OSA's Digital Library

Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 15 — Jul. 28, 2014
  • pp: 18688–18697
« Show journal navigation

Feasibility of resonant metalens for the subwavelength imaging using a single sensor in the far field

Lianlin Li, Fang Li, and Tie Jun Cui  »View Author Affiliations


Optics Express, Vol. 22, Issue 15, pp. 18688-18697 (2014)
http://dx.doi.org/10.1364/OE.22.018688


View Full Text Article

Acrobat PDF (1905 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

This paper investigates the feasibility of the resonant metalens for the imaging beyond the diffraction limit using a single sensor in the far-field. It is shown that the resonant metalens can be related to the super-resonance phenomenon. We demonstrate that the super-resonance supports the enhancement of the information capacity of an imaging system, which is responsible for the subwavelength imaging of the probed objects by using a single sensor in combination with a broadband illumination. Such imaging concept has its unique advantage of producing real-time data when an object is illuminated by broadband waves, without the harsh requirements such as near-field scanning, mechanical scanning, or antenna arrays. The proposed method is expected to find its applications in nanolithography, detection, sensing, and subwavelength imaging in the near future.

© 2014 Optical Society of America

1. Introduction

The theory of superoscillatory indicates that over a finite interval, a waveform oscillates arbitrarily faster than its highest component in its operational spectrum, and thus makes it possible to encode fine details of the probed objects into the field of view. In light of this theory, several optical devices have been built up to achieve super-resolution imaging from far-field measurements [e.g., 3

3. E. T. F. Rogers and N. I. Zheludev, “Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging,” J. Opt. 15(9), 094008 (2013). [CrossRef]

]. Although the superoscillatory allows us to circumvent the near-field scanning, the obtainable enhancement in resolution is determined by the signal-noise ratio (SNR) to some extent, and requires a huge-size mask with enough fabrication finesse.

More recently, Lemoult and associates made efforts towards the direction of super-resolution imaging from the far-field measurements. They introduced the concept of metalens made of an array of resonators, which supports a collection of the eigenmodes [4

4. F. Lemoult, J. de Rosny, M. Fink, and G. Lerosey, “Resonant metalenses for breaking the diffraction barrier,” Phys. Rev. Lett. 104(20), 203901 (2010). [CrossRef] [PubMed]

8

8. F. Lemoult, M. Fink, and G. Lerosey, “A polychromatic approach to far-field superlensing at visible wavelengths,” Nat. Commun. 3, 1885 (2012).

]. These modes have their own resonant frequencies and distinct far-field radiation patterns as the feature of resonant frequencies. Such conversion of spatial and temporal degrees holds the promising of subwavelength imaging from far-field measurements using the polychromatic light. Although the resonant metalens is capable of producing super-resolution imaging from far-field measurements, the imaging speed is limited by the need for multiple measurements [9

9. D. Lu and Z. Liu, “Hyperlenses and metalenses for far-field super-resolution imaging,” Nat. Commun. 3, 1205 (2012).

].

2. Methodology

To illustrate the fundamental principle of the resonant metalens for far-field imaging with subwavelength resolution, we study a two-dimensional (2D) and scalar problem, which can be generalized into three-dimensional full-vector case in a straightforward manner. For the sake of simplicity, the resonant metalens is assumed to consist of a lattice of scatterers (referred to Fig. 1
Fig. 1 The sketch map for illustrating the principle of the resonant metalens for the subwavelength imaging from far-field measurements. In this figure, the metalens is made of a lattice of 3 × 10 metallic cylinders characterized by Eq. (1). The distance between two neighboring scatterers is d.
), and these scatterers are characterized by their frequency-dependent electric polarizabilityα(ω)=4Γc02ωp(ω2ωp2iΓω2/ωp), as adopted in [10

10. R. Pierrat, C. Vandenbem, M. Fink, and R. Carminat, “Subwavelength focusing inside an open disordered medium by time reversal at a single point antenna,” Phys. Rev. A. 87, 041801 (2013).

