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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 8, Iss. 6 — Jun. 27, 2013
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Fundaments of optical far-field subwavelength resolution based on illumination with surface waves

Roberto Lopez-Boada, Charles J. Regan, Daniel Dominguez, Ayrton. A. Bernussi, and Luis Grave de Peralta  »View Author Affiliations


Optics Express, Vol. 21, Issue 10, pp. 11928-11942 (2013)
http://dx.doi.org/10.1364/OE.21.011928


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Abstract

We present a general discussion about the fundamental physical principles involved in a novel class of optical superlenses that permit to realize in the far-field direct non-scanning images with subwavelength resolution. Described superlenses are based in the illumination of the object under observation with surface waves excited by fluorescence, the enhanced transmission of fluorescence via coupling with surface waves, and the occurrence of far-field coherence-related fluorescence diffraction phenomena. A Fourier optics description of the image formation based on illumination with surface waves is presented, and several recent experimental realizations of this technique are discussed. Our theoretical approach explains why images with subwavelength resolution can be formed directly in the microscope camera, without involving scanning or numerical post-processing. While resolution of the order of λ/7 has been demonstrated using the described approach, we anticipate that deeper optical subwavelength resolution should be expected.

© 2013 OSA

1. Introduction

The conventional wisdom, formed from the Abbe’s theory of image formation and Fourier optics, is that traditional optical microscopy is diffraction-limited to spatial periods (p) larger than ~λo/NA [1

1. S. Durant, Z. Liu, J. M. Steele, and X. Zhang, “Theory of the transmission properties of an optical far-field superlens for imaging beyond the diffraction limit,” J. Opt. Soc. Am. B 23(11), 2383–2392 (2006). [CrossRef]

,2

2. E. Hetcht, Optics, 3rd edition (Addison Wesley, 1998).

], or separation between two points (Δx) larger than λo/(2NA) [1

1. S. Durant, Z. Liu, J. M. Steele, and X. Zhang, “Theory of the transmission properties of an optical far-field superlens for imaging beyond the diffraction limit,” J. Opt. Soc. Am. B 23(11), 2383–2392 (2006). [CrossRef]

3

3. M. Born and E. Wolf, Priciples of Optics, 5th edition (Pergamon Press, 1975).

], where NA is the numerical aperture of the microscope objective lens, and λo is the free space wavelength of the illuminating light. This explain why the theoretical prediction that a simple thin layer of metal could be used to realize a practical superlens [4

4. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

] was warmly received by the optical community searching for novel imaging techniques with subwavelength resolution for nanotechnology and subcellular biological applications. Since then, optical subwavelength resolution in the near-field using metallic superlenses has been demonstrated [5

5. D. O. S. Melville, R. J. Blaikie, and C. R. Wolf, “Submicron imaging with a planar silver lens,” Appl. Phys. Lett. 84(22), 4403–4405 (2004). [CrossRef]

,6

6. W. Srituravanich, N. Fang, C. Sun, Q. Luo, and X. Zhang, “Plasmonic Nanolithography,” Nano Lett. 4(6), 1085–1088 (2004). [CrossRef]

], exciting new near-field superlenses without metal have been also proposed [7

7. O. E. Gawhary, N. J. Schilder, A. C. Assafrao, S. F. Pereira, and H. P. Urbach, “Restoration of s-polarized evanescen waves and subwavelength imaging by a single dielectric slab,” New J. Phys. 14(5), 053025 (2012). [CrossRef]

,8

8. Y. Zhang, C. Arnold, P. Offermans, and J. Rivas, “Surface wave sensors based on nanometric layers of strongly absorbing materials,” Opt. Express 20(9), 9431–9441 (2012). [CrossRef] [PubMed]

], and the original concept was extended to the far-field [9

9. Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7(2), 403–408 (2007). [CrossRef] [PubMed]

13

13. C. J. Regan, R. Rodriguez, S. C. Gourshetty, L. Grave de Peralta, and A. A. Bernussi, “Imaging nanoscale features with plasmon-coupled leakage radiation far-field superlenses,” Opt. Express 20(19), 20827–20834 (2012). [CrossRef] [PubMed]

]. Some demonstrated far-field superlenses require intensive numerical image post processing [1

1. S. Durant, Z. Liu, J. M. Steele, and X. Zhang, “Theory of the transmission properties of an optical far-field superlens for imaging beyond the diffraction limit,” J. Opt. Soc. Am. B 23(11), 2383–2392 (2006). [CrossRef]

,9

9. Z. Liu, S. Durant, H. Lee, Y. Pikus, N. Fang, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical superlens,” Nano Lett. 7(2), 403–408 (2007). [CrossRef] [PubMed]

], some require a structure much more complicated than the original Pendry’s proposal [10

10. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007). [CrossRef] [PubMed]

], but until recently, all demonstrated far-field superlenses require a structure with at least a thin metal layer. The unavoidable inherent losses associated with the presence of the metal layer constitute a drawback characteristic of these superlenses [14

14. N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88(20), 207403 (2002). [CrossRef] [PubMed]

]; therefore, alternative methods for mitigation of the metal-related losses are of great interest.

Recently, the first far-field optical superlens without metal has been demonstrated. The details of the experiments will be published elsewhere [15

15. C. J. Regan, D. Dominguez, A. A. Bernussi, and L. Grave de Peralta, “Far-field optical superlens without metal,” J. Appl. Phys. ((to be published).

