## Interaction of highly focused vector beams with a metal knife-edge |

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

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]

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]

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

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]

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]

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]

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]

## 2. The setup, principle of measurement and experimental results

### 2.1. Setup

_{00}) 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.

*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

*μ*m) 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).

### 2.2. Principle of the measurement

_{00}-mode. We investigate two polarization directions of the incoming beam relative to the knife-edge (in the

*x*–

*y*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 where

*S*is the

_{z}*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. 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]

*d*and

_{s}*d*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 (

_{p}*d*=

_{s}*d*). Thus, a non zero value for

_{p}*d*–

_{s}*d*indicates the presence of the polarization effect.

_{p}*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.

*μ*m. Measurements on wider knife-edges (approximately 3 and 4

*μ*m) do not result in any quantitative change within the experimental accuracy. Hence, in what follows we solely discuss results obtained for the 2

*μ*m structures.

### 2.3. Experimental results

*h*= 130 nm. The beam profiling results

*d*and

_{s}*d*are presented in Fig. 3 (a) for two polarization states (

_{p}*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. 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]

*d*and

_{s}*d*can be interpreted intuitively as the width of the single stripe if no polarization effects are present.

_{p}*d*and

_{s}*d*. 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

_{p}*d*depends on the wavelength and monotonically decreases as the wavelength increases. To characterize the polarization effect on the measurement we plot the value

_{s}*d*–

_{s}*d*which also depends on the wavelength (see Fig. 3 (b)).

_{p}*d*–

_{s}*d*for Au are presented in Fig. 3 (b) and (c). The profiling results of Au samples for different thickness shows a zero shift

_{p}*d*–

_{s}*d*at approx.

_{p}*λ*= 550 nm for

*h*= 100 and 130 nm and also for the whole wavelength range investigated at

*h*= 70 nm. Additionally, samples made from Ni and Ti were investigated. The results for Ni and Ti are very similar to each other (see Fig. 3 (d)). As compared with Au of

*h*= 130 nm (see Fig. 3 (b)) no zero shift was observed for Ni and Ti samples in the investigated wavelength range. Thus, the polarization effect depends not only on the thickness but also on the material of the knife-edge. Due to the roughness of the knife-edge and the photodiode surface (see Fig. 1 (b)) the method is position dependent, what is the main source of systematical errors. The theoretical curves, which were obtained using theory developed in the next part of the paper, are also presented in the Fig. 3 (b)–(d). Slight discrepancies between the experimental and theoretical results are due to the fact that in the experiment the edge position was determined as a point of maximum derivative of the probe current. Furthermore, the sample in the experiment had a non-zero slope angle (see Fig. 1 (b)). Our numerical simulations reveal a slight asymmetry in the reconstructed beam profile (see discussion in the next section). So, the position of the maximum of the photocurrent derivative does not coincide with the position of the half level of the photocurrent curve. However, if many measurements are averaged to reduce the influence of noise the curves become more inversion symmetric. Points of maximum derivative and those of half transmission start to merge. In fact, if we add noise to the numerical results and average afterwards the agreement between experimental and numerical results is even improved (see Fig 3 (b)). We see a good qualitative agreement of our experimental and numerical findings for all samples. Only at a wavelength of

*λ*= 532 (see Fig. 3 (c)) and a knife-edge thickness of

*h*= 100 nm and 130 nm we observe stronger discrepancies.

*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 (

*d*–

_{s}*d*) 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

_{p}*h*= 130 nm, compare with Fig. 3 (b).

*h*= 130 nm), see Fig. 4 (a) and No. 9 (

*h*= 70 nm), see Fig.4 (b)). The measurements on these samples were performed under comparable conditions as described before. The results for these samples are similar to the results obtained with non-tempered periodically placed knife-edges and the differences were found to be in the range of the experimental errors. Such outcome is expected in our theoretical model, see next Section. The theoretical curves reveal changes of the same degree as in the experiment (see Fig.4 (c)).

*w*/

_{s}*λ*and

*w*/

_{p}*λ*, which should be constant. As we see from the experimental data, for samples of thickness

*h*= 130 nm, the measured FWHM remains indeed nearly constant for p-polarization (see Fig. 5 (a)) but decreases for s-polarization with growing wavelength (see Fig. 5 (b)). The theoretical predictions are also shown in the pictures. With increasing thickness

*h*of the knife-edge the width of the reconstructed profile increases (see Fig. 5). This accounts to the theoretical prediction, that the slit acts as a complex spatial frequency filter which damps modes with higher spatial frequency. In order to accurately measure the beam, the spatial spectrum of the signal has to be preserved, while the beam is blocked by the knife-edge. Obviously this can be accomplished by a knife-edge of small enough thickness. Next, as the wavelength increases, the reconstructed width of the beam slowly approaches its theoretical value. However, the reconstruction of the p-polarized beam reveals, that the reconstructed width of the beam projection remains larger than its real value over the investigated wavelength range. Due to the rather complicated interplay of the underlying effects, we can only suggest, that this is caused by the remaining power-flow through the wall of the knife (see next section for more details). Nevertheless the agreement between the experiment and the simulations is good, however, the FWHM values obtained from the Debye integrals [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]

## 3. Theoretical model

### 3.1. Realistic knife-edge versus perfect knife-edge

**10**, 2775–2776 (1971). [PubMed]

*S*(

*k*) traveling at different angles

_{x}*α*= arcsin

*k*/

_{x}*k*. Here

*k*=

*ω*/

*c*

_{0}is the wave vector, with

*k*and

_{x}*k*being the transverse and longitudinal components of the wave vector,

_{z}*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**= (

*k*,

_{x}*k*) =

_{z}*k*(

*±*sin

*α,–*cos

*α*) of the spatial spectrum

*S*(

*k*) 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 (

_{x}*r*= 1) from the wall of the knife-edge, however its amplitude remains the same and is therefore also detected properly.

*r*≈ 1 (see Fig. 6 (a)). At the side wall the boundary conditions for s- and p-polarizations are

*ε*is a dielectric constant of the knife-edge material. The fields in the knife-edge are decaying exponentially as exp(−

*κkx*), where

*κ*= Im

*ε*

^{1/2}. They still contribute to the total detected power

*T*as an additional term

*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 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

*d*−

_{s}*d*.

_{p}*r*(

*k*) from the side wall of the knife-edge (see Fig. 6 (a)). For s- and p-polarization the amplitudes of the reflected plane wave parts are

_{x}*E*=

_{re f}*E*exp (i

_{inc}r*δ*), here

*δ*is an angle-dependent relative phase of the reflected plane wave. In the paraxial case it holds:

*r*≈ 1

*–Cα*and

*δ*≈

*Dα*, where

*C*and

*D*are different for s- (⊥) and p- (

*||*) polarization.

*C*and

*D*are obtained from Fresnel’s equations and

*k*=

_{x}*kα*. Thus, the reflected part of the plane wave can be written as

*E*= exp (

_{re f}*–*i

*kxα*+ ln

*r –*i

*δ*). Next we use ln

*r*≈ −

*Cα*. The single plane wave component affected by the reflection can be written as exp(i

*kxα*) + exp[−(i

*kx*+

*C*+ i

*D*)

*α*]. In the perfect case (Fig. 6 (a)) the spatial spectra of the signal blocked by the knife-edge at the photodiode is

*S*(

_{sig}*kα*) =

*S*(

*kα*)exp(−i

*kx*

_{0}

*α*). However due to the reflection, the plane wave component –

*k*is replaced by the aforementioned expression. After some mathematical operations it can be revealed that the reflection from the wall of the knife-edge introduces further modifications to the signal integrated by the detector. The signal experiences a polarization dependent shift Δ

_{x}*x*=

*–D*/(2

*k*), it is spatially filtered and the shape of the spectral profile is deformed.

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

### 3.2. Field representations and eigenvalue problem

*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]. 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

*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.

**E**in this case has only one nonvanishing component of the electric field

*E*parallel (p-polarization) to the slit walls Here

_{y}*β*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

_{e}*Z*

_{0}is the vacuum impedance.