, 13

13. P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70(2), 447–466 (1998). [CrossRef]

], where Γ=1011s1is the linewidth, ωp=3.14×1012s1 is the plasma frequency, and c0 is the light speed in vacuum.

We start the discussion from the Green’s function which plays a fundamental role in analyzing the imaging resolution of the system. As sketched in Fig. 1, the Green’s function G(rd,rs;ω)establishes connection from a probed source located at rs to the field at rd, which is determined by the following coupled-dipole equations [10

10. R. Pierrat, C. Vandenbem, M. Fink, and R. Carminat, “Subwavelength focusing inside an open disordered medium by time reversal at a single point antenna,” Phys. Rev. A. 87, 041801 (2013).

12

12. P. C. Chaumet, K. Belkebir, and A. Rahmani, “Coupled-dipole method in time domain,” Opt. Express 16(25), 20157–20165 (2008). [CrossRef] [PubMed]

]:
G(rd,rs;ω)=G0(rd,rs;ω)+k2α(ω)n=1NG0(rd,rn;ω)gn
(1)
andgm=G0(rm,rs;ω)+k2α(ω)n=1,nmNG0(rm,rn;ω)gn(m=1, 2, ,N)
(2)
Here, N denotes the number of scatterers, G0(rm,rn;ω)=i4H0(1)(k|rmrn|)is the Green’s function in free space, H0(1)is the zero-order Hankel function of the first kind, and k=ω/c0is the free-space wave number. Then the compact-form solution to Eqs. (1) and (2) reads:
G(rd,rs;ω)=G0(rd,rs;ω)+k2α(ω)G0lensfar(rd;ω)× (Ik2α(ω)RgG0lenslens(ω))1G0sourcelens(rs;ω)
(3)
in which G0lensfar(rd;ω) is an N-length row vector formed by evaluating the function G0(rd,rn;ω) for varying locationsrn(n = 1,2,…,N), G0sourcelens(rs;ω)is an N-length column vector by calculating G0(rm,rs;ω) for varying locations rm(m = 1,2,…,N), and RgG0lenslens(ω) is an N×N matrix, whose entries are from G0(rm,rn;ω) but has all diagonal elements being zero.

2.1 The degree of freedom (DoF) of a single-sensor imaging system

In this subsection, we demonstrate that the DoF of far-field measurements can be efficiently driven up by using the resonant metalens, in combination with a broadband illumination, since the temporal measurements in the far field not only depend on the lens size, but also the physical property of the lens.

It is demonstrated from Eq. (3) that the DoF of G(rd,rs;ω) with respect tordfor a monochromatic illumination is mostly restricted by G0lensfar(rd;ω) with the feature of a naturally spatial low-pass filter [14

14. F. Simonetti, M. Fleming, and E. A. Marengo, “Illustration of the role of multiple scattering in subwavelength imaging from far-field measurements,” J. Opt. Soc. Am. A 25(2), 292–303 (2008). [CrossRef] [PubMed]

, 15

15. O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antenn. Propag. 37(7), 918–926 (1989). [CrossRef]

]. Actually, the DoF in relation toG(rd,rs;ω)is merely determined by the sizes of scanning aperture and lens, regardless of the lens physical property [14

14. F. Simonetti, M. Fleming, and E. A. Marengo, “Illustration of the role of multiple scattering in subwavelength imaging from far-field measurements,” J. Opt. Soc. Am. A 25(2), 292–303 (2008). [CrossRef] [PubMed]

]. As a result, for a scanning aperture of a given size, the degree of improvement on imaging resolution by the use of the lens is limited by the lens size: the bigger the lens size is, the higher the resolution is. Furthermore, the information capacity of a monochromatic imaging system is of the orderO(T×B), where B is the spatial bandwidth and T is the size of scanning aperture [16

16. I. J. Cox and C. R. Sheppard, “Information capacity and resolution in an optical system,” J. Opt. Soc. Am. A 3(8), 1152–1158 (1986). [CrossRef]

]. Usually, B is fixed for a given T and operational frequencyω. Consequently, the use of any lens (either conventional or man-made) is limited in improving physically the resolution of a monochromatic imaging system in the far-field region, which is common knowledge.