]. While investigating the physical principles responsible for the subwavelength resolution capabilities of the far-field superlenses that resemble more closely the original Pendry’s proposal [12

12. R. Rodriguez, C. J. Regan, A. Ruiz-Columbié, W. Agutu, A. A. Bernussi, and L. Grave de Peralta, “Study of plasmonic crystals using Fourier-plane images obtained with plasmon tomography far-field superlenses,” J. Appl. Phys. 110(8), 083109 (2011). [CrossRef]

,13

13. C. J. Regan, R. Rodriguez, S. C. Gourshetty, L. Grave de Peralta, and A. A. Bernussi, “Imaging nanoscale features with plasmon-coupled leakage radiation far-field superlenses,” Opt. Express 20(19), 20827–20834 (2012). [CrossRef] [PubMed]

], we realized that the specific nature of the surface plasmon polaritons (SPP) [16

16. M. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

] excited in the superlens was not a requirement for obtaining subwavelength resolution. This idea had been discussed previously in the context of near-field superlenses [7

7. O. E. Gawhary, N. J. Schilder, A. C. Assafrao, S. F. Pereira, and H. P. Urbach, “Restoration of s-polarized evanescen waves and subwavelength imaging by a single dielectric slab,” New J. Phys. 14(5), 053025 (2012). [CrossRef]

,17

17. G. Christou and C. Mias, “Critique of optical negative refraction superlensing,” Plasmonics 6(2), 307–309 (2011). [CrossRef]

] but its application to realize far-field superlenses remains unexplored. In this work we show that the same physical principles are responsible for the optical subwavelength resolution capabilities of demonstrated far-field superlens without metal [15

15. C. J. Regan, D. Dominguez, A. A. Bernussi, and L. Grave de Peralta, “Far-field optical superlens without metal,” J. Appl. Phys. ((to be published).

] and plasmonic superlenses [12

12. R. Rodriguez, C. J. Regan, A. Ruiz-Columbié, W. Agutu, A. A. Bernussi, and L. Grave de Peralta, “Study of plasmonic crystals using Fourier-plane images obtained with plasmon tomography far-field superlenses,” J. Appl. Phys. 110(8), 083109 (2011). [CrossRef]

,13

13. C. J. Regan, R. Rodriguez, S. C. Gourshetty, L. Grave de Peralta, and A. A. Bernussi, “Imaging nanoscale features with plasmon-coupled leakage radiation far-field superlenses,” Opt. Express 20(19), 20827–20834 (2012). [CrossRef] [PubMed]

]. In what follows, we group both types of far-field superlenses in the concept of “surface wave illumination” superlenses (SWIS). We show that the optical far-field subwavelength resolution obtained using SWIS is based on three common principles: the illumination of the object under observation with surface waves excited by fluorescence, the enhanced transmission of fluorescence via coupling with surface waves, and the occurrence of far-field coherence-related fluorescence diffraction phenomena. Therefore, SWIS are not ideal superlenses in the original sense of being capable of infinite resolution [4

4. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef] [PubMed]

,14

14. N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88(20), 207403 (2002). [CrossRef] [PubMed]

,17

17. G. Christou and C. Mias, “Critique of optical negative refraction superlensing,” Plasmonics 6(2), 307–309 (2011). [CrossRef]

], but they are far-field superlenses because SWIS allow direct imaging in the far-field with subwavelength resolution, without involving scanning or numerical post-processing. We present here, for the first time, a comprehensive Fourier Optics description of the image formation in a SWIS-microscope arrangement. The theory of image formation using SWIS proposed in this wok represents a generalization to far-field superlenses of the well-known Abbe’s theory of image formation. This makes the design of far-field superlenses easier and more intuitive by expanding the range of applicability of the conventional wisdom about the image formation in optical instruments. For instance, we show using these ideas how it might be possible to obtain deep optical subwavelength resolution with SWIS.

This paper is organized as follows: In Section 2 we make a general description of a SWIS and discuss the phenomena of enhanced transmission of fluorescence via excitement of surface waves. Section 3 is dedicated to the Fourier optics description of the formation of the Fourier plane (FP) images obtained in a SWIS-microscope arrangement. Several simple examples are presented, and the analytical expressions are derived and compared with experimental FP images. In Section 4, a Fourier optics description of the formation of the surface emission (SE) images is presented, and the origin of the SWIS subwavelength resolution is elucidated. Finally, the conclusions of this work are given in Section 5.

2. Enhanced transmission of fluorescence coupled to surface waves

Figure 1(a)
Fig. 1 (a) Cross-section schematic illustration of a sample used in a typical SWIS. (b)-(e) illustrates different excitation schemes: conventional surface wave excitation [(b) and (d)] and the excitation of the surface waves in SWIS [(c) and (e)] for superlenses with [(b) and (c)] and without [(d) and (e)] a thin metal layer.
shows a schematic illustration of the rather simple transversal structure of a typical SWIS. The SWIS substrate is formed by a ~150 μm thick glass cover slip. At the top of the SWIS is a ~150 nm thick layer of PolyMethylMethAcrylate (PMMA) doped with Rhodamine-6G (R6G) with peak emission at λo~568 nm wavelength. As shown in Fig. 1(a), an optional ~50 nm thick gold layer may be present between the substrate and the top layer. The object under observation (not shown in Fig. 1(a)) should be embedded in the PMMA + R6G layer. The SWIS with the object under observation is then placed in a traditional inverse microscope with an immersion oil high NA objective lens. A detailed description of a typical SWIS-microscope arrangement have being previously published [12

12. R. Rodriguez, C. J. Regan, A. Ruiz-Columbié, W. Agutu, A. A. Bernussi, and L. Grave de Peralta, “Study of plasmonic crystals using Fourier-plane images obtained with plasmon tomography far-field superlenses,” J. Appl. Phys. 110(8), 083109 (2011). [CrossRef]

,13

13. C. J. Regan, R. Rodriguez, S. C. Gourshetty, L. Grave de Peralta, and A. A. Bernussi, “Imaging nanoscale features with plasmon-coupled leakage radiation far-field superlenses,” Opt. Express 20(19), 20827–20834 (2012). [CrossRef] [PubMed]

,15

15. C. J. Regan, D. Dominguez, A. A. Bernussi, and L. Grave de Peralta, “Far-field optical superlens without metal,” J. Appl. Phys. ((to be published).

