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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 8 — Apr. 11, 2011
  • pp: 7244–7261
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Interaction of highly focused vector beams with a metal knife-edge

P. Marchenko, S. Orlov, C. Huber, P. Banzer, S. Quabis, U. Peschel, and G. Leuchs  »View Author Affiliations


Optics Express, Vol. 19, Issue 8, pp. 7244-7261 (2011)
http://dx.doi.org/10.1364/OE.19.007244


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Abstract

We investigate the interaction of highly focused linearly polarized optical beams with a metal knife-edge both theoretically and experimentally. A high numerical aperture objective focusses beams of various wavelengths onto samples of different sub-wavelength thicknesses made of several opaque and pure materials. The standard evaluation of the experimental data shows material and sample dependent spatial shifts of the reconstructed intensity distribution, where the orientation of the electric field with respect to the edge plays an important role. A deeper understanding of the interaction between the knife-edge and the incoming highly focused beam is gained in our theoretical model by considering eigenmodes of the metal-insulator-metal structure. We achieve good qualitative agreement of our numerical simulations with the experimental findings.

© 2011 OSA

1. Introduction

The growing interest in highly focused optical vector beams requires a proper treatment of the polarization state of the beam, which strongly influences the size of the focal spot [1

1. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]

]. In particular, the role of azimuthal and radial polarization has been investigated both theoretically [2

2. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000). [CrossRef] [PubMed]

] and experimentally [3

3. R. Dorn, S. Quabis, and G. Leuchs, “The focus of light-linear polarization breaks the rotational symmetry of the focal spot,” J. Mod. Opt. 50, 1917–1926 (2003).

,4

4. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003). [CrossRef] [PubMed]

] and optimization strategies were proposed thereafter [5

5. G. Leuchs and S. Quabis, “Tailored polarization patterns for performance optimization of optical devices,” J. Mod. Opt. 53, 787–797 (2006). [CrossRef]

]. Furthermore, when focusing with high numerical aperture (NA) objectives, the symmetry of the focal spot is broken [3

3. R. Dorn, S. Quabis, and G. Leuchs, “The focus of light-linear polarization breaks the rotational symmetry of the focal spot,” J. Mod. Opt. 50, 1917–1926 (2003).

] for a linearly polarized beam and strong longitudinal components appear [4

4. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003). [CrossRef] [PubMed]

]. Therefore, a precise characterization of tightly focused laser beams is not just a challenge but is essential for further applications. In the literature many methods for beam characterization are described as e.g. the knife-edge method [6

6. J. A. Arnaud, W. M. Hubbard, G. D. Mandeville, B. de la Claviere, E. A. Franke, and J. M. Franke, “Technique for fast measurement of Gaussian laser beam parameters,” Appl. Opt. 10, 2775–2776 (1971). [PubMed]

, 7

7. A. H. Firester, M. E. Heller, and P. Sheng, “Knife-edge scanning measurements of subwavelength focussed light beams,” Appl. Opt. 16, 1971–1974 (1977). [CrossRef] [PubMed]

], a point scan method [8

8. M. B. Schneider and W. W. Webb, “Measurement of submicron laser beam radii,” Appl. Opt. 20, 1382–1388 (1981). [CrossRef] [PubMed]

] or a slit method [9

9. R. L. McCally, “Measurement of Gaussian beam parameters,” Appl. Opt. 23, 2227–2227 (1984). [CrossRef] [PubMed]

, 10

10. O. Mata-Mendez, “Diffraction and beam-diameter measurement of Gaussian beams at optical and microwave frequencies,” Opt. Lett. 16, 1629–1631 (1991). [CrossRef] [PubMed]

] etc. The data provided by the knife-edge method can be evaluated after the experiment in various ways as by employing an inversion algorithm involving a linear least-square method to measure the beam’s diameter [11

11. J. M. Khosrofian and B. A. Garetz, “Measurement of a Gaussian laser beam diameter through the direct inversion of knife-edge data,” Appl. Opt. 22, 3406–3410 (1983). [CrossRef] [PubMed]

], by performing a numerical differentiation of data in order to directly obtain the beam profile [12

12. G. Brost, P. D. Horn, and A. Abtahi, “Convenient spatial profiling of pulsed laser beams,” Appl. Opt. 24, 38–40 (1985). [CrossRef] [PubMed]

] or by using a direct fit with an error function [13

13. H. R. Bilger and T. Habib, “Knife-edge scanning of an astigmatic Gaussian beam,” Appl. Opt. 24, 686–690 (1985). [CrossRef] [PubMed]

]. Also other numerical approaches [14

14. M. A. de Araujo, R. Silva, E. de Lima, D. P. Pereira, and P. C. de Oliveira, “Measurement of Gaussian laser beam radius using the knife-edge technique: improvement on data analysis,” Appl. Opt. 48, 393–396 (2009).

] are used to characterize the beam. The knife-edge method is also applied to measure mechanical displacements in the nanometer range [15

15. D. Karabacak, T. Kouh, C. C. Huang, and K. L. Ekinci, “Optical knife-edge technique for nanomechanical displacement detection,” Appl. Phys. Lett. 88, 193122 (2006).

]. The application of an optical beam profiler without any moving parts using liquid-crystal displays is another example of the application of the knife-edge method [16

16. M. Gentili and N. A. Riza, “Wide-aperture no-moving-parts optical beam profiler using liquid-crystal displays,” Appl. Opt. 46, 506–512 (2007). [CrossRef] [PubMed]

] with a micro knife-edge scanner fabricated in a silicon-on-insulator substrate [17

17. Y. Chiu and J.-H. Pan, “Micro knife-edge optical measurements device in a silicon-on-insulator substrate,” Opt. Express 15, 6367–6373 (2007). [CrossRef] [PubMed]

]. Beside a single knife-edge, also a periodical array of slits was studied recently, demonstrating the transmission anomalies of TM-polarized light [18

18. Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Transmission of light through periodic arrays of sub-wavelength slits in metallic hosts,” Opt. Express 14, 6400–6413 (2006). [CrossRef] [PubMed]

].

