## Subwavelength optical imaging of evanescent fields using reflections from plasmonic slabs

Optics Express, Vol. 15, Issue 18, pp. 11542-11552 (2007)

http://dx.doi.org/10.1364/OE.15.011542

Acrobat PDF (286 KB)

### Abstract

Reflection can significantly improve the quality of subwavelength near-field images, which is explained by appropriate interference between forward and reflected waves. Plasmonic slabs may form approximate super-mirrors. This paper develops general theory in both spectral and spatial representations that allows the reflector position and permittivity to be determined for optimum image uniformity. This elucidates previous observations and predicts behaviour for some other interesting regimes, including interferometric lithography.

© 2007 Optical Society of America

## 1. Introduction

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

2. J. S. Wei and F. X. Gan, “Dynamic readout of subdiffraction-limited pit arrays with a silver superlens,” Appl. Phys. Lett. **87**, art. no. 211101 (2005). [CrossRef]

3. T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science **313**, 1595–1595 (2006). [CrossRef] [PubMed]

4. D. O. S. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express **13**, 2127–2134 (2005) [CrossRef] [PubMed]

5. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science **308**, 534–537 (2005). [CrossRef] [PubMed]

6. R. J. Blaikie, M. M. Alkaisi, S. J. McNab, and D. O. S. Melville, “Nanoscale optical patterning using evanescent fields and surface plasmons,” Int. J. Nanoscience **3**, 405–417 (2004) [CrossRef]

7. D. B. Shao and S. C. Chen, “Surface-plasmon-assisted nanoscale photolithography by polarized light,” Appl. Phys. Lett. **86**, art. no. 253107 (2005) [CrossRef]

8. D. B. Shao and S. C. Chen, “Numerical simulation of surface-plasmon-assisted nanolithography,” Opt. Express **13**, 6964–6973 (2005). [CrossRef] [PubMed]

6. R. J. Blaikie, M. M. Alkaisi, S. J. McNab, and D. O. S. Melville, “Nanoscale optical patterning using evanescent fields and surface plasmons,” Int. J. Nanoscience **3**, 405–417 (2004) [CrossRef]

10. M. Schrader, M. Kozubek, S. W. Hell, and T. Wilson, “Optical transfer functions of 4Pi confocal microscopes: theory and experiment,” Opt. Lett. **22**, 436–438 (1997). [CrossRef] [PubMed]

6. R. J. Blaikie, M. M. Alkaisi, S. J. McNab, and D. O. S. Melville, “Nanoscale optical patterning using evanescent fields and surface plasmons,” Int. J. Nanoscience **3**, 405–417 (2004) [CrossRef]

*y-z*plane—a periodic binary absorber for the example in Fig. 1—with the object’s exit plane located at

*x=d*. A reflector of some kind is located at

*x*=0 and the system is illuminated from the right with light of wavelength

*λ*. It is the evolution of the optical fields in the imaging region, 0<

*x*<

*d*, that is of interest here, both with and without the reflector. We consider a nonperturbative imaging medium, such as a weakly absorbing photoresist filling the imaging region [4

4. D. O. S. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express **13**, 2127–2134 (2005) [CrossRef] [PubMed]

8. D. B. Shao and S. C. Chen, “Numerical simulation of surface-plasmon-assisted nanolithography,” Opt. Express **13**, 6964–6973 (2005). [CrossRef] [PubMed]

3. T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science **313**, 1595–1595 (2006). [CrossRef] [PubMed]

*x*=

*d*, which may also be considered an ‘exact’ image plane (i.e. it is the fields at this plane that we wish to transfer throughout our imaging volume). The fields evolve right-to-left along the longitudinal axis

*x*, forming an extended near-field image. Any reflected fields (associated with an apparent reflected image) combine with the forward-going fields, and we will call the result ‘the image’ for brevity. With no-reflection the image is longitudinally asymmetric (Fig. 2(a)), but reflection increases symmetry (Fig. 2(b)) which can improve uniformity. This extended image can be summarized in terms of intensity profiles on medial planes (Fig. 2(c) & (d)) under light (

*I*

_{1}) and dark (

*I*

_{0}) features of the object, which can be used to calculate various image metrics.