*H*and electric field components are perpendicular (s-polarization) to the walls of the knife-edges Here

_{y}*β*is the propagation constant of the magnetic field mode in the slit and

_{m}*ε*(

*x*) = 1 for 0

*< x < l*and

*ε*(

*x*) =

*ε*

_{2}elsewhere.

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

*x*∉ (0,

*l*).

*β*

_{(}

_{e,m}_{)}of equation (7) are complex for complex dielectric constants. Therefore, propagating modes are attenuated in the slit. However, the eigenmodes with the modal number

*ν > ν*, where

_{cr}*ν*is estimated from the relation

_{cr}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]

*δ*is the Kronecker delta and

_{νμ}*G*is an overlap integral of the electric and magnetic transverse fields of the eigenmode (8), see Ref. [22

_{νν}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]

*G*is in general a complex-valued one, therefore, we normalize our eigenmodes by introducing a real-valued norma

_{ν ν}*N*, which we define as here,

_{ν}*υ*(

*x*) =

*ε*(

*x*)

^{−1}for TM and

*υ*(

*x*) = 1 for TE polarization. An asterisk denotes a complex conjugation. Those expressions are readily evaluated to here,

*υ*=

*ε*

_{2}for TM and

*υ*= 1 for TE polarization. The normalization is, in general, not necessary, however, this procedure ensures a slightly better convergance in our numerics.

*π*(

_{ν}*x*) can be written as a sum of two images. The image Φ

*represents the field in the walls, and Ψ*

_{ν}*- the field in the slit*

_{ν}### 3.3. Boundary value problem at the interfaces and energy flow between regions

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

*S*(

*k*,

_{x}*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

*S*, the second part the field reflected from the slit

_{in}*S*Here

_{re f}*β*≥ 0, Im

*β*≥ 0. We assume an incoming wave with Gaussian distribution of the

*y*component in the transverse plane. So at the entrance the field

*E*(

_{y}*H*) is here

_{y}*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

*w*of the total electric field for the TM polarization is larger than

*w*

_{0}due to the presence of the

*E*component. The spatial Fourier spectra of the field (15) is

_{z}*S*here

_{trans}*π*. Thus, the resulting field is represented by the sum with the expansion coefficients

_{ν}*a*, which describe a field propagating down to the substrate, whereas the

_{ν}*b*represent a reflected field.

_{ν}*P*is the power of the incoming beam.

_{beam}*P*is the power transmitted into the region III through the slit.

_{trans}*P*is the power reflected from the slit and

_{re f}*P*is the power losses in the slit. After some math the reflection

_{abs}*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

*υ*= 1 for TE field and

_{i}*a*and

_{ν}*b*of the field inside the slit along with the spatial spectra

_{ν}*S*,

_{trans}*S*are the unknowns.

_{re f}*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. 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]

*E*and

_{y}*H*, the derivative of the electric field and the derivative ε

_{y}^{−1}∂

*H*/∂

_{y}*z*of the magnetic field are continuous as well. With the help of equations (13–14) and (18) a pair of equations is obtained at the first interface (

*z*= 0): where the sums run from

*μ*= 0 for TM and from

*μ*= 1 for TE polarization to infinity.

*S*(

_{trans}*k*) and

_{x}*a*,

_{μ}*b*where

_{μ}*defined by equation (12). The integral*

_{ν}*S*is an overlap integral of the incident beam and the eigenmode of the slit and the integral

_{ν}*F*and

_{μ}*∂F*/

_{μ}*∂z*are We substitute those expressions into the second equations of the equations system (21–22) and get

**N**being defined as

**a**and

**b**.

*S*and

_{trans}*S*. In particular, the transmitted power

_{re f}*T*, see (20), can be expressed as where

*β*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

_{μ}*k*. This means, that the slit acts as a complex spatial filter, acting differently on the spatial frequencies

_{x}*k*according to the damping of the respective mode excited in the slit. The higher the value of the initial

_{x}*k*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

_{x}*h*is exceeded only the first mode will reach the exit of Region II, effectively leaving mainly two spatial frequencies