,18

18. S. P. Frisbie, C. Chesnutt, M. E. Holtz, A. Krishnan, L. de Peralta, and A. A. Bernussi, “Image formation in wide-field microscopes based on leakage of surface plasmon-coupled fluorescence,” IEEE Photon. J. 1(2), 153–162 (2009). [CrossRef]

,19

19. L. Grave de Peralta, C. J. Regan, and A. A. Bernussi, “SPP tomography: a simple wide-field nanoscope,” Scanning n/a (2012), doi:. [CrossRef] [PubMed]

].

pmin=λoNA.
(1)

Equation (1) corresponds to a minimum resolvable separation between two points of Δx~0.61pmin [1

1. S. Durant, Z. Liu, J. M. Steele, and X. Zhang, “Theory of the transmission properties of an optical far-field superlens for imaging beyond the diffraction limit,” J. Opt. Soc. Am. B 23(11), 2383–2392 (2006). [CrossRef]

,19

19. L. Grave de Peralta, C. J. Regan, and A. A. Bernussi, “SPP tomography: a simple wide-field nanoscope,” Scanning n/a (2012), doi:. [CrossRef] [PubMed]

]. However, it is less known that Eq. (1) is derived assuming that the sample is illuminated by a coherent monochromatic plane wave that propagates perpendicularly to the surface of the sample under observation. The electric field associated with the surface waves has a more prominent component, E, which is perpendicular to the propagation direction and can be perpendicular or parallel to the medium/PMMA interface; therefore, in SWIS the illumination source can be considered, in a first approximation, a superposition of plane waves propagating in all directions along the medium/PMMA interface. As a consequence, one should expect that Eq. (1) could not apply to SWIS. The component E of the electric field associated with a plane wave propagating along the medium/PMMA interface in the direction defined by the wavevector, kw is described by the following expression:
E(r,t)=Eosin(kwrωt),
(2)
where,

|kw|=kw=2πλ=koneff,λ=λoneff,ω=2πν.
(3)

Eo, λ, and ν are the amplitude, wavelength, and frequency of the plane wave, respectively; neff is the effective refractive index experienced by the surface wave, and r=(x,y) is contained in the plane z = 0 (medium /PMMA interface). The surface waves are continuously leaking to the glass substrate of the sample while propagating through the medium/PMMA interface [25

25. A. Drezet, A. Hohenau, D. Koller, A. Stepanov, H. Ditlbacher, B. Steinberger, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Leakage radiation microscopy of surface plasmon polaritons,” Mater. Sci. Eng. B 149(3), 220–229 (2008). [CrossRef]

27

27. A. Houk, R. Lopez-Boada, A. Ruiz-Columbie, S. Park, A. A. Bernussi, and L. Grave de Peralta, “Erratum: some consequences of experiments with a plasmonic quantum eraser for plasmon tomography,” J. Appl. Phys. 109(11), 119901 (2011). [CrossRef]

]. This is the light used for imaging in a SWIS-microscope arrangement [12

12. R. Rodriguez, C. J. Regan, A. Ruiz-Columbié, W. Agutu, A. A. Bernussi, and L. Grave de Peralta, “Study of plasmonic crystals using Fourier-plane images obtained with plasmon tomography far-field superlenses,” J. Appl. Phys. 110(8), 083109 (2011). [CrossRef]

,13

13. C. J. Regan, R. Rodriguez, S. C. Gourshetty, L. Grave de Peralta, and A. A. Bernussi, “Imaging nanoscale features with plasmon-coupled leakage radiation far-field superlenses,” Opt. Express 20(19), 20827–20834 (2012). [CrossRef] [PubMed]

,15

15. C. J. Regan, D. Dominguez, A. A. Bernussi, and L. Grave de Peralta, “Far-field optical superlens without metal,” J. Appl. Phys. ((to be published).

,18

18. S. P. Frisbie, C. Chesnutt, M. E. Holtz, A. Krishnan, L. de Peralta, and A. A. Bernussi, “Image formation in wide-field microscopes based on leakage of surface plasmon-coupled fluorescence,” IEEE Photon. J. 1(2), 153–162 (2009). [CrossRef]

,19

19. L. Grave de Peralta, C. J. Regan, and A. A. Bernussi, “SPP tomography: a simple wide-field nanoscope,” Scanning n/a (2012), doi:. [CrossRef] [PubMed]

].

As shown in Fig. 2
Fig. 2 Light leaks in the direction defined bykl, which forms an angle θl with respect to the axis z. Leaked light is contained in the plane ρ, which also contains the z axis and the vectors kwand kl.
, for a given value of kw, leakage radiation is very directional. It occurs in a very narrow angle interval centered at a direction forming an angle θl respect to the normal to the medium/PMMA interface [20

20. I. Gryczinski, J. Malicka, K. Nowaczyk, Z. Gryczynski, and J. Lacowicz, “Effects of sample thickness on the optical properties of surface plasmon-coupled emission,” J. Phys. Chem. B 108(32), 12073–12083 (2004). [CrossRef]

,21

21. I. Gryczynski, J. Malicka, Z. Gryczynski, and J. R. Lakowicz, “Surface plasmon-coupled emission with gold films,” J. Phys. Chem. B 108(33), 12568–12574 (2004). [CrossRef] [PubMed]

, 25

25. A. Drezet, A. Hohenau, D. Koller, A. Stepanov, H. Ditlbacher, B. Steinberger, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Leakage radiation microscopy of surface plasmon polaritons,” Mater. Sci. Eng. B 149(3), 220–229 (2008). [CrossRef]

]. Therefore, the light leaking in the direction θl can be described to a good approximation as a plane wave [25

25. A. Drezet, A. Hohenau, D. Koller, A. Stepanov, H. Ditlbacher, B. Steinberger, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Leakage radiation microscopy of surface plasmon polaritons,” Mater. Sci. Eng. B 149(3), 220–229 (2008). [CrossRef]

], which electric field (El) is given by the following expression:
El(s,t)=Elosin(klsωt),
(4)
where

|kl|=k2+k2=2πλs,λs=λons.
(5)

Here ns is the refractive index of the substrate,s=(x,y,z), and Elois proportional to Eo [25

25. A. Drezet, A. Hohenau, D. Koller, A. Stepanov, H. Ditlbacher, B. Steinberger, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Leakage radiation microscopy of surface plasmon polaritons,” Mater. Sci. Eng. B 149(3), 220–229 (2008). [CrossRef]

28

28. J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B Condens. Matter 33(8), 5186–5201 (1986). [CrossRef] [PubMed]

]. kand kare the components of the wavevector of the leakage radiation (kl) in the direction perpendicular and parallel to the medium/PMMA interface, respectively. kis also parallel tokw [26