*γ(x)≡I*at distance

_{1}(x)/I_{0}(x)*d-x*from the object, and global contrast

*Γ*=min(

*I*

_{1})/max(

*I*

_{0}) over the whole imaging domain 0≤

*x*≤

*d*. An alternative measure of contrast is visibility

*V*=(

*Γ*-1)/(

*Γ*+1), and the depth-of-field

*D*is given by the distance over which

*V*>0, as shown in Fig. 2(c). Imaging applications such as lithography usually have minimum requirements for both global contrast and depth-of-field, but evanescent decay results in competition between these two important metrics. In this paper we use reflection to improve the tradeoff between global contrast and depth-of-field.

## 2. Theory of reflection imaging

*H*, which allows us to show some generic aspects of field evolution including reflection. We make a number of assumptions including 2D spatial dependence of the three-component fields with in-plane propagation and isotropic homogeneous media. These assumptions allow us to show important behavior with simpler theory and do not necessarily limit the results to those shown here.

*H(x, y)*in the imaging region bounded by 0≤

*x*≤

*d*, assuming knowledge of the object field

*H*(

*d, y*). The field can be rewritten as a superposition of sinusoidal plane-waves (implicitly exp(-

*iωt*)), which may be reflected at

*x*=0 due to differences in the wave-number

*k*in different media. The lateral propagation constant

*β*is conserved, and longitudinal propagation is

*β*>

*k*then

*α*is purely imaginary corresponding to evanescent components. The spectral components are represented by complex wave amplitudes, strictly written

*h*(

*x,β*), but henceforth we shall omit

*β*for spectral functions.

*β*↔

*y*, and ∗ denotes spatial convolution of the forward object

*H*(

^{f}*d, y*) with a line-spread function (LSF)

*G*that is a property of the imaging system (separation

_{H}*d*and mirror reflectivity

*r*). The Fourier transform of this LSF is an optical transfer function (OTF)

*g*, which we can use to write Eq. 1 in the spectral domain such that

^{f}_{H}*h*(

^{f}*x*)=

*g*(

^{f}_{H}h^{f}*d*).

*h*(

*d*)=

*h*(

^{f}*d*)+

*h*(

^{r}*d*). Explicitly we wish to find an OTF

*g*such that

_{H}*h*(

*x*)=

*g*(

_{H}h*d*), and it is not hard to show that

*g*=

_{H}*g*(

^{f}_{H}/g^{f}_{H}*x*=

*d*) where the denominator is evaluated at

*x*=

*d*. Rewriting in real-space, we can use

*H*(

*x, y*)=

*G*∗

_{H}*H*(

*d, y*) determine the total field throughout the imaging region in terms of the total field at the object (

*H*(

*d, y*)) and the new LSF (

*H*).

_{G}*r*is complex-valued quantity which, for convenience, can be mapped as a translation by a “distance” -2

*X*such that

*r*=±

*e*

^{-iα2X}, and thus the OTF is rewritten

*X*is constant and real, two special planes can be identified from the OTF. For a positive reflection

*x*=

*X*is a symmetry plane, and

*x*=-

*d*+2

*X*is the corresponding exact image plane due to reflection. Logically, for negative reflections we have anti-symmetry and an anti-image. The reflected image may actually lie outside the imaging region itself, but this virtual image still affects the fields inside the imaging region. If

*X*is not real but

*α*is either purely real or imaginary, it can be shown that Re(

*X*) is a conjugate symmetry plane where the magnitude is symmetric but the phase relationship of the images is complicated. Note carefully that the reflected image is exact so long as the conditions above are met, particularly the requirement for constant

*X*.

*X*from the reflectance

*r*and propagation constant

*α*is

*α*real) have multiple values of Re(

*X*), meaning there are multiple images interspersed with anti-images, resulting in the usual longitudinal standing waves which are generally undesirable. Evanescent components (

*α*imaginary) have only one value, which simplifies control of imaging.

*X*=

*d*/2), placing the exact reflected image of the object plane fields on the reflector surface (

*x*=0). However, note that the reflection mapping has an explicit dependence on

*α*(and hence on

*β*) which means that

*X*generally varies with spatial frequency. Hence it may not be possible to achieve exact images, but a possible strategy is to choose a symmetry plane based on the dominant spatial frequencies in the object. Alternatively, the special case

*r*=±1 sets the symmetry plane at the reflector surface (

*X*=0) regardless of

*α*.