*k*= ±i

_{x}*γ*

_{1},

*k*= ±i

_{x}*γ*

_{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

*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

*β*for a perfectly conducting wall were used as an initial guess. For plasmonic modes, an initial guess was the propagation constant

_{ν}^{−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]. We carefully choose free parameters of the theory (

*w*

_{0}

*= 0.9*

_{p}*w*

_{0}

*= 0.55*

_{s}*λ*) to achieve the best possible agreement with our experimental data. The FWHM of the squared electric fields in the model are

*w*= 0.8

_{p}*w*and slightly differ from the expected in our setup [23

_{s}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]

*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 **24**3159–3167 (2007). [CrossRef]

*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.

*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.

*ω*by fitting a Gaussian function

_{trans}*S*(

_{trans}*k*,

_{x}*x*

_{0}). The spectral profile remains symmetric for an perfect knife-edge and the dependence of the angular width Δ

*k*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.

_{x}## 4. Conclusions

**91**, 233901 (2003). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. |

2. | K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express |

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. |

4. | R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. |

5. | G. Leuchs and S. Quabis, “Tailored polarization patterns for performance optimization of optical devices,” J. Mod. Opt. |

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. |

7. | A. H. Firester, M. E. Heller, and P. Sheng, “Knife-edge scanning measurements of subwavelength focussed light beams,” Appl. Opt. |

8. | M. B. Schneider and W. W. Webb, “Measurement of submicron laser beam radii,” Appl. Opt. |

9. | R. L. McCally, “Measurement of Gaussian beam parameters,” Appl. Opt. |

10. | O. Mata-Mendez, “Diffraction and beam-diameter measurement of Gaussian beams at optical and microwave frequencies,” Opt. Lett. |

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. |

12. | G. Brost, P. D. Horn, and A. Abtahi, “Convenient spatial profiling of pulsed laser beams,” Appl. Opt. |

13. | H. R. Bilger and T. Habib, “Knife-edge scanning of an astigmatic Gaussian beam,” Appl. Opt. |

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. |

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

16. | M. Gentili and N. A. Riza, “Wide-aperture no-moving-parts optical beam profiler using liquid-crystal displays,” Appl. Opt. |

17. | Y. Chiu and J.-H. Pan, “Micro knife-edge optical measurements device in a silicon-on-insulator substrate,” Opt. Express |

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 |

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. |

20. | S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” Appl. Phys. |

21. | B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B |

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 |

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 |

24. | A. E. Karbowiak, “Theory of imperfect waveguides: the effect of wall impedance,” Proc. IEEE Part B: Radio Electron. Engin. |

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 |

26. | P. Lalanne, J. P. Hugonin, and J. C. Rodier, “Approximate model for surface-plasmon generation at slit apertures,” J. Opt. Soc. Am. A |

27. | A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt |

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 |

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 |

30. | M. J. Weber, |

31. | Y. Z. Umul, “Scattering of a Gaussian beam by an impedance half-plane,” J. Opt. Soc. Am. A |

**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

- S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]
- K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000). [CrossRef] [PubMed]
- 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).
- R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003). [CrossRef] [PubMed]
- G. Leuchs and S. Quabis, “Tailored polarization patterns for performance optimization of optical devices,” J. Mod. Opt. 53, 787–797 (2006). [CrossRef]
- 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]
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- R. L. McCally, “Measurement of Gaussian beam parameters,” Appl. Opt. 23, 2227–2227 (1984). [CrossRef] [PubMed]
- O. Mata-Mendez, “Diffraction and beam-diameter measurement of Gaussian beams at optical and microwave frequencies,” Opt. Lett. 16, 1629–1631 (1991). [CrossRef] [PubMed]
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- G. Brost, P. D. Horn, and A. Abtahi, “Convenient spatial profiling of pulsed laser beams,” Appl. Opt. 24, 38–40 (1985). [CrossRef] [PubMed]
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- 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).
- 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]
- 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]
- 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]
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- 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]
- A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt 34, 511–538 (1987). [CrossRef]
- 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]
- 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]
- M. J. Weber, Handbook of optical materials (CRC Press, 2003).
- 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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