26. L. Grave de Peralta, R. Lopez-Boada, A. Ruiz-Columbie, S. Park, and A. A. Bernussi, “Some consequences of experiments with a plasmonic quantum eraser for plasmon tomography,” J. Appl. Phys. 109(2), 023101 (2011). [CrossRef]

], and its magnitude is given by the following expression [16

16. M. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

,18

18. S. P. Frisbie, C. Chesnutt, M. E. Holtz, A. Krishnan, L. de Peralta, and A. A. Bernussi, “Image formation in wide-field microscopes based on leakage of surface plasmon-coupled fluorescence,” IEEE Photon. J. 1(2), 153–162 (2009). [CrossRef]

,25

25. A. Drezet, A. Hohenau, D. Koller, A. Stepanov, H. Ditlbacher, B. Steinberger, F. R. Aussenegg, A. Leitner, and J. R. Krenn, “Leakage radiation microscopy of surface plasmon polaritons,” Mater. Sci. Eng. B 149(3), 220–229 (2008). [CrossRef]

]:

k=klsinθl=kw.
(6)

3. Formation of the Fourier plane images

3.1 Homogeneous sample

For a SWIS without object under observation, the optical disturbance (U) at the medium/PMMA interface (z = 0) associated to El is given by the following expression [24

24. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

]:
Uφ(x,y,z=0)=Cφeikw·r=Cφei(xkwcosφ+ykwsinφ),
(8)
where Cφ is a real constant, i is the imaginary unit, and φ is the angle between kwand the x-axis (see Fig. 2). kw=0 corresponds to a plane wave traversing perpendicularly and undisturbed through the plane z = 0; therefore, the phase factoreikw·rcarries the information about the directionality of the leakage radiation associated to a plane wave propagating in the direction defined by kw along the medium/PMMA interface. The optical disturbance at the Fourier plane (FP), Uφ(kx,ky), is given by the Fourier Transform (FT) of Uφ(x,y,z = 0) [24

24. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

], i.e.:

Uφ(kx,ky)FT[Uφ(x,y,z=0)]Uφ(x,y,z=0)ei(xkx+yky)dxdy.
(9)

Here k = 2π/Λ and 1/Λ is the spatial frequency. Using Eqs. (8) and (9) results, for the excitation in a homogeneous sample of the surface wave described by Eq. (2), the following expression:

Uφ(kx,ky)Cφei[(kxkwcosφ)x+(kykwsinφ)y]dxdyδ(kxkwcosφ,kykwsinφ).
(10)

Where δ(k) is the delta of Dirac function [24

24. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

]. The intensity distribution at the FP is proportional to the square of the absolute value ofUφ(kx,ky), i.e.:

Iφ(kx,ky)|Uφ(kx,ky)|2.
(11)

From Eqs. (9)-(11) then follows that:

Iφ(kx,ky)δ2(kxkwcosφ,kykwsinφ).
(12)

The intensity distribution in the FP images obtained in the SWIS-microscope arrangement [12

12. R. Rodriguez, C. J. Regan, A. Ruiz-Columbié, W. Agutu, A. A. Bernussi, and L. Grave de Peralta, “Study of plasmonic crystals using Fourier-plane images obtained with plasmon tomography far-field superlenses,” J. Appl. Phys. 110(8), 083109 (2011). [CrossRef]

,13

13. C. J. Regan, R. Rodriguez, S. C. Gourshetty, L. Grave de Peralta, and A. A. Bernussi, “Imaging nanoscale features with plasmon-coupled leakage radiation far-field superlenses,” Opt. Express 20(19), 20827–20834 (2012). [CrossRef] [PubMed]

,15

15. C. J. Regan, D. Dominguez, A. A. Bernussi, and L. Grave de Peralta, “Far-field optical superlens without metal,” J. Appl. Phys. ((to be published).

,18

18. S. P. Frisbie, C. Chesnutt, M. E. Holtz, A. Krishnan, L. de Peralta, and A. A. Bernussi, “Image formation in wide-field microscopes based on leakage of surface plasmon-coupled fluorescence,” IEEE Photon. J. 1(2), 153–162 (2009). [CrossRef]

,19

19. L. Grave de Peralta, C. J. Regan, and A. A. Bernussi, “SPP tomography: a simple wide-field nanoscope,” Scanning n/a (2012), doi:. [CrossRef] [PubMed]

] is the part of Iφ(kx,ky)inside of the reciprocal space region captured by the microscope objective lens, |k|koNA; i.e.:

Iφ,FPI(kx,ky)=0,|kw|>koNAδ2(kxkwcosφ,kykwsinφ),|kw|koNA,
(13)

Therefore, the intensity distribution in a FP image, corresponding to the excitation in a homogeneous sample of the surface wave described by Eq. (2), is a bright spot at the reciprocal space point (kwcosφ,kwsinφ) at the extreme of the vectork=kw. Basically, Iφ,FPI(kx,ky) is the observed far-field diffraction pattern, which is produced by the interference between photons that leaks from the medium/PMMA interface in the direction defined bykw. The interference occurs because it is impossible to know the path of the photons arriving to the FP image at the spot(kwcosφ,kwsinφ). I.e., photons that leaks in the direction, φ, through different points in the medium/PMMA interface, and arrive to the same spot in the FP image, are undistinguishable; therefore, they interfere [31

31. B. Schumacher, Quantum mechanics: the Physics of the Microscopic World (The Teaching Company, 2009).

33

33. L. Grave de Peralta, “Phenomenological quantum description of the ultra fast response of arrayed waveguide gratings,” J. Appl. Phys. 108(10), 103110 (2010). [CrossRef]

]. However, photons that leaks in different directions can be distinguished by the direction of their momenta; thus, they cannot interfere [31

31. B. Schumacher, Quantum mechanics: the Physics of the Microscopic World (The Teaching Company, 2009).

33

33. L. Grave de Peralta, “Phenomenological quantum description of the ultra fast response of arrayed waveguide gratings,” J. Appl. Phys. 108(10), 103110 (2010). [CrossRef]

]. In consequence, the total intensity at the FP, IT(kx,ky), must be calculated by adding the intensity contributionsIφ(kx,ky)corresponding to all the surface waves illumination directions φ; i.e, IT(kx,ky) is described by the following expression:

