## Corrections to the knife-edge based reconstruction scheme of tightly focused light beams |

Optics Express, Vol. 21, Issue 21, pp. 25069-25076 (2013)

http://dx.doi.org/10.1364/OE.21.025069

Acrobat PDF (873 KB)

### Abstract

The knife-edge method is an established technique for profiling light beams. It was shown, that this technique even works for tightly focused beams, if the material and geometry of the probing knife-edges are chosen carefully. Furthermore, it was also reported recently that this method fails, when the knife-edges are made from pure materials. The artifacts introduced in the reconstructed beam shape and position depend strongly on the edge and input beam parameters, because the knife-edge is excited by the incoming beam. Here we show, that the actual beam shape and spot size of tightly focused beams can still be derived from knife-edge measurements for pure edge materials and different edge thicknesses by adapting the analysis method of the experimental data taking into account the interaction of the beam with the edge.

© 2013 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]

3. 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]

4. J. Kindler, P. Banzer, S. Quabis, U. Peschel, and G. Leuchs, “Waveguide properties of single subwavelength holes demonstrated with radially and azimuthally polarized light,” Appl. Phys. B **89**, 517–520 (2007). [CrossRef]

7. P. Banzer, U. Peschel, S. Quabis, and G. Leuchs, “On the experimental investigation of the electric and magnetic response of a single nano-structure,” Opt. Express **18**, 10905–10923 (2010). [CrossRef] [PubMed]

8. A. H. Firester, M. E. Heller, and P. Sheng, “Knife-edge scanning measurements of subwavelength focussed light beams,” Appl. Opt. **16**, 197-1–1974 (1977). [CrossRef]

13. 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). [CrossRef] [PubMed]

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

15. 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). [CrossRef] [PubMed]

_{00}-mode (in the

*xy*-plane) are investigated (compare [15

15. 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). [CrossRef] [PubMed]

*P*detected by the photodiode and is recorded for each beam position

*x*

_{0}with respect to the knife-edge (see Fig. 1(b)) where

*P*

_{0}is a proportionality coefficient and

*I*is the electric field intensity. In the conventional knife-edge method the derivative

*∂P/∂x*

_{0}of the photocurrent curve with respect to the beam position

*x*

_{0}(see Fig. 1(c)) reconstructs a projection of the intensity onto the

*xz*-plane at

*z*= 0 (projection onto the x-axis) [8

8. A. H. Firester, M. E. Heller, and P. Sheng, “Knife-edge scanning measurements of subwavelength focussed light beams,” Appl. Opt. **16**, 197-1–1974 (1977). [CrossRef]

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

*d*and

_{s}*d*define the distance between the maxima (minima) of the reconstructed intensity projections for s- and p-polarized input beams respectively, see Fig. 1(c).

_{p}**S**both being proportional to each other because longitudinal electric field components are negligible in this limit. The latter is not necessarily true anymore in the case of tightly focused light beams (non-paraxial propagation), which can exhibit quite strong longitudinal electric field components resulting in different distributions of |

**E**(

*x*,

*y*)|

^{2}and

*S*(

_{z}*x*,

*y*). Therefore, the question arises which distribution is meant by intensity in this case. It was believed, that the integral Eq. (1) borrowed/adopted from the conventional knife-edge method for retrieving the beam projection of paraxial light beams allows for the reconstruction of the beam profile in terms of its total electric energy density distribution |

**E**(

*x*,

*y*)|

^{2}also in case of tightly focused vectorial beams. It was shown that this assumption holds true only if special edge materials, thicknesses and certain wavelengths are chosen [2]. Nevertheless, it was also shown just recently [15

15. 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). [CrossRef] [PubMed]

*d*≠

_{s}*d*≠

_{p}*d*

_{0}with

*d*

_{0}the width of the metal pad; see Fig. 1) and asymmetrically deformed also causing deviations in the retrieved beam diameters

*w*and

_{s}*w*due to interactions between the knife-edge and the beam, Fig. 1(c).