### 2.1 Plasmonic reflectors

*x*<

*d*) having permittivity

*ε*and a semi-infinite mirror with permittivity

*ε*, so the relationship between media (

_{r}*k*) and vacuum (

*k*

_{0}) wave-numbers is

*k*

^{2}=

*εk*

^{2}

_{0}. In this case only TM polarization allows evanescent amplification, in which case the magnetic field is

*H*as given by Eq. (1) above. Actually, we are primarily interested in electric fields, and the

*x*and

*y*components of the electric field can be similarly calculated by their respective OTFs:

*e*(

_{y}*d*)=

*h*(

*d*)

*α/ωε*and

*e*(

_{x}*d*)=

*h*(

*d*)

*β/ωε*. Note that the spatial field

*E*has the same type of symmetry as

_{x}*H*, but

*E*is the opposite. Then the reflectivity is

_{y}*r*subscript indicates the reflector medium, and lack of a subscript indicates the imaging medium.

*r*) and an imaging-specific solution (in terms of symmetry plane

*X*).

*ε*and object-mirror separation

_{r}*d*are shown in Fig. 3. Analysis of Eq. (6) shows that the permittivity for the negative solution is usually smooth and relatively flat across a range of

*β*whereas the positive solution is not, so unless otherwise stated, only the negative solutions are plotted in Fig. 3. Referring to Fig. 3(a), the far evanescent limit (

*β*→∞) is given by

*ε*→-

_{r}*ε*which is independent of

*d*and is the same condition as required for plasmonic super-lensing [1

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

_{β→0}

*ε*=-

_{r}*ε*tan

^{2}(

*kX*), which determines the response for the low-spatial frequency (dc) image components; if this is matched to the far-evanescent limit we find a reasonable choice for the symmetry plane at

*X*=

*π*/(4

*k*), which implies quarter-wavelength for

*d*. We can also consider the dispersion of the image plane under fixed permittivity, shown in Fig. 3(b). It can be shown that there is a pole corresponding to the surface plasmon,

*ε*|→∞ which could be provided by metals, absorbers or high-index dielectrics, and the negative solution requires |

_{r}*ε*|→0 corresponding to metals at the plasma frequency.

_{r}## 3. Representative results

*H*

^{(1)}is the Hankel function of the first kind. This LSF corresponds to the sum of cylindrical contributions from the object and the mirror image. Unfortunately there is no simple expression for total objects or general interfacial reflections, and so we now look at numerical results generated via FFT of Eq. (4). In general the LSF should primarily be laterally narrow and secondarily be longitudinally uniform.

*d*we calculated a LSF in (

*x,y*) space, drew a contour level with the longitudinal minimum along the central ridge, and took the maximum width of the region enclosed by the contour. First, comparing the free-space case (

*r*=0) to the other cases, the overall width can be made to decrease significantly for a range of

*d*by the use of a reflector, highlighting the main point of this paper that reflections can improve image quality. But note that the width can also become much greater than for the no-reflection case if the imaging geometry is not chosen correctly. The no-reflection case (dashed) scales as width/

*d*~1, whereas with reflection (dotted) scaling goes like width/(

*d-X*)~1, at least up to

*d-X*~λ/4 where standing wave effects begin to make reflection undesirable. As expected, maximum performance is attained when the exact reflected image lies on the reflector surface (

*X*=

*d*/2), although there is some improvement when the symmetry plane lies on the reflector surface (

*X*=0). Now considering the

*ε*=-

_{r}*ε*case, the concept of an exact image is broken due to the dispersion of the symmetry plane. There is a virtual singularity at

*x*=-

*d*which has some characteristics of an image, but it is severely distorted so symmetry considerations are most useful for characterizing the depth of the extended image. The symmetry plane dispersion runs across 0≤

*X*≤

*λ*/8, so thicknesses in the range

*λ*/8≤

*d*≤

*λ*/4 are expected to be ideal. Indeed, when compared to no reflection there is significant improvement in uniformity and width, and performance often approaches that of the perfect case.

*E*have been shown. In general all types of transfer functions exhibit similar behavior (i.e. reflection narrowing) in some regime, but this is highly dependent on the symmetry of reflection for that component and in general each component should be considered separately. The choice of the particular type of transfer function depends on the nature of the physical object, but there is good justification of using total-to-total transfers. These can be directly applied to information provided by many simulations and we shall see soon that this is most appropriate for some common types of object.

_{y}*β/k*~1 are significantly damped and far-evanescent frequencies are actually least affected. Hence subdiffraction-limited exposures are not likely to be significantly degraded.

### 3.1 Binary absorber objects

**3**, 405–417 (2004) [CrossRef]

*ε*=-

_{r}*ε*have a critical effect on the dark plane, and we now show that this explains the strong correlation between absorber width

*a*and optimum

*d*that was previously observed [9].

*d*-

*a*curves (Fig. 6) by approximating the longitudinal evolution of the image along the medial dark-plane. Image evolution was estimated by convolving the total-to-total LSF for

*E*with an idealized total object field that is assumed to be independent of reflection. Two model objects were used in this work: one with delta functions at absorber edges approximating edge diffraction, and the other is a rectangular function representing geometric transmission (see Fig. 5(b)). Note that for numerical efficiency we have used only one edge in the models, which therefore correspond to a semi-infinite absorber screen. This is justified provided that the interaction between neighboring features is small, and also that the fields are symmetric about the center-line of the absorber. Given a coupling distance

_{y}*d*, these models allow us to simultaneously estimate behavior for all values of absorber width using a single convolution. We now describe how the optimization was accomplished by way of the edge diffraction example of Fig 6(a).

*x*=

*d*=0.125

*λ*and the mirror is at

*x*=0. If we use a single delta function for an edge object and ignore contributions from any neighboring edges, then Fig 6(a) also represents the blurring of the edge at (

*x,y*)=(0.125,0) into the region below the absorber (

*y*>0). Given an arbitrary absorber width

*a*, then the line

*y*=

*a*/2 is the dark plane for that absorber, and the global dark intensity for (

*d,a*) is the maximum intensity along the line. Minimizing global dark intensity against

*a*yields optimum

*a*for the given

*d*, which in this case is represented by the dashed line in Fig. 6(a) and the point at the intersection of the dashed lines in Fig. 6(b). The blue curve in Fig. 6(b) shows optimum

*a*as a function of

*d*for this edge diffraction model, and the cyan curve shows the result for a geometric model using a similar process (the LSF was convolved with a step function).

*d,a*) space. This shows that the model predicts the minimum absorber width at a given depth-of-field, or alternatively the maximum depth-of-field for a given absorber width. The map also shows that the coupling distance is less critical for wider absorbers, and that there is a second optimum coupling for narrower absorbers. While these additional features are not conveyed by the model curves shown, the model does indeed predict them.

### 3.2 Interferometric imaging

*β*, as shown in Fig. 7. In this case both electric field components should be considered (

*I*=|

*E*|

_{x}^{2}+|

*E*|

_{y}^{2})), and it can be shown that the image has planes of lateral symmetry that are null planes of either component. For integral order

*m*the field is given by

*E*) and the lower to normal-only planes (

_{y}*E*).

_{x}*r*=-

*e*

^{-iα2X}) yields

*X*is predicted, allowing maximum local contrast which is not achieved without reflection. Uniformity and hence global contrast is maximised when

*X*=

*d*/2, requiring an appropriate reflectance that could be achieved using strategies such as described by Eq. (6). Maximum depth-of-field with reflection (corresponding to

*Γ*=1) is found by solving:

*β/k*=1.2,

*d*=

*D*for the right-most image). Here we see the different lateral positioning of the dark nulls due to the dominance of either the normal field (positive reflection, left) or the tangential field (negative reflection, right). In this particular case the positive reflection has the best global contrast, followed closely by negative reflection, both of which significantly outperform the non-reflecting case. We note that tuning the permittivity to the required value could be realized by appropriate choice of metal for the reflector and illumination wavelength.

*β*>

*k*). The positive case (+

*r*) is slightly better because |

*β*|>|

*α*| when

*β*>

*k*/2, but as expected the negative case (-

*r*) has less dispersion. As there is only one spatial frequency involved, perfect reflection is readily achievable, but low dispersion is probably still helpful.