To investigate the capability of the super-resonance lens in resolving two points located atrs1andrs2, subject to|rs1rs2|λ/8~λ/10, we introduce two notations as
ΔG(ω)=G(rs1;ω)G(rs2;ω),
ΔG0sourcelens(ω)=G0sourcelens(rs1;ω)G0sourcelens(rs2;ω)
Then one can deduce from Eq. (6) that
 d dωΔG(ω)=dk2α(ω)dω (Ik2α(ω)RgG0lenslens(ω))1(ΔG(ω)ΔG0sourcelens(ω))
(7)
It should be highlighted that bothΔG(ω)andΔG0sourcelens(ω) capture the difference of the two fields inside the super-resonance lens emerged from the two sources located atrs1andrs2. When the frequency is in the range of the super resonances of the lens, |dk2α(ω)dω| becomes extremely large. More importantly, in this rangeIk2α(ω)RgG0lenslens(ω)is strongly ill-posed, leading to the extremely large entries in its reverse. Thus, there are two factors to amplifyΔG(ω)andΔG0sourcelens(ω): (Ik2α(ω)RgG0lenslens(ω))1and dk2α(ω)dω. On the other hand, to ensure bothΔG(ω) andΔG0sourcelens(ω) are non-zero, bothrs1andrs2should be in the vicinity of the super-resonance lens. Due to the use of the super-resonance lens, we observe thatΔG(ω)is remarkably larger thanΔG0sourcelens(ω). Hence we conclude thatΔG(ω)is more sensitive to the working frequencies in the super resonance range of the lens, leading to the capability of resolving two objects separated in a subwavelength distance with a broadband illumination.

2.2 The reconstruction algorithm

3. Simulation results

In this section, we verify the proposed subwavelength far-field imaging with the use of the super-resonance lens along with a single sensor, as sketched in Fig. 1. Here, the simulation parameters are set as:d = 5.8 μm (around 0.01λ, λ is the central wavelength of the illumination pulse),x0 = 11.7μm, N = 20×20,rd = (6mm,6mm), and the operational wavelength varies from 578μm to 585μm with a step of 0.002μm.

First, we will investigate the super-resonance property. As discussed previously, the far-field response is highly sensitive to the working frequencies when the super-resonance lens is used. Thus, it is expected that the frequency-dependent measurement acquired at rd exhibits a large amount of peaks within a certain frequency range when a broadband point source is set in the vicinity of the super-resonance lens. To show this clearly, we excite the super-resonance lens with a z-polarized line source centered at (−11.7, 0)µm, where this source emits a pulse with the frequency-dependent amplitude being unity for the operational wavelength ranging from 578µm to 585µm while being zero otherwise. The frequency-dependent response acquired at rdis provided in Fig. 2(a)
Fig. 2 The normalized amplitude of the frequency-dependent (a) and time-dependent (b) responses acquired atrd. Here, a point source (as shown in the inset in Fig. 2(b)) is centered at (−11.7, 0) µm. This set of figures shows many abrupt changes within a very small frequency separation, implying that the response is highly sensitivity to frequency. The results are generated by applying the full-wave solver to the Maxwell’s equations, i.e., the coupled dipole method.
, and its corresponding time-dependent response, as shown in Fig. 2(b), can be calculated by performing the Fourier transform. It is noted that the time-domain response is complex-valued instead of being real since the response in the frequency domain is not conjugate symmetric. Therefore, only the real part of the response in the time domain is plotted in Fig. 2(b). Additionally, the excitation pulse in the time domain is plotted in the inset of Fig. 2(b). This set of figures demonstrates that a broadband pulse will show rich peaks in the frequency domain after experiencing the super-resonance lens; accordingly, it will be fully expanded in the time domain with a factor of more than 100. Consequently, from the respect of the information capacity, the DoF of the measurements will be considerably driven up with a factor of 100 and beyond.