IT(kx,ky)=φIφ(kx,ky).
(14)

3.2 Periodic sample

For a sample with a sinusoidal profile of period p along the x axes, the optical disturbance (U) at the gold/PMMA interface (z = 0) associated to El is proportional to the Bloch wavefunction corresponding to the periodical structure, i.e [36

36. C. J. Regan, O. Thiabgoh, L. Grave de Peralta, and A. A. Bernussi, “Probing photonic Bloch wavefunctions with plasmon-coupled leakage radiation,” Opt. Express 20(8), 8658–8666 (2012). [CrossRef] [PubMed]

]:

Uφ,p(x,y,z=0)=Cφ[1+sin(2πpx)]eikw·r=Cφ[1+sin(2πpx)]ei(xkwcosφ+ykwsinφ).
(15)

From Eqs. (9) and (15) follows that the optical disturbance at the FP is now given by the following expression:

Uφ,p(kx,ky)Cφ[1+sin(2πpx)]ei[(kxkwcosφ)x+(kykwsinφ)y]dxdy.
(16)

Thus:
Uφ,p(k){δ[kkw]+δ[k(G+kw)]+δ[k(G+kw)]},
(17)
where Gis the reciprocal lattice vector corresponding to the grating, i.e.:

|G|=G=2πp.
(18)

Uφ,p(k)is a shifted version of the optical disturbance corresponding to the far-field two-dimensional diffraction pattern that would be obtained under traditional out-of-plane perpendicular illumination [24

24. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

]. Therefore, Eq. (16) can be rewritten as:
Uφ,p(k)=U,p(kkw),
(19)
where:

U,p(k){δ[k]+δ[kG]+δ[k+G]}.
(20)

Consequently, if kw+GkoNA, the BFP image is formed by three bright spots at the reciprocal space points(kwcosφ,kwsinφ), (kwcosφ+G,kwsinφ), and (kwcosφG,kwsinφ) at the extreme of the vectors kw, G+kw, and G+kw, respectively. Substituting Eq. (19) into Eq. (14) permits to calculate the total intensity distribution at the FP, IT(kx,ky), corresponding to the incoherent superposition of numerous plane waves propagating in all directions along the medium/PMMA interface.

Figure 4
Fig. 4 FP image corresponding to a plasmonic sample with a periodic profile along the x-axis, with p = 4 μm. When the direction φ of the surface waves illumination changes from 0 to 2π, the extreme of each vector kw describes a circumference.
shows a FP image corresponding to a plasmonic sample with a periodic profile along the x-axis, with p = 4 μm, for a case wherekw+G<koNA [18

18. S. P. Frisbie, C. Chesnutt, M. E. Holtz, A. Krishnan, L. de Peralta, and A. A. Bernussi, “Image formation in wide-field microscopes based on leakage of surface plasmon-coupled fluorescence,” IEEE Photon. J. 1(2), 153–162 (2009). [CrossRef]

]. As shown in Fig. 4, when the direction φ of the surface waves illumination changes from 0 to 2π, the vectors kw with origin at the points(0,0),(+G,0), and(G,0)rotate simultaneously around their origins. As a consequence, the three spots(kwcosφ,kwsinφ), (kwcosφ+G,kwsinφ), and (kwcosφG,kwsinφ) simultaneously describe circumferences of radius kw centered at the points (0,0),(+G,0), and(G,0), respectively. Due to the non-sinusoidal profile of the sample, in addition to the zero and first order rings, two arc-segments of the second order rings centered at the points (+2G,0), and(2G,0) can be seen in the FP image shown in Fig. 4. The second order rings do not appear complete because in the FP image only appears the fraction of the total intensity distribution IT(kx,ky), which is inside of the reciprocal space region |k|koNA.

3.3 General case

In general, the optical disturbance at the medium/PMMA interface corresponding to a single plane wave propagating along the medium/PMMA interface is given by the following expression:
Uφ(x,y,z=0)=f(x,y)eikw·r=f(x,y)ei(xkwcosφ+ykwsinφ),
(21)
where f(x,y) is the optical disturbance at the medium/PMMA interface that would exist under traditional perpendicular out-of-plane illumination. The corresponding optical disturbance at the FP is then given by the following expression:

Uφ(kx,ky)f(x,y)ei[(kxkwcosφ)x+(kykwsinφ)y]dxdy.
(22)

Therefore, Uφ(k)is proportional to the two-dimensional FT of f(x,y) after a translation in the reciprocal space by kw, i.e.:
Uφ(k)=U(kkw),
(23)
where, U(k)would be the optical disturbance at the medium/PMMA interface if the sample was traditionally studied with perpendicular out-of-plane illumination i.e.:

U(k)FT[f(x,y)]f(x,y)eik·rdxdy.
(24)

In general,Uφ(k)is different than zero in a large area, therefore Uφ(k)is not an slide in the direction φ of the BFP image as it was proposed in Ref. 18

18. S. P. Frisbie, C. Chesnutt, M. E. Holtz, A. Krishnan, L. de Peralta, and A. A. Bernussi, “Image formation in wide-field microscopes based on leakage of surface plasmon-coupled fluorescence,” IEEE Photon. J. 1(2), 153–162 (2009). [CrossRef]

. The intensity distribution Iφ(k) at the FP corresponding toUφ(k) is given by the Eq. (11). The total intensity distribution at the FP,IT(k), must be obtained by adding the intensity contributionsIφ(k)corresponding to all the surface waves illumination directions φ; therefore, it is described by the Eq. (14). Consequently, from Eqs. (11) and (14), and Eq. (24) we obtain:

IT(k)φaφ|U(kxkwcosφ,kykwsinφ)|2.
(25)

Each term aφ|U(kxkwcosφ,kykwsinφ)|2in Eq. (25) is a shifted version with weight aφof the FP image that would be observed under traditional perpendicular out-of-plane illumination. Therefore, one can find, from the distribution of points where|U(k)|0, the distribution of reciprocal space points (kx,ky) whereIT(k)0. In order to achieve this, one can imagine, as shown in Fig. 4, a vector kw attached to each point (kx',ky') such that |U(kx',kx')|0 (represented by red dots in Fig. 4). When the direction φ of the surface wave illumination changes from 0 to 2π, the vectors kw rotate simultaneously around their origins. This resulting in each point (kx',ky')being substituted by a ring of radius kwcentered in that point. The collection of all these rings form the distribution of reciprocal space points (kx,ky) whereIT(kx,ky)0. The total intensity distribution in the FP image, IT,FPI(kx,ky), is the part of IT(kx,ky)captured by the microscope high NA objective lens; i.e.:

IT,FPI(kx,ky)=0,|k|>koNA=IT(kx,ky),|k|koNA.
(26)

It should be pointed out here that the observation of full rings in the FP images (see instance in Fig. 4) is conditioned by the existence of surface waves traveling in all directions in the medium/PMMA interface. However, this is not always the case. It is well known that two dimensional plasmonic and photonic crystals may have directional gaps [37

37. C. J. Regan, L. Grave de Peralta, and A. A. Bernussi, “Equifrequency curve dispersion in dielectric-loaded plasmonic crystals,” J. Appl. Phys. 111(7), 073105 (2012). [CrossRef]

39

39. J. D. Joannopoulus, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2008).

]. Directional gaps are formed when the propagation in the crystal of surface waves are prohibited in a range of directions. When this happen the weight coefficient in Eq. (25) is equal to zero for all prohibited directions.

4. Origin of subwavelength resolution

In the SWIS-microscope arrangement, each surface wave illumination direction φ produces an independent image of the object under observation,Iφ,SEI(x,y), at the image plane of the microscope. The intensity distribution of this image is given by the following expression:
Iφ,SEI(x,y)|Uφ,IP(x,y)|2,
(27)
whereUφ,IP(x,y)is the optical disturbance at the image plane corresponding toUφ,FP(kx,ky), which is the part ofUφ(k)captured by the microscope lenses, i.e.:

Uφ,FP(kx,ky)=0,|k|>koNA=Uφ(kx,ky),|k|koNA.
(28)

Therefore, Uφ,IP(x,y)is given by the inverse FT of Uφ,FP(kx,ky), i.e [24

24. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

]:

Uφ,IP(x,y)FT1[Uφ,FP(kx,ky)]Uφ,FP(kx,ky)ei(xkx+yky)dkxdky.
(29)

The final intensity distribution in the SE image obtained with the SWIS-microscope arrangement [12

12. R. Rodriguez, C. J. Regan, A. Ruiz-Columbié, W. Agutu, A. A. Bernussi, and L. Grave de Peralta, “Study of plasmonic crystals using Fourier-plane images obtained with plasmon tomography far-field superlenses,” J. Appl. Phys. 110(8), 083109 (2011). [CrossRef]

,13

13. C. J. Regan, R. Rodriguez, S. C. Gourshetty, L. Grave de Peralta, and A. A. Bernussi, “Imaging nanoscale features with plasmon-coupled leakage radiation far-field superlenses,” Opt. Express 20(19), 20827–20834 (2012). [CrossRef] [PubMed]

,15

15. C. J. Regan, D. Dominguez, A. A. Bernussi, and L. Grave de Peralta, “Far-field optical superlens without metal,” J. Appl. Phys. ((to be published).

,18

18. S. P. Frisbie, C. Chesnutt, M. E. Holtz, A. Krishnan, L. de Peralta, and A. A. Bernussi, “Image formation in wide-field microscopes based on leakage of surface plasmon-coupled fluorescence,” IEEE Photon. J. 1(2), 153–162 (2009). [CrossRef]

,19

19. L. Grave de Peralta, C. J. Regan, and A. A. Bernussi, “SPP tomography: a simple wide-field nanoscope,” Scanning n/a (2012), doi:. [CrossRef] [PubMed]

],ISEI(x,y), is formed by addition of the images corresponding to all directions, i.e.:

ISEI(x,y)=φIφ,SEI(x,y).
(30)

The sum of the intensities corresponding to different directions at the medium/PMMA interface was previously introduced for plasmonic crystals by showing that plasmon-coupled leakage radiation superlenses can be used to directly probe the Bloch wavefunctions of the photons in a two-dimensional crystal [36

36. C. J. Regan, O. Thiabgoh, L. Grave de Peralta, and A. A. Bernussi, “Probing photonic Bloch wavefunctions with plasmon-coupled leakage radiation,” Opt. Express 20(8), 8658–8666 (2012). [CrossRef] [PubMed]

]. The correspondence between the SE image and the electric field distribution in the medium/PMMA interface is due to the proportionality between Elo in Eq. (4) and Eoin Eq. (2). The origin of the subwavelength resolution capabilities of SWIS [13

13. C. J. Regan, R. Rodriguez, S. C. Gourshetty, L. Grave de Peralta, and A. A. Bernussi, “Imaging nanoscale features with plasmon-coupled leakage radiation far-field superlenses,” Opt. Express 20(19), 20827–20834 (2012). [CrossRef] [PubMed]

,15

15. C. J. Regan, D. Dominguez, A. A. Bernussi, and L. Grave de Peralta, “Far-field optical superlens without metal,” J. Appl. Phys. ((to be published).

,19

19. L. Grave de Peralta, C. J. Regan, and A. A. Bernussi, “SPP tomography: a simple wide-field nanoscope,” Scanning n/a (2012), doi:. [CrossRef] [PubMed]

] can be better illustrated for simple periodical structures. As discussed above, whenkw+GkoNA, the zero and first order rings are observed in the FP image corresponding to a one dimensional periodic structure (see Fig. 4); therefore, using Eqs. (17) and (19), and Eqs. (28)-(29), results:

Uφ,IP(x,y)FT1[U,p(kkw)]FT1{δ[kkw]+δ[k(kw+G)]+δ[k(kwG)]}.
(31)

Therefore, it follows from the shift property of FT and Eq. (31) that for any illumination direction φ, the absolute value of the optical disturbance at the image plane Uφ,IP(x,y)is proportional to the optical disturbance that would be at the object plane under traditional perpendicular out-of-plane illumination, i.e.:

|Uφ,IP(x,y)|[1+sin(2πpx)].
(32)

From Eqs. (30) and (32) follows that every SE image obtained using directional surface waves illumination,Iφ,SEI(x,y), independently of the illumination direction φ, matches the image that would be obtained under traditional perpendicular out-of-plane illumination. Therefore, in correspondence with experimental results [18

18. S. P. Frisbie, C. Chesnutt, M. E. Holtz, A. Krishnan, L. de Peralta, and A. A. Bernussi, “Image formation in wide-field microscopes based on leakage of surface plasmon-coupled fluorescence,” IEEE Photon. J. 1(2), 153–162 (2009). [CrossRef]

], from Eq. (30) follows than in the casekw+GkoNA, the total intensity distribution in the SE image closely matches the image that would be obtained under perpendicular out-of-plane illumination.