The background of the conventional knife-edge method is the scalar diffraction theory, so the standard knife-edge method’s evaluation scheme is polarization independent. Thus, theoretically one dimensional beam scans by a knife-edge can be used in a variety of algorithms to retrieve two dimensional beam profiles with a Radon backward transform. However, when a highly focused two dimensional beam is profiled with a knife-edge, it is natural to ask ourselves, how well the conventional knife-edge method performs for vector-beams. The role of polarization in such beam measurements was studied theoretically for the knife-edge made from an ideal conductor [10

10. O. Mata-Mendez, “Diffraction and beam-diameter measurement of Gaussian beams at optical and microwave frequencies,” Opt. Lett. 16, 1629–1631 (1991). [CrossRef] [PubMed]

]. The first precise measurements of the highly focused beams using the knife-edge method were performed by R. Dorn et al [3

3. R. Dorn, S. Quabis, and G. Leuchs, “The focus of light-linear polarization breaks the rotational symmetry of the focal spot,” J. Mod. Opt. 50, 1917–1926 (2003).

]. However, this work already shows a very first experimental indication that without careful optimization of material and knife-edge parameters the knife-edge method can be polarization sensitive and the conventional evaluation may fail. Thus, motivated by this work we performed a systematical study on how careful various parameters of knife-edges for beam reconstruction have to be selected.

The structure of our paper is as follows. We start with the description of our experimental setup and explain the principle of our measurement. After that, we proceed with the discussion of our experimental findings. In the third chapter we discuss the physical mechanisms in the realizable knife-edge method and their effects on the outcome of measurements. Finally, we develop a theoretical model and present results of numerical simulations, which are compared with experimental findings.

2. The setup, principle of measurement and experimental results

2.1. Setup

Most of the experiments were performed at wavelengths from 500 up to 700 nm using a tunable femtosecond laser system from TOPTICA. Additionally, we performed measurements at a wavelength of 780 nm using a laser diode-based cw-system. The collimated linearly polarized Gaussian (TEM00) laser beam was focused onto the sample using a microscope objective with NA of 0.9. The full width at the half maximum (FWHM) of the intensity of the incoming beam was 3.1 mm filling 86% of the entrance pupil of the microscope objective. The sample was mounted onto a piezo stage to control its 3D-position with nanometer accuracy.

As sensors we used in-house fabricated p-i-n photodiodes. Opaque and pure material films were thermally deposited on a photodiode. Therefore periodic stripe-like structures were patterned using electron-beam lithography, where the slit and the stripe width is both equal (see Fig. 1). The thickness h was approx. 130 nm. For the Au samples we additionally patterned structures with thicknesses of approx. h = 70,100,190 nm. Forming structures directly on the detector surface allows the detection of a large solid angle in transmission. The materials of the opaque films were gold (Au), titanium (Ti) and nickel (Ni). To obtain the geometric parameters of the structures after finishing all measurements we cut each structure (knife-edge) by using the focused-ion-beam (FIB) technique at three positions and average values of its width at the bottom (d = 2.005 μm) and slope angle (α = 15°) were estimated (see Fig. 1). The film thickness was measured with high accuracy by means of an atomic force microscope (AFM).

Fig. 1 Electron-micrographs of one of the gold samples investigated in the experiments. The knife-edge width and film thickness were determined by performing cuts with a focused ion beam (FIB) machine. The width of the investigated knife-edge is d = 2.0 μm, the slit width is l = 2.0 μm.

2.2. Principle of the measurement

The principle of the measurement is depicted in Fig. 2. The experiments are performed using a highly focused linearly polarized TEM00-mode. We investigate two polarization directions of the incoming beam relative to the knife-edge (in the xy plane). At first the electric field is oriented perpendicularly (s-polarization) and then parallel (p-polarization) to the wall of the knife-edge (see Fig 2 (a), (b)). The investigated laser beam is blocked stepwise by the edge of an opaque metal stripe that is building a single knife-edge. The photocurrent generated inside the photodiode is recorded for each sample position x 0 (see Fig. 2 (c)). It is proportional to the power P of the diffracted field detected by the photodiode
Fig. 2 Schematic depiction of the knife-edge method for a two-dimensional beam (a,b). Typical beam profiling data (c) and their derivatives (d). The state of polarization always refers to the orientation of the electric field.
P=0Sz(x,x0)dx,
(1)
where Sz is the z-component of the Poynting vector of the field at the photodiode. In the conventional knife-edge method the derivative of the photocurrent curve with respect to the sample position x 0 (see Fig. 2 (d)) reconstructs a beam projection on one axis [6

6. J. A. Arnaud, W. M. Hubbard, G. D. Mandeville, B. de la Claviere, E. A. Franke, and J. M. Franke, “Technique for fast measurement of Gaussian laser beam parameters,” Appl. Opt. 10, 2775–2776 (1971). [PubMed]

, 7

7. A. H. Firester, M. E. Heller, and P. Sheng, “Knife-edge scanning measurements of subwavelength focussed light beams,” Appl. Opt. 16, 1971–1974 (1977). [CrossRef] [PubMed]

], so the width and position of the projection can be determined.

Since the measurements with s- and p-polarized beams are performed one after another, a thermal drift of the sample between measurements for each polarization could introduce a systematic error. Therefore we profile the beam by two adjacent knife-edges, so the effects of sample drifts between measurements with different polarizations cancel out (see Fig. 2 (c)). After that the intensity of the beam projection is reconstructed from the recorded photocurrent and both the position and the width of the beam are evaluated. Two parameters ds and dp define the distance between the peaks of the reconstructed beam profiles (see Fig. 2 (d)). In the conventional knife-edge method no polarization effects are present, so the distance between the peaks for s- and p-polarized beam projections is equal (ds = dp). Thus, a non zero value for dsdp indicates the presence of the polarization effect.

Measurements for each polarization are performed at 22 different positions of the knife-edge. Each scan line in x-direction consists of 400 steps with a step size of 10 nm. For each position of the edge 10 data points are measured and averaged to improve the signal-to-noise ratio. The photocurrent data was additionally filtered using a Savitzky-Golay smoothing algorithm (11 points) before the photocurrent curve was differentiated.

Most scans are performed at knife-edges with a width of approx. 2 μm. Measurements on wider knife-edges (approximately 3 and 4 μm) do not result in any quantitative change within the experimental accuracy. Hence, in what follows we solely discuss results obtained for the 2 μm structures.

During the measurement the relative position of the sample is automatically calibrated every 15 min by the determination of the position of a 500 nm hole. The hole is etched into one of the metal structures which is situated some microns away from the knife-edges. The detection of the hole position is estimated by the maximum of transmission through it. Thus long term stability is obtained and the actual sample position is always determined with a precision better than 50 nm.