The super-resonance lens is spatial-temporal dispersive, which is capable of encoding the spatial details of the object under consideration in the subwavelength scale into the time domain [6

6. F. Lemoult, M. Fink, and G. Lerosey, “Revisiting the wire medium: an ideal resonant metalens,” Waves in Random and Complex Media 21(4), 591–613 (2011). [CrossRef]

8

8. F. Lemoult, M. Fink, and G. Lerosey, “A polychromatic approach to far-field superlensing at visible wavelengths,” Nat. Commun. 3, 1885 (2012).

]. It is expected that parts of the evanescent waves emerged from the objects under consideration can be converted into the propagating waves after experiencing the super-resonance lens. To show this point clearly, we perform another set of simulations. We assume that the super-resonance lens is illuminated by a plane wave characterized by Einc=z^ei(ykin, y+xkin,x),where kin,x=k02kin,y2 andk0denotes the wave number in vacuum. Note that the case|kin,y|>k0corresponds to the illumination of an evanescent wave, in contrast to|kin,y|<k0for a propagating wave. Figure 3
Fig. 3 The dependence of the amplitude of electrical field scattered from the super-resonance lens on kin,y/kp and ω/ωp. The electrical field is acquired at rd. In this figure, the x-axis is kin,y/kp and the y-axis denotes ω/ωp. This set of results is generated by applying the full-wave solver to the Maxwell’s equations, i.e., the coupled dipole method.
shows the dependence of the electric field scattered from the super-resonance lens acquired at rdon kin,y/kp(kp=2π/ωp) and ω/ωp. From this set of figures, we can see that the illumination of the evanescent waves can be efficiently captured by the sensor at the far field, implying the evanescent waves have been converted into propagating waves after experiencing the resonant aperture, as illustrated by the bright region in this figure. Note that the amplitude of the field scattered from the super-resonance lens is comparable to the illumination field, as demonstrated in Fig. 3.

Finally, we will examine the effects of both the distance of the detector and the SNR level on the reconstruction quality. Figure 5
Fig. 5 The reconstruction results using the super-resonance (SR) lens for four different working distances of sensor, where the simulation parameters are the same as those used in Fig. 4. The ground truth is also provided.
compares the reconstruction results for different distances of the acquiring sensor, where the rest of simulation parameters are the same as those used in Fig. 4. From this set of curves, one can see that with the considered resonant metalens the reconstruction quality is not very sensitive to the working distance of the single sensor overall, and that it will be decreased more or less with the growth of the distance of the sensor. The possible reason is as follows. The whole propagating waves acquired by the sensor in the far field consist mainly of two parts: one is from the original propagating illumination illuminated onto the metalens, and the other is converted from the evanescent waves. However, the latter is weaker in contrast to the former more or less, as illuminated in Fig. 3, especially for large distance since the latter are emerged from the interaction between the resonant metalens and the evanescent waves illuminated on it. More in-depth physical investigation about it should be carried out in the near future.

Figure 6
Fig. 6 The reconstruction results using the super-resonance (SR) lens for different noise levels of 40dB, 35dB, 30dB, 25dB and 20dB, where the simulation parameters are the same as those used in Fig. 4. The ground truth is also provided.
presents the comparison of the reconstructions achieved by the proposed imaging methodology for different SNR levels, where the rest of simulation parameters are the same as those in Fig. 4. We have added white Gaussian noise (WGN) with signal-to-noise ratio values of 40dB, 35dB, 30dB, 25dB and 20 dB to the simulated field acquired by a single sensor. It is observed that the higher the SNR is, the better the reconstruction quality is, and that the proposed imaging system of a single sensor is robust to the noise up to the SNR being 30dB.

4. Summary

References and links

1.

L. Rayleigh,“On pin-hole photography,” The London, Edinburg and Dublin philosophical magazine and journal of science, 5, 31 (1891).