## 4. Conclusion

## Acknowledgments

## References and links

1. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

2. | J. S. Wei and F. X. Gan, “Dynamic readout of subdiffraction-limited pit arrays with a silver superlens,” Appl. Phys. Lett. |

3. | T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science |

4. | D. O. S. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express |

5. | N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science |

6. | R. J. Blaikie, M. M. Alkaisi, S. J. McNab, and D. O. S. Melville, “Nanoscale optical patterning using evanescent fields and surface plasmons,” Int. J. Nanoscience |

7. | D. B. Shao and S. C. Chen, “Surface-plasmon-assisted nanoscale photolithography by polarized light,” Appl. Phys. Lett. |

8. | D. B. Shao and S. C. Chen, “Numerical simulation of surface-plasmon-assisted nanolithography,” Opt. Express |

9. | M. D. Arnold and R. J. Blaikie, “Using surface-plasmon effects to improve process latitude in near-field optical lithography,” in Proceedings of the International Conference on Nanoscience and Nanotechnology, Brisbane, Australia, IEEE Press 06EX1411C , 548–551 (2006). |

10. | M. Schrader, M. Kozubek, S. W. Hell, and T. Wilson, “Optical transfer functions of 4Pi confocal microscopes: theory and experiment,” Opt. Lett. |

11. | B. W. Smith, Y. Fan, J. Zhou, N. Lafferty, and A. Estroff, “Evanescent wave imaging in optical lithography,” Proc. SPIE6154 (2006) |

**OCIS Codes**

(100.6640) Image processing : Superresolution

(110.5220) Imaging systems : Photolithography

(240.6680) Optics at surfaces : Surface plasmons

(260.3910) Physical optics : Metal optics

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: June 14, 2007

Revised Manuscript: August 20, 2007

Manuscript Accepted: August 21, 2007

Published: August 27, 2007

**Citation**

Matthew D. Arnold and Richard J. Blaikie, "Subwavelength optical imaging of evanescent fields using reflections from plasmonic slabs," Opt. Express **15**, 11542-11552 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-18-11542

Sort: Year | Journal | Reset

### References

- J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
- J. S. Wei and F. X. Gan, "Dynamic readout of subdiffraction-limited pit arrays with a silver superlens," Appl. Phys. Lett. 87, 211101 (2005). [CrossRef]
- T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, R. Hillenbrand, "Near-field microscopy through a SiC superlens," Science 313, 1595-1595 (2006). [CrossRef] [PubMed]
- D. O. S. Melville and R. J. Blaikie, "Super-resolution imaging through a planar silver layer," Opt. Express 13, 2127-2134 (2005) [CrossRef] [PubMed]
- N. Fang, H. Lee, C. Sun, X. Zhang, "Sub-diffraction-limited optical imaging with a silver superlens," Science 308, 534-537 (2005). [CrossRef] [PubMed]
- R. J. Blaikie, M. M. Alkaisi, S. J. McNab, D. O. S. Melville, "Nanoscale optical patterning using evanescent fields and surface plasmons," Int. J. Nanoscience 3, 405-417 (2004) [CrossRef]
- D. B. Shao and S. C. Chen, "Surface-plasmon-assisted nanoscale photolithography by polarized light," Appl. Phys. Lett. 86, 253107 (2005) [CrossRef]
- D. B. Shao and S. C. Chen, "Numerical simulation of surface-plasmon-assisted nanolithography," Opt. Express 13, 6964-6973 (2005). [CrossRef] [PubMed]
- M. D. Arnold and R. J. Blaikie, "Using surface-plasmon effects to improve process latitude in near-field optical lithography," in Proceedings of the International Conference on Nanoscience and Nanotechnology, Brisbane, Australia, IEEE Press 06EX1411C, 548-551 (2006).
- M. Schrader, M. Kozubek, S. W. Hell, T. Wilson, "Optical transfer functions of 4Pi confocal microscopes: theory and experiment," Opt. Lett. 22, 436-438 (1997). [CrossRef] [PubMed]
- B. W. Smith, Y. Fan, J. Zhou, N. Lafferty, A. Estroff, "Evanescent wave imaging in optical lithography," Proc. SPIE 6154 (2006)

## 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.