The more interesting case of kw+G>koNA but kwGkoNAis depicted in Fig. 5
Fig. 5 Schematic illustration of the transversal structure of a (a) plasmonic SWIS, and (b) SWIS without metal. (c) FP image corresponding to the plasmonic SWIS. (d) Intensity distribution inside of the zero order ring resulting from subtracting the FP image corresponding to a homogeneous SWIS without metal, from the FP image corresponding to the SWIS without metal with a periodic patterned structure.
. A schematic illustration of the transversal structure of a plasmonic SWIS (SWIS without metal) is shown in Fig. 5(a) (Fig. 5(b)). The fabrication details of these samples have been published elsewhere [15

15. C. J. Regan, D. Dominguez, A. A. Bernussi, and L. Grave de Peralta, “Far-field optical superlens without metal,” J. Appl. Phys. ((to be published).

,19

19. L. Grave de Peralta, C. J. Regan, and A. A. Bernussi, “SPP tomography: a simple wide-field nanoscope,” Scanning n/a (2012), doi:. [CrossRef] [PubMed]

]. The plasmonic SWIS is a plasmonic crystal with square symmetry and period of p = 300 nm formed by patterning the top PMMA + R6G layer; therefore, the patterned air holes with a diameter of 100 nm were the object under observation in this sample [19

19. L. Grave de Peralta, C. J. Regan, and A. A. Bernussi, “SPP tomography: a simple wide-field nanoscope,” Scanning n/a (2012), doi:. [CrossRef] [PubMed]

]. The SWIS without metal is a photonic crystal with square symmetry and period of p = 220 nm formed by ~35 nm thick chromium (Cr) deposited over a ~150 μm thick coverslip substrate. A 110 nm thick layer of PMMA doped with R6G was then spun on top of the whole structure, and finally, in order to increase the subwavelength resolution, a drop of water was placed on top of the fabricated SWIS; therefore, the Cr pillars were the object under observation in this sample [15

15. C. J. Regan, D. Dominguez, A. A. Bernussi, and L. Grave de Peralta, “Far-field optical superlens without metal,” J. Appl. Phys. ((to be published).

]. Cr was used here because it does not exhibit a well-defined plasmonic signature, can be simply patterned after deposition on a glass substrate, and it provides high SE contrast images. Therefore we denote the sample sketched in Fig. 5(b) as an SWIS without metal.

Figure 5(c) shows the FP image obtained with a SWIS-microscope arrangement corresponding to the plasmonic SWIS with the structure sketched in Fig. 5(a). The zero order ring and arc-segments of the first order rings distributed with square symmetry are clearly seen in this image. A similar FP image was obtained using the SWIS without metal with the structure sketched in Fig. 5(b) [15

15. C. J. Regan, D. Dominguez, A. A. Bernussi, and L. Grave de Peralta, “Far-field optical superlens without metal,” J. Appl. Phys. ((to be published).

]; however, the first order rings were faint in the image. This indicates that the in-plane scattering of the surface waves was more efficient in the plasmonic SWIS than in the SWIS without metal [34

34. Y. Chen, D. Zhang, L. Han, G. Rui, X. Wang, P. Wang, and H. Ming, “Surface-plasmon-coupled emission microscopy with a polarization converter,” Opt. Lett. 38(5), 736–738 (2013). [CrossRef] [PubMed]

]. Nevertheless, the presence of the faint first order rings is revealed in Fig. 5(d), which shows the intensity distribution inside of the zero order ring, resulting from subtracting the FP image corresponding to the homogeneous SWIS without metal with the structure sketched in Fig. 1(a) (see Fig. 3(b)), from the FP image corresponding to the SWIS without metal with the structure sketched in Fig. 5(b). Similarly to the FP image shown in Fig. 5(c), arc-segments of the thick first order rings distributed with square symmetry are clearly seen in Fig. 5(d).

The graphical compositions shown in Fig. 6
Fig. 6 Graphical composition illustrating how the FP images shown in (a) Fig. 5(c), and 5(b) Fig. 5(d) are formed. The red spots represent the diffraction pattern that would be observed using traditional perpendicular out-of-plane illumination. Similar fractions of the first order rings are observed in both images.
illustrate how the FP images shown in Fig. 5 are formed by the superposition of rings of radius kwcentered at the spots corresponding to the diffraction pattern that would be obtained using traditional perpendicular out-of-plane illumination. Only the arc-segments of these rings captured by the high NA objective lens are observed in the FP image. Neighbors spots are separated in the reciprocal space a distance G = 2π/p. A well-known result of the Abbe’s theory of image formation is that the periodic structure of the sample would not appear in the image if the first order spots were not captured by the microscope lenses [2

2. E. Hetcht, Optics, 3rd edition (Addison Wesley, 1998).

]. This is what happens in the FP images shown in Figs. 5 and 6; therefore, the periodic structures existing in the samples sketched in Fig. 5(a) and 5(b) could not be observed in the SE images that were obtained using traditional perpendicular out-of-plane illumination. This is in correspondence with the minimum observable period of pmin~381 nm calculated from (1) with NA = 1.49. However, the periodic structures with period of p = 300 nm and 220 nm sketched in Fig. 5(a) and 5(b), respectively, have been successfully imaged (not shown here) using a SWIS-microscope arrangement [15

15. C. J. Regan, D. Dominguez, A. A. Bernussi, and L. Grave de Peralta, “Far-field optical superlens without metal,” J. Appl. Phys. ((to be published).