2.

http://en.wikipedia.org/wiki/Near-field_scanning_optical_microscope

3.

E. T. F. Rogers and N. I. Zheludev, “Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging,” J. Opt. 15(9), 094008 (2013). [CrossRef]

4.

F. Lemoult, J. de Rosny, M. Fink, and G. Lerosey, “Resonant metalenses for breaking the diffraction barrier,” Phys. Rev. Lett. 104(20), 203901 (2010). [CrossRef] [PubMed]

5.

F. Lemoult, M. Fink, and G. Lerosey, “Far-field sub-wavelength imaging and focusing using a wire medium based resonant metalens,” Waves in Random and Complex Media 21(4), 614–627 (2011). [CrossRef]

6.

F. Lemoult, M. Fink, and G. Lerosey, “Revisiting the wire medium: an ideal resonant metalens,” Waves in Random and Complex Media 21(4), 591–613 (2011). [CrossRef]

7.

F. Lemoult, M. Fink, and G. Lerosey, “Dispersion in media containing resonant inclusions: where does it come from,” 2012 Conference on, Lasers and Electro-Optics (2012). [CrossRef]

8.

F. Lemoult, M. Fink, and G. Lerosey, “A polychromatic approach to far-field superlensing at visible wavelengths,” Nat. Commun. 3, 1885 (2012).

9.

D. Lu and Z. Liu, “Hyperlenses and metalenses for far-field super-resolution imaging,” Nat. Commun. 3, 1205 (2012).

10.

R. Pierrat, C. Vandenbem, M. Fink, and R. Carminat, “Subwavelength focusing inside an open disordered medium by time reversal at a single point antenna,” Phys. Rev. A. 87, 041801 (2013).

11.

P. C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(3), 036606 (2004). [CrossRef] [PubMed]

12.

P. C. Chaumet, K. Belkebir, and A. Rahmani, “Coupled-dipole method in time domain,” Opt. Express 16(25), 20157–20165 (2008). [CrossRef] [PubMed]

13.

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. 70(2), 447–466 (1998). [CrossRef]

14.

F. Simonetti, M. Fleming, and E. A. Marengo, “Illustration of the role of multiple scattering in subwavelength imaging from far-field measurements,” J. Opt. Soc. Am. A 25(2), 292–303 (2008). [CrossRef] [PubMed]

15.

O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antenn. Propag. 37(7), 918–926 (1989). [CrossRef]

16.

I. J. Cox and C. R. Sheppard, “Information capacity and resolution in an optical system,” J. Opt. Soc. Am. A 3(8), 1152–1158 (1986). [CrossRef]

17.

I. Tolstoy, “Superresonant systems of scatters I,” J. Acoust. Soc. Am. 80(1), 282–294 (1986). [CrossRef]

18.

G. S. Sammelmann and R. H. Hackman, “Acoustic scattering in a homogeneous waveguide,” J. Acoust. Soc. Am. 82(1), 324–336 (1987). [CrossRef]

19.

The super-resonance is mathematically that the matrix B=Ik2α(ω)RgG0lenslens(ω)in Eq. (3) is strongly ill-posed, which means that the ratio σ1σN(i.e., the condition number) is very large, whereσ1is the first singular value (the maximum) of the matrix, and σNis the final (the minimum) singular value.

20.

L. Li and B. Jafarpour, “Effective solution of nonlinear subsurface flow inverse problems in sparse bases,” Inverse Probl. 26(10), 105016 (2010). [CrossRef]

21.

M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing (Springer Press 2010).

