Opt. Soc. Am. A 20, 569 (2003).
[Crossref]

S. Fan and J. D. Joannopoulos, Phys. Rev. B 65, 235112 (2002).
[Crossref]

Fan, Y.

Y. Fan, N.-H. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, Adv. Opt. Mater. 4, 1824 (2016).
[Crossref]

Fedyanin, A. A.

I. V. Soboleva, V. V. Moskalenko, and A. A. Fedyanin, Phys. Rev. Lett. 108, 123901 (2012).
[Crossref]

Flach, S.

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, Rev. Mod. Phys. 82, 2257 (2010).
[Crossref]

Fu, Q.

Y. Fan, N.-H. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, Adv. Opt. Mater. 4, 1824 (2016).
[Crossref]

Gao, L.

Genevet, P.

Giessen, H.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, Nat. Mater. 9, 707 (2010).
[Crossref]

Goos, F.

F. Goos and H. Hänchen, Ann. Phys. 436, 333 (1947).
[Crossref]

Halas, N. J.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, Nat. Mater. 9, 707 (2010).
[Crossref]

Hänchen, H.

F. Goos and H. Hänchen, Ann. Phys. 436, 333 (1947).
[Crossref]

Hao, J.

Hollas, J. M.

J. M. Hollas, Modern Spectroscopy, 4th ed. (Wiley, 2004).

Hsu, C. W.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, Nat. Rev. Mater. 1, 16048 (2016).
[Crossref]

Jepsen, P. U.

P. U. Jepsen, D. G. Cooke, and M. Koch, Laser Photon. Rev. 5, 124 (2011).
[Crossref]

Joannopoulos, J. D.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, Nat. Rev. Mater. 1, 16048 (2016).
[Crossref]

S. Fan, W. Suh, and J. D. Joannopoulos, J. Opt. Soc. Am. A 20, 569 (2003).
[Crossref]

S. Fan and J. D. Joannopoulos, Phys. Rev. B 65, 235112 (2002).
[Crossref]

Khorasaninejad, M.

Kivshar, Y. S.

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, Nat. Photonics 11, 543 (2017).
[Crossref]

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, Rev. Mod. Phys. 82, 2257 (2010).
[Crossref]

I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, Appl. Phys. Lett. 83, 2713 (2003).
[Crossref]

Koch, M.

P. U. Jepsen, D. G. Cooke, and M. Koch, Laser Photon. Rev. 5, 124 (2011).
[Crossref]

Kong, W.

Y. Wan, Z. Zheng, W. Kong, X. Zhao, and J. Liu, IEEE Photon. J. 5, 7200107 (2013).
[Crossref]

Koschny, T.

Y. Fan, N.-H. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, Adv. Opt. Mater. 4, 1824 (2016).
[Crossref]

Lane, A. M.

J.-P. Connerade and A. M. Lane, Rep. Prog. Phys. 51, 1439 (1988).
[Crossref]

Li, H.

Y. Fan, N.-H. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, Adv. Opt. Mater. 4, 1824 (2016).
[Crossref]

J. Hao, H. Li, C. Yin, and Z. Cao, J. Opt. Soc. Am. B 27, 1305 (2010).
[Crossref]

Limonov, M. F.

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, Nat. Photonics 11, 543 (2017).
[Crossref]

Liu, J.

Y. Wan, Z. Zheng, W. Kong, X. Zhao, and J. Liu, IEEE Photon. J. 5, 7200107 (2013).
[Crossref]

Lochbihler, H.

Luk’yanchuk, B.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, Nat. Mater. 9, 707 (2010).
[Crossref]

Ma, P.

Magnusson, R.

Maier, S. A.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, Nat. Mater. 9, 707 (2010).
[Crossref]

Mendis, R.

W. Zhang, A. Charous, M. Nagai, D. M. Mittleman, and R. Mendis, Opt. Express 26, 13195 (2018).
[Crossref]

R. Mendis, M. Nagai, W. Zhang, and D. M. Mittleman, Sci. Rep. 7, 5909 (2017).
[Crossref]

Miroshnichenko, A. E.

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, Rev. Mod. Phys. 82, 2257 (2010).
[Crossref]

Mittleman, D. M.

W. Zhang, A. Charous, M. Nagai, D. M. Mittleman, and R. Mendis, Opt. Express 26, 13195 (2018).
[Crossref]

R. Mendis, M. Nagai, W. Zhang, and D. M. Mittleman, Sci. Rep. 7, 5909 (2017).
[Crossref]

Morris, G. M.

Moskalenko, V. V.

I. V. Soboleva, V. V. Moskalenko, and A. A. Fedyanin, Phys. Rev. Lett. 108, 123901 (2012).
[Crossref]

Nagai, M.

W. Zhang, A. Charous, M. Nagai, D. M. Mittleman, and R. Mendis, Opt. Express 26, 13195 (2018).
[Crossref]

R. Mendis, M. Nagai, W. Zhang, and D. M. Mittleman, Sci. Rep. 7, 5909 (2017).
[Crossref]

Nordlander, P.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, Nat. Mater. 9, 707 (2010).
[Crossref]

Norton, S. M.

Poddubny, A. N.

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, Nat. Photonics 11, 543 (2017).
[Crossref]

Rybin, M. V.

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, Nat. Photonics 11, 543 (2017).
[Crossref]

Shadrivov, I. V.

I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, Appl. Phys. Lett. 83, 2713 (2003).
[Crossref]

Shen, N.-H.

Y. Fan, N.-H. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, Adv. Opt. Mater. 4, 1824 (2016).
[Crossref]

Soboleva, I. V.

I. V. Soboleva, V. V. Moskalenko, and A. A. Fedyanin, Phys. Rev. Lett. 108, 123901 (2012).
[Crossref]

Soljacic, M.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, Nat. Rev. Mater. 1, 16048 (2016).
[Crossref]

Soukoulis, C. M.

Y. Fan, N.-H. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, Adv. Opt. Mater. 4, 1824 (2016).
[Crossref]

Stone, A. D.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, Nat. Rev. Mater. 1, 16048 (2016).
[Crossref]

Suh, W.

Theisen, M. J.

M. J. Theisen and T. G. Brown, J. Mod. Opt. 62, 244 (2015).
[Crossref]

Wan, Y.

Y. Wan, Z. Zheng, W. Kong, X. Zhao, and J. Liu, IEEE Photon. J. 5, 7200107 (2013).
[Crossref]

Wang, L.-G.

Wang, S. S.

Wei, Z.

Y. Fan, N.-H. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, Adv. Opt. Mater. 4, 1824 (2016).
[Crossref]

Xiao, P.

Yin, C.

Zhang, F.

Y. Fan, N.-H. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, Adv. Opt. Mater. 4, 1824 (2016).
[Crossref]

Zhang, P.

Y. Fan, N.-H. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, Adv. Opt. Mater. 4, 1824 (2016).
[Crossref]

Zhang, W.

W. Zhang, A. Charous, M. Nagai, D. M. Mittleman, and R. Mendis, Opt. Express 26, 13195 (2018).
[Crossref]

R. Mendis, M. Nagai, W. Zhang, and D. M. Mittleman, Sci. Rep. 7, 5909 (2017).
[Crossref]

Zhao, Q.

Y. Fan, N.-H. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, Adv. Opt. Mater. 4, 1824 (2016).
[Crossref]

Zhao, X.

Y. Wan, Z. Zheng, W. Kong, X. Zhao, and J. Liu, IEEE Photon. J. 5, 7200107 (2013).
[Crossref]

Zharov, A. A.

I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, Appl. Phys. Lett. 83, 2713 (2003).
[Crossref]

Zheludev, N. I.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, Nat. Mater. 9, 707 (2010).
[Crossref]

Zhen, B.

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, Nat. Rev. Mater. 1, 16048 (2016).
[Crossref]

Zheng, Z.

Y. Wan, Z. Zheng, W. Kong, X. Zhao, and J. Liu, IEEE Photon. J. 5, 7200107 (2013).
[Crossref]

Zhu, S.-Y.

Adv. Opt. Mater. (1)

Y. Fan, N.-H. Shen, F. Zhang, Z. Wei, H. Li, Q. Zhao, Q. Fu, P. Zhang, T. Koschny, and C. M. Soukoulis, Adv. Opt. Mater. 4, 1824 (2016).
[Crossref]

Ann. Phys. (2)

F. Goos and H. Hänchen, Ann. Phys. 436, 333 (1947).
[Crossref]

K. Artmann, Ann. Phys. 437, 87 (1948).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, Appl. Phys. Lett. 83, 2713 (2003).
[Crossref]

IEEE Photon. J. (1)

Y. Wan, Z. Zheng, W. Kong, X. Zhao, and J. Liu, IEEE Photon. J. 5, 7200107 (2013).
[Crossref]

J. Mod. Opt. (1)

M. J. Theisen and T. G. Brown, J. Mod. Opt. 62, 244 (2015).
[Crossref]

J. Opt. (1)

K. Y. Bliokh and A. Aiello, J. Opt. 15, 014001 (2013).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (2)

Laser Photon. Rev. (1)

P. U. Jepsen, D. G. Cooke, and M. Koch, Laser Photon. Rev. 5, 124 (2011).
[Crossref]

Nat. Mater. (1)

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, Nat. Mater. 9, 707 (2010).
[Crossref]

Nat. Photonics (1)