,19

19. L. Grave de Peralta, C. J. Regan, and A. A. Bernussi, “SPP tomography: a simple wide-field nanoscope,” Scanning n/a (2012), doi:. [CrossRef] [PubMed]

]. Consequently, the subwavelength resolution capabilities of SWIS have already been unambiguously demonstrated.

In what follows we will show that the subwavelength resolution capabilities of SWIS can be easily understood using the Fourier optics description of the image formation in a SWIS-microscope arrangement presented above. The diffraction pattern produced by illuminating the samples with surface waves propagating in the direction φ is formed by the five spots pointed by the vectorskw, which are represented by red arrows in Fig. 6. The rings are formed when the vectors kw rotated around the red spots shown in Fig. 6. Only the diffraction spots captured by the high NA objective lens are observed in the FP image. When φ~3π/4 in the instance illustrated in Fig. 6(b), only the diffraction spot at the reciprocal space point S = (22kw,22kw)is contained in the region |k|koNA; therefore, using Eq. (29), results:

U5π4,IP(x,y)FT1{δ[k(22kw,22kw)]}.
(33)

5. Conclusions

We have presented, for the first time, a Fourier optics description of the image formation in a SWIS-microscope arrangement. We found an excellent correspondence between the analytical expressions obtained using the Fourier optics approach and the experimental images. We have shown that the same physical principles are responsible for the demonstrated subwavelength resolution capabilities of both plasmonic SWIS and SWIS without metal. In general, SWIS are based in the illumination of the object under observation with surface waves excited by fluorescence, the enhanced transmission of fluorescence via coupling with surface waves, and the occurrence of far-field coherence-related fluorescence diffraction phenomena. Our theoretical approach explains why images with subwavelength resolution can be formed directly in the microscope camera, without involving scanning or numerical post-processing. Finally, we have suggested alternative solutions to obtain deep optical subwavelength resolution with a SWIS-microscope arrangement.

Acknowledgments

This work was partially supported by the NSF CAREER Award (ECCS-0954490).

References and links

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O. E. Gawhary, N. J. Schilder, A. C. Assafrao, S. F. Pereira, and H. P. Urbach, “Restoration of s-polarized evanescen waves and subwavelength imaging by a single dielectric slab,” New J. Phys. 14(5), 053025 (2012). [CrossRef]

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R. Rodriguez, C. J. Regan, A. Ruiz-Columbié, W. Agutu, A. A. Bernussi, and L. Grave de Peralta, “Study of plasmonic crystals using Fourier-plane images obtained with plasmon tomography far-field superlenses,” J. Appl. Phys. 110(8), 083109 (2011). [CrossRef]

13.

C. J. Regan, R. Rodriguez, S. C. Gourshetty, L. Grave de Peralta, and A. A. Bernussi, “Imaging nanoscale features with plasmon-coupled leakage radiation far-field superlenses,” Opt. Express 20(19), 20827–20834 (2012). [CrossRef] [PubMed]

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N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens,” Phys. Rev. Lett. 88(20), 207403 (2002). [CrossRef] [PubMed]

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C. J. Regan, D. Dominguez, A. A. Bernussi, and L. Grave de Peralta, “Far-field optical superlens without metal,” J. Appl. Phys. ((to be published).

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M. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

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S. P. Frisbie, C. Chesnutt, M. E. Holtz, A. Krishnan, L. de Peralta, and A. A. Bernussi, “Image formation in wide-field microscopes based on leakage of surface plasmon-coupled fluorescence,” IEEE Photon. J. 1(2), 153–162 (2009). [CrossRef]

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L. Grave de Peralta, C. J. Regan, and A. A. Bernussi, “SPP tomography: a simple wide-field nanoscope,” Scanning n/a (2012), doi:. [CrossRef] [PubMed]

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I. Gryczinski, J. Malicka, K. Nowaczyk, Z. Gryczynski, and J. Lacowicz, “Effects of sample thickness on the optical properties of surface plasmon-coupled emission,” J. Phys. Chem. B 108(32), 12073–12083 (2004). [CrossRef]

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Y. Chen, D. Zhang, L. Han, G. Rui, X. Wang, P. Wang, and H. Ming, “Surface-plasmon-coupled emission microscopy with a polarization converter,” Opt. Lett. 38(5), 736–738 (2013). [CrossRef] [PubMed]

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C. J. Regan, O. Thiabgoh, L. Grave de Peralta, and A. A. Bernussi, “Probing photonic Bloch wavefunctions with plasmon-coupled leakage radiation,” Opt. Express 20(8), 8658–8666 (2012). [CrossRef] [PubMed]

37.

C. J. Regan, L. Grave de Peralta, and A. A. Bernussi, “Equifrequency curve dispersion in dielectric-loaded plasmonic crystals,” J. Appl. Phys. 111(7), 073105 (2012). [CrossRef]

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C. J. Regan, A. Krishnan, R. Lopez-Boada, L. Grave de Peralta, and A. A. Bernussi, “Direct observation of photonic Fermi surfaces by plasmon tomography,” Appl. Phys. Lett. 98(15), 151113 (2011). [CrossRef]

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OCIS Codes
(110.0180) Imaging systems : Microscopy
(240.6680) Optics at surfaces : Surface plasmons
(240.6690) Optics at surfaces : Surface waves
(260.2510) Physical optics : Fluorescence
(260.6970) Physical optics : Total internal reflection

ToC Category:
Optics at Surfaces

History
Original Manuscript: March 5, 2013
Revised Manuscript: April 28, 2013
Manuscript Accepted: May 3, 2013
Published: May 8, 2013

Virtual Issues
Vol. 8, Iss. 6 Virtual Journal for Biomedical Optics

Citation
Roberto Lopez-Boada, Charles J. Regan, Daniel Dominguez, Ayrton. A. Bernussi, and Luis Grave de Peralta, "Fundaments of optical far-field subwavelength resolution based on illumination with surface waves," Opt. Express 21, 11928-11942 (2013)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-10-11928


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References

  1. S. Durant, Z. Liu, J. M. Steele, and X. Zhang, “Theory of the transmission properties of an optical far-field superlens for imaging beyond the diffraction limit,” J. Opt. Soc. Am. B23(11), 2383–2392 (2006). [CrossRef]
  2. E. Hetcht, Optics, 3rd edition (Addison Wesley, 1998).
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