2.3. Experimental results

We start our discussion with experimental results for a Au knife-edge with a thickness (or height) of h = 130 nm. The beam profiling results ds and dp are presented in Fig. 3 (a) for two polarization states (s and p) and six particular wavelengths. In the conventional knife-edge theory, the zero crossing of the second derivative of the photocurrent curve coincides with the position x in the photocurrent curve, where half of the maximum photocurrent is reached. It physically indicates the position of the knife’s edge [6

6. J. A. Arnaud, W. M. Hubbard, G. D. Mandeville, B. de la Claviere, E. A. Franke, and J. M. Franke, “Technique for fast measurement of Gaussian laser beam parameters,” Appl. Opt. 10, 2775–2776 (1971). [PubMed]

, 7

7. A. H. Firester, M. E. Heller, and P. Sheng, “Knife-edge scanning measurements of subwavelength focussed light beams,” Appl. Opt. 16, 1971–1974 (1977). [CrossRef] [PubMed]

]. So, the total distances between the peaks of the reconstructed profiles ds and dp can be interpreted intuitively as the width of the single stripe if no polarization effects are present.

Fig. 3 Distance between the peaks ds, dp versus wavelength λ derived from the experimental data for the s- and p-polarizations. The actual knife-edge width measured by SEM is shown by the gray bar (a). Difference of the peak positions of the photocurrent’s derivative dsdp versus wavelength λ for various Au samples (b). Difference of the peak positions dsdp of the photocurrent’s derivative versus sample height h for Au samples at various wavelengths (c). Difference of the peak positions dsdp of the photocurrent’s derivative versus wavelength λ for Ti and Ni samples (d). The colored lines represent results of numerical simulations. The dash-dotted lines in (b) represent a situation with noise artificially added to the photocurrent and smoothed with the same filter as in the experiment. The colored points represent experimental results.

As one can see in Fig. 3 (a), the optical beam profiling reveals the polarization and wavelength dependence of the derived values ds and dp. It should be noted that the obtained knife-edge width for the p-polarized beam remains almost constant as the wavelength changes, although it is actually about 100 nm larger than the value determined by the SEM. Concerning s-polarization the knife-edge width ds depends on the wavelength and monotonically decreases as the wavelength increases. To characterize the polarization effect on the measurement we plot the value dsdp which also depends on the wavelength (see Fig. 3 (b)).

To check the reproducibility of the results several Au samples were produced. The relative shifts obtained in the measurements are presented in Fig. 4 (a) for h = 130 nm (sample No. 1–7) and (b) for h = 70 nm (sample No. 8, 9). The results on sample No. 1 were already shown in Fig. 3 (b). Samples No. 1–3 were fabricated using the same fabrication steps. Furthermore we investigated the dependency of the measured relative shift on the recrystallization of the knife-edge material (Au) by additionally tempering (for 30 s. at 400 C°) the sample after fabrication. By tempering the grainy metal surface becomes smooth and the surface roughness decreases. The results for the tempered Au samples are shown in Fig. 4 (a) (sample No. 4–6). In Fig. 4 (a) one can see that by tempering the knife-edges the relative shifts (dsdp) derived from the measurements are slightly decreased for short wavelengths, although, the overall qualitative wavelength dependency is preserved (compare samples 4–6 with 1–3). The agreement between the experiment and the theory is slightly better for the tempered sample of the thickness h = 130 nm, compare with Fig. 3 (b).

Fig. 4 Difference in the peak positions dsdp of the photocurrent’s derivative vs wavelength λ for various Au samples of 130 (a) and 70 (b) nm thickness. (T) means tempered samples (30 sec. 400C°). (S) means a sample with a standing-alone structure to investigate a possible influence of the opposite wall on the knife-edge measurement. In this case other structures are missing (a,b). Theoretical difference of the peak positions of the photocurrent’s derivative (dsdp) versus slit width l for Au sample (h = 130 nm) at various wavelengths (c).

As one can see measurement results depend on many parameters including the surface quality and roughness of the knife-edge. Therefore it is important to perform measurements of the relative shift at different positions of the knife-edge, which was done in all measurements performed here.

Fig. 5 Dependence of the ratios wp/λ (a), ws/λ (b) on the wavelength λ for Au samples of varying thickness. Dependence of the ratios ws/λ, ws/λ on the on the wavelength λ for Ni (c) and Ti (d) samples of comparable thickness of the opaque film h = 130 nm. The colored curves represent results of numerical simulations, the colored points - of experimental measurements. Parameters of the model are w 0 p/λ = 0.9w 0 s/λ = 0.55. The dashed lines represent the FWHM’s of the squared electric field estimated from the Debye integrals [23].

3. Theoretical model

3.1. Realistic knife-edge versus perfect knife-edge

We start the theoretical discussion with a brief reminder of the knife-edge method basics. In the original work [6

6. J. A. Arnaud, W. M. Hubbard, G. D. Mandeville, B. de la Claviere, E. A. Franke, and J. M. Franke, “Technique for fast measurement of Gaussian laser beam parameters,” Appl. Opt. 10, 2775–2776 (1971). [PubMed]

] the following assumptions were made: the incident beam is paraxial, the knife-edge is made from a perfect conductor and no losses are present. For sake of simplicity we consider a two-dimensional situation. The incident field in the spectral domain is represented as plane waves with amplitudes S(kx) traveling at different angles α = arcsinkx/k. Here k = ω/c 0 is the wave vector, with kx and kz being the transverse and longitudinal components of the wave vector, ω is the frequency and c 0 is the speed of light in vacuum. In the first part of Figure 6 (a) a single plane wave component k = (kx, kz) = k (± sinα,–cosα) of the spatial spectrum S(kx) of the Gaussian beam is shown. Three possible interaction scenarios are shown in different colors. First, the plane wave part, represented by the red rays is blocked by the knife-edge and reflected from the metal surface. Second, the partial ray shown in green is not affected by the knife-edge and is properly detected. Furthermore, the wave part shown in blue experiences reflection (r = 1) from the wall of the knife-edge, however its amplitude remains the same and is therefore also detected properly.

Fig. 6 Schematic depiction of a single plane wave component k = (±kx, kz) impinging on a knife-edge. Here k 1 = (kx, kz), k 2 = (−kx, kz) and r = (x, z) (a). Sketch of the considered structure (b).

Next, we introduce a finite conductivity of the knife-edge but still consider paraxial beams, i.e. reflection r ≈ 1 (see Fig. 6 (a)). At the side wall the boundary conditions for s- and p-polarizations are Ex=εEx(2) and Ey=Ey(2), where superscript 2 denotes the electric field component in the knife-edge and ε is a dielectric constant of the knife-edge material. The fields in the knife-edge are decaying exponentially as exp(−κkx), where κ = Imε 1/2. They still contribute to the total detected power T as an additional term exp(x02d02)Reυκk, where d 0 is the FWHM of the intensity |E|2 of the scalar beam, υ = ε −1 for a TM incoming field and υ = 1 for the TE field. The differentiation of the transmission gives us
Tx0=exp(x02d02)2x0d02exp(x02d02)Reυκk
(2)
and can be interpreted as the beam projection (compare with Section 2). The second term leads to an asymmetric profile and results in a shift of the projection maximum, which therefore does not longer correspond to the half maximum of the transmission curve T. The parameter υ depends on the polarization relative to the edge. Thus, the shift of the maximum in the projection curve is polarization dependent in this simple model, which leads, in general, to a non-zero relative shift dsdp.

3.2. Field representations and eigenvalue problem

We consider the structure represented in Fig. 6 (b), where two adjacent knife-edges build a wide (l > λ) slit. The space is divided into three regions, of which region I (half space z > 0) and III (half space z < −h) are assumed to be dielectric and homogeneous with dielectric constants ε 1 and ε 3, respectively. The intermediate region II (0 ≥ z–h) of thickness h consists of an opaque (i.e. the skin-effect depth is smaller than the height h) material (for x ≤ 0 and x > l) with a dielectric constant ε 2 and empty space (for l > x > 0), with the same dielectric constant as region I, which we put to unity ε 1 = 1. The considered structure is piece-wise homogeneous and in each homogeneous part of the structure solutions of the Helmholtz equation describe the field propagation. A continuous solution up to first order of the derivatives is derived by combining solutions of homogeneous parts using appropriate boundary conditions. In the general case, in the two-dimensional waveguide structure with ohmic losses in the walls the coupling between transverse electric (TE) and transverse magnetic (TM) modes was reported [24

24. A. E. Karbowiak, “Theory of imperfect waveguides: the effect of wall impedance,” Proc. IEEE Part B: Radio Electron. Engin. 102, 698 – 708 (1955).

]. So, as a further simplification, we restrict our consideration to a planar approximation, assuming that the incident field does not change in y direction, so the Helmholtz equation in each homogeneous part can be written in it’s two-dimensional form
(2x2+2z2+kε2){E(x,z)H(x,z)}=0,kε=ωεc0,
(3)
ε denotes the respective dielectric constant. The solution of equation (3) consists of two independent classes: transverse electric and transverse magnetic modes, propagating in z direction.

We start our consideration with TE modes of the Region II. The solution of the two-dimensional Helmholtz equation (3) for the electric field E in this case has only one nonvanishing component of the electric field Ey parallel (p-polarization) to the slit walls
Hx=βec0ωZ0Ey,Hz=ic0ωZ0Eyx,Ex=Ez=Hy=0.
(4)
Here βe is the effective propagation constant of the electric field mode in the empty part of region II (for the sake of brevity we will call it slit) and Z 0 is the vacuum impedance.

Next, the transverse magnetic (TM) modes of the metal slit have one nonvanishing component of the magnetic field Hy and electric field components are perpendicular (s-polarization) to the walls of the knife-edges
Ex=βmωε0ε(x)Hy,Ez=iωε0ε(x)Hyx,Hx=Hz=Ey=0.
(5)
Here βm is the propagation constant of the magnetic field mode in the slit and ε(x) = 1 for 0 < x < l and ε(x) = ε 2 elsewhere.

In general, the eigenmodes of the metal-insulator-metal waveguide can be divided into the two classes: the localized and nonlocalized solutions of Eq. 3, see Ref. [21

21. B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007). [CrossRef]

]. The first class has a discrete spectrum of the eigenvalues and represents the field localized in the slit. The second class has a continuous spectrum of the eigenvalues and is necessary to ensure the boundary conditions on the metal surface x ∉ (0, l).

In what follows we will evaluate boundary conditions between the different regions I and II or II and III in Fourier space. Therefore we derive the Fourier transform of the eigenfunctions in the slit. The Fourier image of the eigenfunction πν(x) can be written as a sum of two images. The image Φν represents the field in the walls, and Ψν - the field in the slit
Πν(kx)=Φν(kx)+Ψν(kx),Φν(kx)=[1(1)νeγ1*l]γ2*+kxi(1)νeikxl(γ2*kxi)2πNν(γ2*2+kx2),Ψν(kx)=(γ1*+kxi)(e(γ1*ikx)l1)(1)ν(γ1*kxi)(eγ1*leikxl)2πNν(γ1*2+kx2).
(12)

3.3. Boundary value problem at the interfaces and energy flow between regions

In this section we discuss the boundary value problem at the entrance and exit interfaces of the slit. The eigenfunctions of the Helmholtz equation in regions I and II are those of free space, namely plane waves. On the other hand the eigenfunctions (6) of region II have discrete eigenvalues. We note, that for the optimal convergence to be achieved, the continuity of the tangential components has to be tested differently for TE and TM polarizations [27

27. A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt 34, 511–538 (1987). [CrossRef]

]. However, for the sake of brevity, we employ here an unified and polarization-independent approach. To match the fields in the three regions the problem will be investigated in the Fourier space.

We assume, that the electric (for TE case) or magnetic (for TM case) fields in regions I and III can be expressed as
FI,III(x,z)=12πS(kx,z)exp(ikxx)dkx,
(13)
here S(kx, z) is a Fourier image of the field. The slit is illuminated from the top. Therefore in the region above the first interface, the spatial spectra S can be expressed as consisting of two parts. One part describes the field propagating into the slit Sin, the second part the field reflected from the slit Sre f
S(kx,z)=Sin(kx)exp[iβ1(kx)z]+Sref(kx)exp[iβ1(kx)z].
(14)
Here β1=ε1k2kx2 is the longitudinal component of the wave vector and Reβ ≥ 0, Imβ ≥ 0. We assume an incoming wave with Gaussian distribution of the y component in the transverse plane. So at the entrance the field Ey (Hy) is
FG(x,z=0)=exp[(xx0)2/W02],
(15)
here W 0 is the beam waist radius and x 0 is the displacement from the center of the coordinates, v which in our case coincides with the left wall. The FWHM w 0 can be found as w0=2ln2W0. We note, that the FWHM w of the total electric field for the TM polarization is larger than w 0 due to the presence of the Ez component. The spatial Fourier spectra of the field (15) is
Sin(kx)=W02exp(ikxx0)exp(kx2W02/4).
(16)

In region III light only propagates from the slit into the substrate Strans
S(kx,z)=Strans(kx)exp[iβ3(kx)z],
(17)
here β3=ε3k2kx2.

The electric and magnetic fields in region II consist of forward and backward propagating eigenmodes πν. Thus, the resulting field is represented by the sum
FII(x,z)=ν=1[aνeiβνz+bνeiβν(z+h)]πν(x)=ν=1Fν(z)πν(x),
(18)
with the expansion coefficients aν, which describe a field propagating down to the substrate, whereas the bν represent a reflected field.

The energy conservation law can be written as
Pbeam=Ptrans+Pref+Pabs,
(19)
where Pbeam is the power of the incoming beam. Ptrans is the power transmitted into the region III through the slit. Pre f is the power reflected from the slit and Pabs is the power losses in the slit. After some math the reflection R from the slit and transmission T through the slit can be expressed in terms of the spatial spectra of incoming and outgoing fields as follows
R=k01k01β1(kx)|Sref(kx)|2dkxk01k01β1(kx)|Sin(kx)|2dkx,T=Re(υ3)k03k03β3(kx)|Strans(kx)|2dkxRe(υ1)k01k01β1(kx)|Sin(kx)|2dkx,
(20)
where k012=ε1k0, k032=ε3k0. υi = 1 for TE field and υi=εi1 for TM field. The expansion coefficients aν and bν of the field inside the slit along with the spatial spectra Strans, Sre f are the unknowns.

The problem of finding the unknowns is solved by implying the continuity of electric and magnetic fields at the two interfaces (z = 0 and z = –h). Traditionally, the procedure of the imposition of the continuity is different for TE and TM polarizations (see Ref. [28

28. J. Sumaya-Martinez, O. Mata-Mendez, and F. Chavez-Rivas, “Rigorous theory of the diffraction of Gaussian beams by finite gratings: TE polarization,” J. Opt. Soc. Am. A 20, 827–835 (2003). [CrossRef]

, 29

29. O. Mata-Mendez, J. Avendano, and F. Chavez-Rivas, “Rigorous theory of the diffraction of Gaussian beams by finite gratings: TM polarization,” J. Opt. Soc. Am. A 23, 1889–1896 (2006). [CrossRef]

] for examples). The main reasoning behind that is an optimal convergence. Nevertheless, for the sake of brevity, we will use the same projections for two different polarizations. This will slow the convergence, but a compact and polarization independent approach can be used.

In addition to the continuity of Ey and Hy, the derivative of the electric field and the derivative ε−1Hy/∂z of the magnetic field are continuous as well. With the help of equations (1314) and (18) a pair of equations is obtained at the first interface (z = 0):
Sref(kx)=1iβ1(kx)μFμz(0)Θμ(kx)+Sin(kx),2Sν=μ[iFμz(0)Jμν(1)+Fμ(0)Gμν],
(21)
where the sums run from μ = 0 for TM and from μ = 1 for TE polarization to infinity.

The matching of fields at the second interface with help of Eqs. (17) and (18) gives us a similar system of equations for the unknowns Strans (kx) and aμ, bμ
Strans(kx)=exp[iβ3(kx)h]μFμ(h)ϒμ(kx),0=μ[iFμz(h)Jμν(3)Fμ(h)Gμν],
(22)
where
Θμ(kx)=υ2υ1Φμ(kx)+Ψμ(kx),Jμν(1)=β11(kx)Θμ(kx)Πν(kx)dkx,Sν=Sin(kin)Πν(kx)dkx,ϒμ(kx)=υ1υ3Θμ(kx),Jμν(3)=β31(kx)ϒμ(kx)Πν(kx)dkx,Gμν=δμνϒν(kx)Πν(kx)
(23)
with Πν defined by equation (12). The integral Sν is an overlap integral of the incident beam and the eigenmode of the slit and the integral Jμν(1,3) describes optical and geometrical properties of the slit. The overlap integral defines the behavior of the reflected and transmitted radiation.

The explicit expressions of Fμ and ∂Fμ/∂z are
Fμ(0)=aμ+bμeiβμh,Fμz(0)=iβμ(aμ+bμeiβμh),Fμ(h)=aμeiβμh+bμ,Fμz(h)=iβμ(aμeiβμh+bμ).
(24)
We substitute those expressions into the second equations of the equations system (21–22) and get
2Sν=μ(Gμν+βμJμν(1))aμ+eiβμh(GμνβμJμν(1))bμ,0=μeiβμh(GμνβμJμν(3))aμ+(Gμν+βμJμν(3))bμ.
(25)

Now, we rewrite Eq. (25) in the following matrix form
2S=Na(1)a+Nb(1)b,0=Na(2)a+Nb(2)b,
(26)
with the matrices N being defined as
Na(1)=diag(βμ)J1+G,Nb(1)=diag(eiβμh)[Gdiag(βμ)J1],Na(2)=diag(eiβμh)[diag(βμ)J3G],Nb(2)=diag(βμ)J3G.
(27)
They solve the resulting matrix equations for unknown vectors a and b.

Now, from Eqs. (21) and (22), we obtain the unknown Fourier images Strans and Sre f. In particular, the transmitted power T, see (20), can be expressed as
T=Reυ3Reυ1PbeamReμνiFμz(h)Fμ*(h)Hμν(2),
(28)
where
Hμν(2)=k03k03ϒμ(kx)Πν*(kx)dkx.
(29)

The physical meaning of the derived formulation can be explained as follows. The propagation constants βμ are complex, so each eigenmode is attenuated while it propagates from the entrance to the exit. The incoming field excites eigenmodes in the slit. The strength of the excitation is defined by the overlap of the eigenmode and the incoming field given by a certain distribution of transverse wave vectors kx. This means, that the slit acts as a complex spatial filter, acting differently on the spatial frequencies kx according to the damping of the respective mode excited in the slit. The higher the value of the initial kx the stronger is the interaction of the light with the slit walls. Those modes are damped even stronger for deeper slits. If a certain thickness h is exceeded only the first mode will reach the exit of Region II, effectively leaving mainly two spatial frequencies kx = ±iγ 1, kx = ±iγ 2 in the transmitted signal (see Eq. (12)). This holds true in general for both polarizations with one notable difference. For s-polarization besides the propagating modes there are two plasmonic modes, which are exponentially localized at the slit walls. For narrow slits the plasmons are damped less than all the other modes. Hence, the electric field distribution at the photodiode may develop two distinct spatial peaks - each at it’s wall of the slit.

3.4. Numerical simulations and discussion

In the following section we present and discuss the results of our numerical simulations based on our theoretical model. For a given value of the beam displacement x 0 the procedure of finding the unknowns a and b (see Eq. (26)) was as follows. As a first step, roots of Eqs. (7) were calculated by built-in iterative MatLab procedures. The values of the propagation constant βν for a perfectly conducting wall were used as an initial guess. For plasmonic modes, an initial guess was the propagation constant βp=ωc[ε2/(1+ε2)]1/2 of a single plasmonic mode propagating at the interface between the knife-edge walls and the slit. The matrix dimensions were chosen in such a way that the energy conservation criterion (19) was satisfied with a relative accuracy of 10−8. In general, the typical length of the unknown vectors was about 40–50 elements due to the beamwidth of the incident field close to the wavelength. All dielectric constants were taken from [30

30. M. J. Weber, Handbook of optical materials (CRC Press, 2003).

]. We carefully choose free parameters of the theory (w 0 p = 0.9w 0 s = 0.55λ) to achieve the best possible agreement with our experimental data. The FWHM of the squared electric fields in the model are wp = 0.8ws and slightly differ from the expected in our setup [23

23. B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. R. Soc. A 253, 358 – 379 (1959). [CrossRef]

].

As a numerical result the electric field distribution at a knife-edge maid from gold is presented in Fig. 7 for two different wavelengths and two polarizations (s and p). We remind here, that for small values of z the electric field distribution is approximate, see Section 3.2. The beam is impinging from the top and the thickness of the knife-edge is h = 200 nm. The observed patterns are similar to those observed during the diffraction of the beam from a wedge, see for example [31

31. Y. Z. Umul, “Scattering of a Gaussian beam by an impedance half-plane,” J. Opt. Soc. Am. A 243159–3167 (2007). [CrossRef]

]. For both states of polarization beam diffraction is observed (see Fig. 7). in the case of an s-polarized incoming beam, the excitation of plasmonic modes at the edge is expected and can be seen in Fig. 7 (b,d). In all graphs we see a shift of the electric field maximum introduced by reflection from the side wall of the knife surface, as already suggested in the discussion in Section 3.1. The beam reaches the photodiode at different positions. In all cases some part of the beam penetrates the knife-edge and reaches the photodiode as well. As a main difference, the part of the field attributed to the plasmonic modes is largest at λ = 780 nm for s-polarization, whereas at λ = 532 nm the plasmonic modes are attenuated comparatively fast due to increased absorption. For that reason localization of an s-polarized field is less strong in the latter case.

Fig. 7 Absolute value of the electric field |E|/|E 0| distribution on one of the l = 2 μm wide slit walls for TE (a,c) and TM (b,d) radiation at λ = 532 nm (a,b) and 780 nm (c,d). The center of the incident beam is at x = 0. The Au knife-edge is situated at x < 0 and the beam propagates in negative z-direction. The parameters of the model are w 0 p = 0.9w 0 s = 0.55λ.

In addition to the Au sample we consider Ni and Ti samples of the same thickness (h = 200 nm) and illustrate the importance of plasmonic modes in the transmission of the TM fields (s-polarized) at next. The amount of power transmitted by the plasmonic modes as a function of beam displacement is plotted in Fig. 8 (a). We see that due to their spatial extent into the metal knife-edge plasmonic modes have a noticeable influence on the transmission, even, if most of the beam is blocked. So, for a gold film at λ = 532 nm they account for up to 90% of the transmitted power for a beam impinging on the slit walls. For Ni and Ti samples this value decreases to nearly 70%, while at λ = 780 nm the Au-knife-edge maintains it’s plasmonic properties. The last point we have to note is that still a non-zero amount of energy is transmitted by plasmons even for the beam centered in the slit. This happens due to the excitation of the coupled plasmons pair on both walls of the slit.

Fig. 8 a) Fraction of energy transmitted by plasmons Tp/T as a function of the beam displacement x 0 for various samples of the same thickness h = 200 nm and width l = 2 μ m at λ = 532 nm (solid lines) and λ = 780 (dashed lines). b–d) Dependence of the angular spectral width Δkx of the transmitted signal’s spectra Strans on the beam displacement x 0. Thickness of the Au film is h = 200 nm, the slit width is l = 2 μm, the wavelengths are λ = 532 nm (b), 633 nm (c), 780 nm (d). The spectral FWHM Δkx is determined by fitting a Gaussian function Sfit(kx,x0)=S0exp[2ln2(kxkx0)2/Δkx2] to the spectral distribution Strans (kx, x 0). For comparison the black line represents the same dependence for an perfect knife-edge. Parameters of the theory are w 0 p = 0.9w 0 s = 0.55λ.

As a further result of our theoretical calculations we see that a knife edge acts as a filter of the incoming beams spatial spectrum. To have a reference we start with a pair of perfect knife-edges, see Eq. (1), for which the spatial spectrum of the transmitted signal is Strans(kx,x0)=0lFG(x)exp(ikxx)dx. For convenience, we determine the spectral FWHM ωtrans by fitting a Gaussian function Sfit(kx,x0)=S0exp[2ln2(kxkx0)2/Δkx2] to the resulting spectral distribution Strans (kx, x 0). The spectral profile remains symmetric for an perfect knife-edge and the dependence of the angular width Δkx is presented in Fig. 8 (b–d). Our intention here is to compare the spectral width of the signal chopped by the real knife-edge to the spectral FWHM of the same signal chopped by the perfect knife-edge. During beam pro-filing with a perfect knife-edge the spatial spectrum broadens, when the beam is blocked by the knife-edge. However, the spectral width of the signal blocked by the other knife-edges is slightly larger compared to the perfect case, when the beam is in the middle of the slit. Nevertheless, as the beam approaches the knife-edge, the spectral width becomes much smaller than expected for a perfect knife-edge. This is a direct indication of the filtering of spatial frequencies. So, dependent on the polarization the reconstruction may result in a beam profile wider than expected, if other effects can be neglected.

4. Conclusions

In conclusion, the knife-edge method at nanoscale was investigated in detail. We experimentally as well as theoretically investigated the knife-edge technique for reconstructing the intensity profile of highly focused linearly polarized beams. Different thicknesses and edge materials were studied. It was demonstrated that a variety of polarization sensitive physical effects can influence the successful beam reconstruction: The excitation of plasmonic modes, the polarization dependent reflection from the wall of the knife-edge and a non-zero power-flow through the edge material. Thus, the reconstructed beam profile strongly depends on the used material, the wavelength and the thickness of the knife-edge. The peak position of the conventionally reconstructed beam as well as its width is affected by the interaction of the knife-edge with the measured beam. For the fabrication of knife-edges for highly accurate beam reconstruction a suppression of plasmonic modes is crucial along with minimization of a polarization dependent reflection from the inner side of the knife-edge. For pure materials the suggested workaround can be the use of very thin metal films. Another possibility to minimize the influence of the plas-monic modes and the polarization dependent reflection is the use of films with rough internal structure, where the reflection is diffuse and plasmons are suppressed (compare [3

3. R. Dorn, S. Quabis, and G. Leuchs, “The focus of light-linear polarization breaks the rotational symmetry of the focal spot,” J. Mod. Opt. 50, 1917–1926 (2003).

, 4

4. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003). [CrossRef] [PubMed]

]).

Acknowledgments

The support by the DFG funded Erlangen Cluster of Excellence “Engineering of Advanced Materials” (EAM) is gratefully acknowledged. We thank Stefan Malzer, Isabel Gäßner, Olga Rusina, Irina Harder and Daniel Ploß for their valuable support in preparing the samples. S. Orlov acknowledges the support by the Humboldt Foundation.

References and links

1.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]

2.

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000). [CrossRef] [PubMed]

3.

R. Dorn, S. Quabis, and G. Leuchs, “The focus of light-linear polarization breaks the rotational symmetry of the focal spot,” J. Mod. Opt. 50, 1917–1926 (2003).

4.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003). [CrossRef] [PubMed]

5.

G. Leuchs and S. Quabis, “Tailored polarization patterns for performance optimization of optical devices,” J. Mod. Opt. 53, 787–797 (2006). [CrossRef]

6.

J. A. Arnaud, W. M. Hubbard, G. D. Mandeville, B. de la Claviere, E. A. Franke, and J. M. Franke, “Technique for fast measurement of Gaussian laser beam parameters,” Appl. Opt. 10, 2775–2776 (1971). [PubMed]

7.

A. H. Firester, M. E. Heller, and P. Sheng, “Knife-edge scanning measurements of subwavelength focussed light beams,” Appl. Opt. 16, 1971–1974 (1977). [CrossRef] [PubMed]

8.

M. B. Schneider and W. W. Webb, “Measurement of submicron laser beam radii,” Appl. Opt. 20, 1382–1388 (1981). [CrossRef] [PubMed]

9.

R. L. McCally, “Measurement of Gaussian beam parameters,” Appl. Opt. 23, 2227–2227 (1984). [CrossRef] [PubMed]

10.

O. Mata-Mendez, “Diffraction and beam-diameter measurement of Gaussian beams at optical and microwave frequencies,” Opt. Lett. 16, 1629–1631 (1991). [CrossRef] [PubMed]

11.

J. M. Khosrofian and B. A. Garetz, “Measurement of a Gaussian laser beam diameter through the direct inversion of knife-edge data,” Appl. Opt. 22, 3406–3410 (1983). [CrossRef] [PubMed]

12.

G. Brost, P. D. Horn, and A. Abtahi, “Convenient spatial profiling of pulsed laser beams,” Appl. Opt. 24, 38–40 (1985). [CrossRef] [PubMed]

13.

H. R. Bilger and T. Habib, “Knife-edge scanning of an astigmatic Gaussian beam,” Appl. Opt. 24, 686–690 (1985). [CrossRef] [PubMed]

14.

M. A. de Araujo, R. Silva, E. de Lima, D. P. Pereira, and P. C. de Oliveira, “Measurement of Gaussian laser beam radius using the knife-edge technique: improvement on data analysis,” Appl. Opt. 48, 393–396 (2009).

15.

D. Karabacak, T. Kouh, C. C. Huang, and K. L. Ekinci, “Optical knife-edge technique for nanomechanical displacement detection,” Appl. Phys. Lett. 88, 193122 (2006).

16.

M. Gentili and N. A. Riza, “Wide-aperture no-moving-parts optical beam profiler using liquid-crystal displays,” Appl. Opt. 46, 506–512 (2007). [CrossRef] [PubMed]

17.

Y. Chiu and J.-H. Pan, “Micro knife-edge optical measurements device in a silicon-on-insulator substrate,” Opt. Express 15, 6367–6373 (2007). [CrossRef] [PubMed]

18.

Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Transmission of light through periodic arrays of sub-wavelength slits in metallic hosts,” Opt. Express 14, 6400–6413 (2006). [CrossRef] [PubMed]

19.

O. Mata Mendez, M. Cadilhac, and R. Petit, “Diffraction of a two-dimensional electromagnetic beam wave by a thick slit pierced in a perfectly conducting screen,” J. Opt. Soc. Am. 73, 328–331 (1983). [CrossRef]

20.

S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” Appl. Phys. 98, 011101 (2005). [CrossRef]

21.

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007). [CrossRef]

22.

Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal wavequides,” Phys. Rev. B 79, 035120 (2009). [CrossRef]

23.

B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. R. Soc. A 253, 358 – 379 (1959). [CrossRef]

24.

A. E. Karbowiak, “Theory of imperfect waveguides: the effect of wall impedance,” Proc. IEEE Part B: Radio Electron. Engin. 102, 698 – 708 (1955).

25.

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006). [CrossRef]

26.

P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Approximate model for surface-plasmon generation at slit apertures,” J. Opt. Soc. Am. A 23, 1608–1615 (2006). [CrossRef]

27.

A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt 34, 511–538 (1987). [CrossRef]

28.

J. Sumaya-Martinez, O. Mata-Mendez, and F. Chavez-Rivas, “Rigorous theory of the diffraction of Gaussian beams by finite gratings: TE polarization,” J. Opt. Soc. Am. A 20, 827–835 (2003). [CrossRef]

29.

O. Mata-Mendez, J. Avendano, and F. Chavez-Rivas, “Rigorous theory of the diffraction of Gaussian beams by finite gratings: TM polarization,” J. Opt. Soc. Am. A 23, 1889–1896 (2006). [CrossRef]

30.

M. J. Weber, Handbook of optical materials (CRC Press, 2003).

31.

Y. Z. Umul, “Scattering of a Gaussian beam by an impedance half-plane,” J. Opt. Soc. Am. A 243159–3167 (2007). [CrossRef]

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(240.6680) Optics at surfaces : Surface plasmons
(260.5430) Physical optics : Polarization
(140.3295) Lasers and laser optics : Laser beam characterization
(050.6624) Diffraction and gratings : Subwavelength structures

ToC Category:
Diffraction and Gratings

History
Original Manuscript: January 27, 2011
Revised Manuscript: March 10, 2011
Manuscript Accepted: March 13, 2011
Published: March 31, 2011

Citation
P. Marchenko, S. Orlov, C. Huber, P. Banzer, S. Quabis, U. Peschel, and G. Leuchs, "Interaction of highly focused vector beams with a metal knife-edge," Opt. Express 19, 7244-7261 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-8-7244


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References

  1. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]
  2. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000). [CrossRef] [PubMed]
  3. R. Dorn, S. Quabis, and G. Leuchs, “The focus of light-linear polarization breaks the rotational symmetry of the focal spot,” J. Mod. Opt. 50, 1917–1926 (2003).
  4. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003). [CrossRef] [PubMed]
  5. G. Leuchs and S. Quabis, “Tailored polarization patterns for performance optimization of optical devices,” J. Mod. Opt. 53, 787–797 (2006). [CrossRef]
  6. J. A. Arnaud, W. M. Hubbard, G. D. Mandeville, B. de la Claviere, E. A. Franke, and J. M. Franke, “Technique for fast measurement of Gaussian laser beam parameters,” Appl. Opt. 10, 2775–2776 (1971). [PubMed]
  7. A. H. Firester, M. E. Heller, and P. Sheng, “Knife-edge scanning measurements of subwavelength focussed light beams,” Appl. Opt. 16, 1971–1974 (1977). [CrossRef] [PubMed]
  8. M. B. Schneider and W. W. Webb, “Measurement of submicron laser beam radii,” Appl. Opt. 20, 1382–1388 (1981). [CrossRef] [PubMed]
  9. R. L. McCally, “Measurement of Gaussian beam parameters,” Appl. Opt. 23, 2227–2227 (1984). [CrossRef] [PubMed]
  10. O. Mata-Mendez, “Diffraction and beam-diameter measurement of Gaussian beams at optical and microwave frequencies,” Opt. Lett. 16, 1629–1631 (1991). [CrossRef] [PubMed]
  11. J. M. Khosrofian and B. A. Garetz, “Measurement of a Gaussian laser beam diameter through the direct inversion of knife-edge data,” Appl. Opt. 22, 3406–3410 (1983). [CrossRef] [PubMed]
  12. G. Brost, P. D. Horn, and A. Abtahi, “Convenient spatial profiling of pulsed laser beams,” Appl. Opt. 24, 38–40 (1985). [CrossRef] [PubMed]
  13. H. R. Bilger and T. Habib, “Knife-edge scanning of an astigmatic Gaussian beam,” Appl. Opt. 24, 686–690 (1985). [CrossRef] [PubMed]
  14. M. A. de Araujo, R. Silva, E. de Lima, D. P. Pereira, and P. C. de Oliveira, “Measurement of Gaussian laser beam radius using the knife-edge technique: improvement on data analysis,” Appl. Opt. 48, 393–396 (2009).
  15. D. Karabacak, T. Kouh, C. C. Huang, and K. L. Ekinci, “Optical knife-edge technique for nanomechanical displacement detection,” Appl. Phys. Lett. 88, 193122 (2006).
  16. M. Gentili and N. A. Riza, “Wide-aperture no-moving-parts optical beam profiler using liquid-crystal displays,” Appl. Opt. 46, 506–512 (2007). [CrossRef] [PubMed]
  17. Y. Chiu and J.-H. Pan, “Micro knife-edge optical measurements device in a silicon-on-insulator substrate,” Opt. Express 15, 6367–6373 (2007). [CrossRef] [PubMed]
  18. Y. Xie, A. R. Zakharian, J. V. Moloney, and M. Mansuripur, “Transmission of light through periodic arrays of sub-wavelength slits in metallic hosts,” Opt. Express 14, 6400–6413 (2006). [CrossRef] [PubMed]
  19. O. Mata Mendez, M. Cadilhac, and R. Petit, “Diffraction of a two-dimensional electromagnetic beam wave by a thick slit pierced in a perfectly conducting screen,” J. Opt. Soc. Am. 73, 328–331 (1983). [CrossRef]
  20. S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” Appl. Phys. 98, 011101 (2005). [CrossRef]
  21. B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007). [CrossRef]
  22. Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal wavequides,” Phys. Rev. B 79, 035120 (2009). [CrossRef]
  23. B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. R. Soc. A 253, 358 – 379 (1959). [CrossRef]
  24. A. E. Karbowiak, “Theory of imperfect waveguides: the effect of wall impedance,” Proc. IEEE Part B: Radio Electron. Engin. 102, 698 – 708 (1955).
  25. J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: Towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006). [CrossRef]
  26. P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Approximate model for surface-plasmon generation at slit apertures,” J. Opt. Soc. Am. A 23, 1608–1615 (2006). [CrossRef]
  27. A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt 34, 511–538 (1987). [CrossRef]
  28. J. Sumaya-Martinez, O. Mata-Mendez, and F. Chavez-Rivas, “Rigorous theory of the diffraction of Gaussian beams by finite gratings: TE polarization,” J. Opt. Soc. Am. A 20, 827–835 (2003). [CrossRef]
  29. O. Mata-Mendez, J. Avendano, and F. Chavez-Rivas, “Rigorous theory of the diffraction of Gaussian beams by finite gratings: TM polarization,” J. Opt. Soc. Am. A 23, 1889–1896 (2006). [CrossRef]
  30. M. J. Weber, Handbook of optical materials (CRC Press, 2003).
  31. Y. Z. Umul, “Scattering of a Gaussian beam by an impedance half-plane,” J. Opt. Soc. Am. A 243159–3167 (2007). [CrossRef]

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