_{p}*U*(±

_{E}*x*

_{0}) onto the edge. Obviously, if one does not account for these effects, the standard scheme is not valid without corrections unless the knife-edge parameters are carefully chosen [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]

## 2. Theoretical considerations

*y*-axis reduces the dimensionality of the Helmholtz equation by one, i.e. we do not have to consider the electric field

**E**but its projection

**Ê**onto the

*xz*-plane. Next, due to the symmetry of the knife edge solutions of the Helmholtz equation consist of two independent classes: transverse electric (in our notation p-polarized) and transverse magnetic (s-polarized) modes, see [15

**19**7244–7261 (2011). [CrossRef] [PubMed]

**Ê**in the p-polarized case has only one non-vanishing component of the electric field

*Ê*parallel to the knife-edge. The s-polarized solutions have two non-vanishing components of the electric field projections. The main component

_{y}*Ê*is perpendicular (s-polarization) to the knife-edge while

_{x}*Ê*depends on

_{z}*Ê*as

_{x}*Ê*= −(i/

_{z}*k*

_{0})

*∂Ê*, where

_{x}/∂x*k*

_{0}is the wave vector of the carrier wave [15

**19**7244–7261 (2011). [CrossRef] [PubMed]

17. 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]

**E**

*with the knife-edge, we need to decompose the projection*

_{b}**Ê**

*into s- and p-polarized modes. The highly focused linearly polarized TEM*

_{b}_{00}-mode (which we consider here) has two orientations of the main electric field component relative to the knife-edge, for which this decomposition is trivial. If the beam polarization is parallel to the knife-edge (p-situation), the projection of the electric field

**Ê**

*has only a non-vanishing*

_{b}**Ê**

*component, so it is decomposed only into p-modes. When the beam’s electric field is oriented perpendicularly (s-situation) to the knife edge, it is decomposed only into s-modes. For the sake of brevity we consider further only those two orientations of the beam.*

_{y}*y*-axis and express the projection of the signal at the photodiode in the Fourier-domain as

*Û*(

_{E}*k*,

_{x}*x*

_{0})

*T̂*(

*k*). Here

_{x}*Û*(

_{E}*k*,

_{x}*x*

_{0}) is the Fourier-image of the signal, which we expect to measure (projection of the electric field energy density

*U*(

_{E}*x*) onto

*xz*-plane at the knife-edge).

*T̂*(

*k*) is a spectral representation of the polarization dependent knife-edge interaction operator. Let us introduce the Taylor expansion of

_{x}*T̂*(

*k*) with

_{x}*Û*(

_{E}*k*,

_{x}*x*

_{0}) =

*Û*

_{E,0}(

*k*)e

_{x}^{∓ikxx0}, where

*Û*

_{E,0}(

*k*) is the Fourier image of the electric energy density projection

_{x}*U*

_{E,0}(

*x*) of the beam exactly on the knife-edge (

*x*

_{0}= 0). We substitute Eq. (3) into Eq. (2). We use the relation

*∂*(

^{n}U_{E}*x*)/

*∂x*= ∫ d

^{n}*k*(i

_{x}*k*)

_{x}*(*

^{n}Û_{E}*k*) e

_{x}^{ikxx}, so the resulting expression reads with

*U*

_{E,0}(±

*x*

_{0}) (the classical knife-edge term) plus interaction terms expressed as a knife-edge and polarization dependent sum over higher order derivatives (which we call moments) of the beam projections. The physical meaning behind Eq. (5) is the following. The first term in the sum (

*n*= 1) is due to the local response of the knife-edge to the s- or p-polarized electric field. The second term (

*n*= 2) expresses the local response of the knife-edge to the projection of the electric field gradient and so on.

## 3. Experimental results and adapted fit algorithm

*h*= 130 nm on gallium-arsenide photodiodes at wavelengths from 530 nm to 700 nm. The details of our experimental setup and measurement technique can be found in [15

**19**7244–7261 (2011). [CrossRef] [PubMed]

*w*and

_{s}*w*on the wavelength

_{p}*λ*for various samples is depicted in Fig. 2. As it was already mentioned, we see that conventionally determined beamwidths do not fit to the expectations from the estimations based on the Debye integrals [3

3. 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]

*U*(

_{E}*x*

_{0}). With a high degree of accuracy we can assume a Gaussian distribution of the

*x*(

*y*) components of the s-(p-)polarized beams in the plane of projection. The

*Ê*component of the s-polarized beam depends on

_{z}*Ê*, thus a different ansatz is used. Before fitting, we predetermine the exact position of the knife-edge to reduce the number of free parameters in the fitting-routine and to fix the coordinate frame. For that purpose, we measure the real distances

_{x}*d*

_{0}between both edges using a scanning electron microscope (SEM), find the center

*x*between the peak values of both projections in one scan and finally set the actual positions of both knife-edges to be at

_{c}*x*±

_{c}*d*

_{0}/2. An example of such a fitting procedure is presented in Fig. 3 for two polarization states (s and p respectively). It turns out, that for all investigated experimental samples, we were able to successfully determine the real diameters of the beam projections by simultaneously ensuring

*d*=

_{s}*d*=

_{p}*d*

_{0}.

*w*and

_{s}*w*reconstructed with this technique, see Fig. 4. Here, we compare the theoretically expected values for the beam width calculated using vectorial diffraction theory [3

_{p}3. 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]

*A*, Eq. (5), depend on the edge parameters (thickness, material and optical properties). Hence, once determined, they can be used as fitting parameters also in situations in which, for instance, the focusing NA is smaller or larger than in the case for which they have been retrieved. Furthermore, also for input beams of other geometries, such as cylindrical vector beams etc., a proper retrieval of the beam parameters, such as the beam size in the focal plane can be achieved when choosing appropriate ansatz functions in Eq. (5). A more detailed study on these topics will be presented elsewhere soon.

_{n}## 4. Conclusions

## 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. | R. Dorn, S. Quabis, and G. Leuchs, “The focus of light-linear polarization breaks the rotational symmetry of the focal spot,” J. Mod. Opt. |

3. | B. Richards and E. Wolf, “Electromagnetic diffraction in optical Systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A |

4. | J. Kindler, P. Banzer, S. Quabis, U. Peschel, and G. Leuchs, “Waveguide properties of single subwavelength holes demonstrated with radially and azimuthally polarized light,” Appl. Phys. B |

5. | T. Züchner, A. V. Failla, A. Hartschuh, and A. J. Meixner, “A novel approach to detect and characterize the scattering patterns of single Au nanoparticles using confocal microscopy,” J. Microsc. |

6. | P. Banzer, J. Kindler, S. Quabis, U. Peschel, and G. Leuchs, “Extraordinary transmission through a single coaxial aperture in a thin metal film,” Opt. Express |

7. | P. Banzer, U. Peschel, S. Quabis, and G. Leuchs, “On the experimental investigation of the electric and magnetic response of a single nano-structure,” Opt. Express |

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

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

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

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. | 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. |

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

15. | 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 |

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

17. | Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal wavequides”, Phys. Rev. B |

**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: August 13, 2013

Manuscript Accepted: October 1, 2013

Published: October 14, 2013

**Citation**

C. Huber, S. Orlov, P. Banzer, and G. Leuchs, "Corrections to the knife-edge based reconstruction scheme of tightly focused light beams," Opt. Express **21**, 25069-25076 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-21-25069

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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]
- 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).
- B. Richards and E. Wolf, “Electromagnetic diffraction in optical Systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A253, 358–379 (1959). [CrossRef]
- J. Kindler, P. Banzer, S. Quabis, U. Peschel, and G. Leuchs, “Waveguide properties of single subwavelength holes demonstrated with radially and azimuthally polarized light,” Appl. Phys. B89, 517–520 (2007). [CrossRef]
- T. Züchner, A. V. Failla, A. Hartschuh, and A. J. Meixner, “A novel approach to detect and characterize the scattering patterns of single Au nanoparticles using confocal microscopy,” J. Microsc.229, 337–343 (2008). [CrossRef] [PubMed]
- P. Banzer, J. Kindler, S. Quabis, U. Peschel, and G. Leuchs, “Extraordinary transmission through a single coaxial aperture in a thin metal film,” Opt. Express18, 10896–10904 (2010). [CrossRef] [PubMed]
- P. Banzer, U. Peschel, S. Quabis, and G. Leuchs, “On the experimental investigation of the electric and magnetic response of a single nano-structure,” Opt. Express18, 10905–10923 (2010). [CrossRef] [PubMed]
- A. H. Firester, M. E. Heller, and P. Sheng, “Knife-edge scanning measurements of subwavelength focussed light beams,” Appl. Opt.16, 197-1–1974 (1977). [CrossRef]
- M. B. Schneider and W. W. Webb, “Measurement of submicron laser beam radii,” Appl. Opt.20, 1382–1388 (1981). [CrossRef] [PubMed]
- R. L. McCally, “Measurement of Gaussian beam parameters,” Appl. Opt.23, 2227–2227 (1984). [CrossRef] [PubMed]
- 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]
- G. Brost, P. D. Horn, and A. Abtahi, “Convenient spatial profiling of pulsed laser beams,” Appl. Opt.24, 38–40 (1985). [CrossRef] [PubMed]
- 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). [CrossRef] [PubMed]
- R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91, 233901 (2003). [CrossRef] [PubMed]
- 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. Express197244–7261 (2011). [CrossRef] [PubMed]
- B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures”, Phys. Rev. B76, 125104 (2007). [CrossRef]
- Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal wavequides”, Phys. Rev. B79, 035120 (2009). [CrossRef]

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