M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, Nat. Photonics 11, 543 (2017).
[Crossref]

Nat. Rev. Mater. (1)

C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, Nat. Rev. Mater. 1, 16048 (2016).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Optica (1)

Phys. Rev. B (1)

S. Fan and J. D. Joannopoulos, Phys. Rev. B 65, 235112 (2002).
[Crossref]

Phys. Rev. Lett. (1)

I. V. Soboleva, V. V. Moskalenko, and A. A. Fedyanin, Phys. Rev. Lett. 108, 123901 (2012).
[Crossref]

Rep. Prog. Phys. (1)

J.-P. Connerade and A. M. Lane, Rep. Prog. Phys. 51, 1439 (1988).
[Crossref]

Rev. Mod. Phys. (1)

A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, Rev. Mod. Phys. 82, 2257 (2010).
[Crossref]

Sci. Rep. (1)

R. Mendis, M. Nagai, W. Zhang, and D. M. Mittleman, Sci. Rep. 7, 5909 (2017).
[Crossref]

Other (1)

J. M. Hollas, Modern Spectroscopy, 4th ed. (Wiley, 2004).

Supplementary Material (1)

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» Supplement 1       Supplement 1

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Figures (4)

Fig. 1.
Fig. 1. fk diagram of an optical resonance from a planar system. The resonance (green line) can be measured on a frequency spectrum with a fixed incident angle (along the red line) or on a PM spectrum with a fixed frequency (along the blue line). The linewidths from the two methods are linearly related as δf=ρ·δPM for narrow resonances.
Fig. 2.
Fig. 2. Examples of spatial reshaping from Fano resonances. (a)–(c) Blue curves illustrate the spectra of Fano resonances with q=, 2.41, and 0 as functions of dimensionless PM spectra (Ω). Red curves illustrate the PM spectra of the incident Gaussian beam |F(Ω)|2 whose spectral center is offset from the resonance by Δk (or Δk·2/δPM for the dimensionless PM offset). (d)–(f) Normalized spatial intensity profile of the output beam as a function of the dimensionless PM offset Δk·2/δPM when the input Gaussian beam interacts with the Fano resonances with q=, 2.41, and 0. Spatial reshaping of the beam profile near the resonant condition (Δk·2/δPM=0) can be clearly observed.
Fig. 3.
Fig. 3. FEM simulations of a specific resonant structure: a uniform array of thin metal plates. (a) An incident Gaussian beam illuminates an array of thin (lossless) metal plates with plate widths of 4 mm, plate thicknesses of 0.1 mm, plate separations of d, and an incident angle of 25°. The transmitted beam profile has spatial reshaping due to interaction with the resonance of the structure. (b) FEM simulation of the transmission near the resonance of the structure (d=1.05  mm). (c) Logarithm of the simulated transmitted beam intensities as functions of displacement for various values of d. For large displacement, linear relationships can be clearly observed. Their slopes can be used to extract the linewidth of the resonance. (d) FWHM of the resonance as a function of d extracted from the simulations via the spatial-decay method (red squares), which match the rigorous analytical result (black curve). The dashed horizontal line shows the approximate limit of the linewidth that could be resolved using conventional spectroscopic methods, using the assumed PM resolution for the input beam.
Fig. 4.
Fig. 4. Experimental results on a specific resonant structure: a uniform array of thin metal plates. (a) Experimental transmitted beam intensities as functions of displacement at 177.4 GHz (on resonance, blue dots) and at 146.6 GHz (off resonance, red dots). (b) Slopes of the profiles in (a) as a function of displacement. Off resonance (red dots), the slope changes linearly with the displacement (fitted as the red line), indicating a Gaussian spatial profile. On resonance (blue dots), a plateau is observed for large displacement (fitted as the blue line), indicating an exponential decay in the spatial profile. (c) Experimental FWHM of the same resonance from the spatial-decay method (solid dots) and the traditional spectroscopic method (empty dots) as functions of experimental frequency resolution. The spatial-decay method can accurately determine the intrinsic linewidth of the resonance, even if the linewidth is smaller than the frequency resolution.

Equations (9)

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δf=ρ·δPM,
|H(Ω)|2=D2(q+Ω)21+Ω2.
H(Ω)=12(e2iarccot(q)+1iΩ1+iΩ).
h(x)=11iqδ(x)+δPM2u(x)eik0xexδPM/2,
|g(x)|2=|11iqf(x)+eik0xv(x)exδPM/2|2,
v(x)=π2δPM2w0ew022(δPM2+iΔk)2·erfc[w02(δPM2+iΔk)x2w0]
|g(x)|2C·|f(xΔx)|2,
C=(2Δk+δPMq)2(1+q2)(4Δk2+δPM2)exp(Δx2w02),
Δx=2δPM4Δk2+δPM2.