OCIS Codes
(290.3200) Scattering : Inverse scattering
(290.4210) Scattering : Multiple scattering
(100.3200) Image processing : Inverse scattering

ToC Category:
Scattering

History
Original Manuscript: June 11, 2014
Revised Manuscript: July 14, 2014
Manuscript Accepted: July 14, 2014
Published: July 25, 2014

Virtual Issues
Vol. 9, Iss. 9 Virtual Journal for Biomedical Optics

Citation
Lianlin Li, Fang Li, and Tie Jun Cui, "Feasibility of resonant metalens for the subwavelength imaging using a single sensor in the far field," Opt. Express 22, 18688-18697 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-15-18688


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. L. Rayleigh,“On pin-hole photography,” The London, Edinburg and Dublin philosophical magazine and journal of science, 5, 31 (1891).
  2. http://en.wikipedia.org/wiki/Near-field_scanning_optical_microscope
  3. E. T. F. Rogers and N. I. Zheludev, “Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging,” J. Opt.15(9), 094008 (2013). [CrossRef]
  4. F. Lemoult, J. de Rosny, M. Fink, and G. Lerosey, “Resonant metalenses for breaking the diffraction barrier,” Phys. Rev. Lett.104(20), 203901 (2010). [CrossRef] [PubMed]
  5. F. Lemoult, M. Fink, and G. Lerosey, “Far-field sub-wavelength imaging and focusing using a wire medium based resonant metalens,” Waves in Random and Complex Media21(4), 614–627 (2011). [CrossRef]
  6. F. Lemoult, M. Fink, and G. Lerosey, “Revisiting the wire medium: an ideal resonant metalens,” Waves in Random and Complex Media21(4), 591–613 (2011). [CrossRef]
  7. F. Lemoult, M. Fink, and G. Lerosey, “Dispersion in media containing resonant inclusions: where does it come from,” 2012 Conference on, Lasers and Electro-Optics (2012). [CrossRef]
  8. F. Lemoult, M. Fink, and G. Lerosey, “A polychromatic approach to far-field superlensing at visible wavelengths,” Nat. Commun.3, 1885 (2012).
  9. D. Lu and Z. Liu, “Hyperlenses and metalenses for far-field super-resolution imaging,” Nat. Commun.3, 1205 (2012).
  10. R. Pierrat, C. Vandenbem, M. Fink, and R. Carminat, “Subwavelength focusing inside an open disordered medium by time reversal at a single point antenna,” Phys. Rev. A.87, 041801 (2013).
  11. P. C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.70(3), 036606 (2004). [CrossRef] [PubMed]
  12. P. C. Chaumet, K. Belkebir, and A. Rahmani, “Coupled-dipole method in time domain,” Opt. Express16(25), 20157–20165 (2008). [CrossRef] [PubMed]
  13. P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys.70(2), 447–466 (1998). [CrossRef]
  14. F. Simonetti, M. Fleming, and E. A. Marengo, “Illustration of the role of multiple scattering in subwavelength imaging from far-field measurements,” J. Opt. Soc. Am. A25(2), 292–303 (2008). [CrossRef] [PubMed]
  15. O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antenn. Propag.37(7), 918–926 (1989). [CrossRef]
  16. I. J. Cox and C. R. Sheppard, “Information capacity and resolution in an optical system,” J. Opt. Soc. Am. A3(8), 1152–1158 (1986). [CrossRef]
  17. I. Tolstoy, “Superresonant systems of scatters I,” J. Acoust. Soc. Am.80(1), 282–294 (1986). [CrossRef]
  18. G. S. Sammelmann and R. H. Hackman, “Acoustic scattering in a homogeneous waveguide,” J. Acoust. Soc. Am.82(1), 324–336 (1987). [CrossRef]
  19. The super-resonance is mathematically that the matrix B=I−k2α(ω)RgG0lens→lens(ω)in Eq. (3) is strongly ill-posed, which means that the ratio σ1σN(i.e., the condition number) is very large, whereσ1is the first singular value (the maximum) of the matrix, and σNis the final (the minimum) singular value.
  20. L. Li and B. Jafarpour, “Effective solution of nonlinear subsurface flow inverse problems in sparse bases,” Inverse Probl.26(10), 105016 (2010). [CrossRef]
  21. M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing (Springer Press 2010).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited