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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 21 — Oct. 8, 2012
  • pp: 23875–23886
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Measurement limitations in knife-edge tomographic phase retrieval of focused IR laser beams

Manuel Silva-López, José María Rico-García, and Javier Alda  »View Author Affiliations


Optics Express, Vol. 20, Issue 21, pp. 23875-23886 (2012)
http://dx.doi.org/10.1364/OE.20.023875


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Abstract

An experimental setup to measure the three-dimensional phase-intensity distribution of an infrared laser beam in the focal region has been presented. It is based on the knife-edge method to perform a tomographic reconstruction and on a transport of intensity equation-based numerical method to obtain the propagating wavefront. This experimental approach allows us to characterize a focalized laser beam when the use of image or interferometer arrangements is not possible. Thus, we have recovered intensity and phase of an aberrated beam dominated by astigmatism. The phase evolution is fully consistent with that of the beam intensity along the optical axis. Moreover, this method is based on an expansion on both the irradiance and the phase information in a series of Zernike polynomials. We have described guidelines to choose a proper set of these polynomials depending on the experimental conditions and showed that, by abiding these criteria, numerical errors can be reduced.

© 2012 OSA

1. Introduction

The optical response of devices at micro and nano-scales is governed by the local phase and amplitude fields [1

1. P. S. Carney, B. Deutsch, A. A. Govyadinov, and R. Hillenbrand, “Phase in nanooptics,” ACS Nano 6, 8 – 12 (2012). [CrossRef] [PubMed]

,2

2. G. Volpe, S. Cherukulappurath, R. J. Parramon, G. Molina-Terriza, and R. Quidant, “Controlling the Optical Near Field of Nanoantennas with Spatial Phase-Shaped Beams,” Nano Lett. 93608–3611 (2009). [CrossRef] [PubMed]

]. In particular, phase is central to a complete understanding of plasmonic-based structures: light-matter interaction and the near field response of such devices can be greatly enhanced [3

3. E. C. Kinzel, J. C. Ginn, R. L. Olmon, D. J. Shelton, B. A. Lail, I. Brener, M. B. Sinclair, M. B. Raschke, and G. D. Boreman, “Phase resolved near-field mode imaging for the design of frequency-selective surfaces,” Opt. Express 20, 11986–11993 (2012). [CrossRef] [PubMed]

,4

4. E. Cubukcu, N. Yu, E. J. Smythe, L. Diehl, K. B. Crozier, and F. Capasso, “Plasmonic Laser Antennas and Related Devices,” J. Sel. T. Q. Elec. 14, 1448–1461 (2008). [CrossRef]

]. Indeed, for optical antenna and antenna-coupled detectors, a wide range of applications in sensing or non-linear optics, have arisen [5

5. L. Novotny and N. Van Hulst, “Antennas for light,” Nature Phot. 5, 83–90 (2011). [CrossRef]

8

8. R. L. Olmon, P. M. Krenz, B. A. Lail, L. V. Saraf, G. D. Boreman, and M. B. Raschke, “Determination of electric-field, magnetic-field, and electric-current distributions of infrared optical antennas: A near-field optical vector network analyzer,” Phys. Rev. Lett. 105, 167403 (2010). [CrossRef]

]. Specifically, characterization of antenna-coupled detectors, both at IR and visible frequencies, depends on the spatial response mapping of these devices and also on the phase distribution of the incident beam [9

9. C. Fumeaux, G. D. Boreman, W. Herrmann, F. K. Kneubhl, and H. Rothuizen, “Spatial impulse response of lithographic infrared antennas,” Appl. Opt. 38, 37–46 (1999). [CrossRef]

]. To characterize such devices a probe beam, (a high quality, almost Gaussian focused laser beam) is scanned across the photosensitive region. Thus, the device under test is located under the beam waist plane, where the wavefront is assumed to be represented by a plane wave. The measured output signal is, in general, the convolution of the sensor’s spatial response and the beam spatial distribution [10

10. J. Alda, C. Fumeaux, L. Codreanu, J. A. Schaefer, and G. D. Boreman, “Deconvolution method for two dimensional spatial response mapping of lithographic infrared antennas,” Appl. Opt. 383993–4000 (1999). [CrossRef]

]. Furthermore, it has been shown that their characterization is the largest source of uncertainty in this deconvolution process [11

11. J. M. López-Alonso, B. Monacelli, J. Alda, and G. D. Boreman, “Uncertainty analysis in the measurement of the spatial responsivity of infrared antennas,” Appl. Opt. 44, 4557–4568 (2005). [CrossRef] [PubMed]

]. Therefore, a reliable characterization, both in amplitude and phase, is required.

However, measuring both phase and amplitude of the probe beam at focus is a challenging task. Light sources for these characterization processes and the use of focusing optics yields a beam waist of a few wavelengths in size. Therefore, at IR wavelengths, conventional image-based laser beam characterization is no longer applicable: the available focal plane array detectors have a pixel size comparable to the beam size, if not greater. Consequently low spatial resolution can introduce strong uncertainties in the intensity distribution measurement. On the other hand, phase measurement demands, in principle, an interferometric arrangement to perform a phase measurement at the plane where the detector is going to be probed, which is frequently at focus, as mentioned above. Hence, the same problem arises, and lack of resolution prevents an accurate measurement of the beam wavefront. A well established method to determine phase distributions from irradiance maps is based in the Gerchberg-Saxton method that relates the irradiances at the plane of interest and at the plane where the Fourier transform appears [12

12. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

]. This strategy has been analyzed, refined and adapted even to include the Fresnel diffraction maps or eliminate the irradiance measurement at the plane of interest using a set of lenses [13

13. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]

15

15. W. Chen and X. Chen, “Quantitative phase retrieval of complex-valued specimens based on noninterferometric imaging,” Appl. Opt. 50, 2008–2015 (2011). [CrossRef] [PubMed]

]. An interesting extension of the diffractive imaging techniques has been exploited to find the phase distribution at a given plane of interest for coherent and incoherent x-ray sources [16

16. H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. H. Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200(2006). [CrossRef]

, 17

17. H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nature Phys. 2, 101–104 (2006). [CrossRef]

]. However, these methods require a propagation of the beam from the plane of interest to another acquisition plane where the irradiance is measured by a CCD or similar array detector. When this propagation is not possible, or an array detector is not available, we need to work with the accessible irradiance distributions around the plane of interest, as it happens in this paper.

Tomographic techniques based on the knife-edge technique have been applied to the determination of the intensity distribution of focused beams [18

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

, 19

19. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001). [CrossRef]

]. This method avoids the use of image forming devices which are not suitable under low spatial resolution conditions and allows reconstruction of irradiance planes with sub-microm resolution [20

20. J. M. Rico-García, L. M. Sánchez-Brea, and J. Alda, “Application of tomographic techniques to the spatial-response mapping of antenna-coupled detectors in the visible,” Appl. Opt. 47, 768–775 (2008). [CrossRef] [PubMed]

, 21

21. 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, 7244–7249 (2011). [CrossRef] [PubMed]

]. In this paper we evaluate the goodness of this method. The transport of intensity equation (TIE) was also used to derive the wavefront from these irradiance measurements. As a result, a complete description of the propagating field in the focal region of the lens is obtained, and an analysis of the reliability of the numerical calculation is explained based on the experimental procedure.

This contribution is sectioned as follows: In section 2 the theoretical foundations with the main approximations that will be employed are presented. Section 3 is devoted to the description of the experimental set up. We used a multi angle knife-edge technique to obtain a series of intensity measurements along the path of a laser beam. Section 4 evaluates the goodness of this method and its reliability. Then a tomographic process is performed: the intensity profiles obtained are processed via the Radon transform to form 2D maps of irradiance distribution at planes perpendicular to the beam axis. With these planes of irradiance, measured at known distances, the TIE can derive the phase of the propagating beam. However, before these images are fed into the TIE algorithm, a number of considerations must be taken into account. Our main interest has been to understand and evaluate the different uncertainties of the measurement and processing method. The analysis discussed in section 5 leads to a specific set of polynomials to be employed in the Zernike expansion. Assuming these considerations the results of the wavefronts retrieved are shown in section 6 and the wavefront reconstruction error is evaluated. Finally, a summary of the main conclusions is presented in section 7.

2. Theoretical background

The wavefront of a propagating beam can be recovered using the TIE [22

22. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983). [CrossRef]

, 23

23. F. Roddier, “Wavefront sensing and the irradiance transport equation,” Appl. Opt. 29, 1402–1403 (1990). [CrossRef] [PubMed]

]. Consider the inset of Fig. 1. In a paraxial light beam propagating in a given z-direction the TIE associates the irradiance images at two different planes (IP+1 and IP−1), located symmetrically with respect to the plane of interest, with the phase front φ at IP.

Fig. 1 Experimental setup diagram. The inset shows the working region where the beam path is cut by the knife.

To solve this equation we have used the method derived by Gureyev and Nugent [24

24. T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13(8), 1670–1682 (1996). [CrossRef]

]. A fundamental aspect of this method is that both the data and the solution are expanded into a series of N Zernike polynomials. For instance, the phase is retrieved as decomposition in Zernike terms.
φ=iNφiZi.
(1)

These polynomials are widely recognized as a good estimation of the aberration content of the phase front [25

25. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976). [CrossRef]

, 26

26. J. Alda, J. Alonso, and E. Bernabéu, “Characterization of aberrated laser beams,” J. Opt. Soc. Am. A 14, 2737–2747 (1997). [CrossRef]

]. The TIE is thus reduced to a set of N algebraic equations,
φN=NfM(N)1δzI(N),
(2)
where Nf=πR2λΔz is a dimensionless parameter that combines all the scaling factors of the TIE. These are the laser wavelength λ, the pupil radii R and the distance between planes Δz.

M is a N × N matrix that depends on the irradiance at the plane PF where the phase is to be recovered (see inset of Fig. 1), which elements Mij are defined as [24

24. T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13(8), 1670–1682 (1996). [CrossRef]

]
Mij=02π0RIP(ρ/R,θ)Zi(ρ/R,θ)Zj(ρ/R,θ)ρdρdθ,
(3)
with 2 ≤ i, jN + 1. The derivatives of the Zernike polynomials can be easily calculated following the procedure described in [25

25. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976). [CrossRef]

].

δzI corresponds to the axial derivative of intensity. This can be obtained by applying the central finite difference approximation:
δzI=IP+1(x,y,Δz)IP1(x,y,Δz).
(4)

That is, subtracting the irradiance of two planes and dividing by the distance separating them, (this last factor 1/2Δz was moved to the constant Nf). This is acceptable as long as Δz is small so that the ray deflections are not too large. That is indeed the case within the focal region of a laser beam, and we will check this condition in our experimental data.

δzI is then expanded in a series of N Zernike polynomials, and applying Eq. (2) the N Zernike coefficients of the phase can be retrieved. The reliability of this process depends strongly on the experimental estimation of δzI [27

27. S. Barbero and L. N. Thibos, “Error analysis and correction in wavefront reconstruction from the transport-of-intensity equation,” Opt. Eng. 45, 094001 (2006). [CrossRef]

,28

28. M. Soto, E. Acosta, and S. Rios “Performance analysis of curvature sensors: optimum positioning of measurement planes,” Opt. Express 11, 2577–2588 (2003). [CrossRef] [PubMed]

]. However, the number of polynomials used in a Zernike expansion is typically limited to the secondary order aberrations, around 30 polynomials, which is enough to describe wave aberration functions of optical systems with circular pupils. Nevertheless, we consider that this is not sufficient for our application. We believe there is room for a criterion, and we will based it in our experimental conditions.

3. Experimental set- up

Figure 1 shows the experimental set up. A power-stabilized CO2 laser (LASY-5 Access Laser) is used as a light source. It has an output continuous power of 5 W maximum at a 10.6 μm wavelength, with a high quality gaussian beam (M2 ∼ 1). The beam is directed to a suitable IR lens (ASPH-ZC-50-50 from ISP Optics) with an f = 50.4 mm of focal length. The lens was slightly tilted (∼ 6°) to introduce some aberrations. The knife edge is located at the vicinity of the waist of the beam, and by means of DC-motor stages (PI M-415.DG) it can be moved in any x, y and z-direction. Intensity variations, due to the obstructing knife, are observed in a thermopile based detector (Thorlabs S310C).

Fig. 2 Normalized intensity measurement (dashed line), and deconvolved signal computed with the PSF of the detector (dotted line). Difference between both curves is plotted below.

For a tomographic reconstruction the measurements must be performed at different angles. Thus, the knife is also attached to a rotary stage (New Focus 8410M). This allows a change in its orientation, around z–axis, so that the beam can be cut from any angle θ. We have performed 9 cuts along different orientations with Δθ = 20° separation. Clearly the more cuts the more reliable the tomographic reconstruction will result. We will discuss the influence of the number of cuts on the angular resolution in section 5. A set of normalized measurements at a function of angle are illustrated in Fig. 3(a). These are 9 intensity profiles, arranged in columns (representation also known as sinogram). Once differentiated, these profiles will form a set of projections of the beam intensity. Because the projections at 0° and 180° are mirror images of each other, the measurements are carried out to the last angle increment before 180°. The sinogram clearly shows a strong asymmetry in the beam shape produced by the misaligned lens.

Fig. 3 (a) Normalized intensity profiles obtained at different angles (arranged in columns), also called sinogram. (b) Beam irradiance map generated from this sinogram via the inverse Radon transform. Astigmatism is clearly seen.

Since the detector is a slow thermal sensor (tr ≈ 2.3 s rise time) the knife edge must move accordingly. The speed of the knife (v) was thus set to 3.4 μm s−1 and recording each intensity profile took around 4 mins (knife-edge going back and forth). The power meter console (Thor-labs PM100D), allowed a sampling frequency of ∼ 13 Hz, and was connected via USB to a computer where data was processed using MATLAB.

4. Measurement constraints and limitations

This resolution can be improved by reducing speed at a cost of increasing measuring time. For high v the bandwidth of the detector sets the limit. We have derived the point spread function (PSF) of the detector by measuring the response to a step impulse. Then, by applying a deconvolution procedure to the PSF and a measured intensity profile, we obtained a restored signal. For these calculations we used the Lucy-Richardson algorithm in an iterative process that converges to the maximum likelihood solution [10

10. J. Alda, C. Fumeaux, L. Codreanu, J. A. Schaefer, and G. D. Boreman, “Deconvolution method for two dimensional spatial response mapping of lithographic infrared antennas,” Appl. Opt. 383993–4000 (1999). [CrossRef]

]. The restored, deconvoluted signal and the measured signal are plotted in Fig. 2. The difference between these two plots is also represented. This difference shows a relative error of ∼ 1%, and corresponds with an uncertainty in the beam width of about 0.5 μm, well below Δρ.

Figure 3(b) shows a 2D irradiance map reconstructed from the data shown in the sinogram of Fig. 3(a). It corresponds to the beam irradiance at a location close to the focal plane in a square region of 400 μm2 with a step grid size of ∼ 1.4 μm. The elliptic shape of the beam is due to astigmatism. Some artifacts from the transform, as rays emanating from the center, can be observed.

5. Zernike expansion considerations

An important property of the Zernike polynomials is that they can be written as a product of a radial and an azimuthal part [31

31. M. Born and E. Wolf, Principles of Optics, (Pergamon Press, 1980).

]
Znm(ρ,θ)=Rnm(ρ)Radialpart{2cos|m|θ(m>0)1(m=0)2sin|m|θ(m<0),Angularpart
(5)
where the indices n and m are mode ordering numbers which denote the radial and the azimuthal orders respectively.

As we mention before, Eq. (2) is based on an expansion in Zernike polynomials. Since the polynomials are slow variant functions, a small amount of them may not provide the spatial spectral components necessary to reconstruct the wavefront we wish to represent. On the other hand increasing the number of terms can simply propagate high frequency noise generated in different processing stages, such as the artifacts generated by the inverse Radon transform. Thus, it is apparent that a compromise must be made in order to preserve the information that we pass to the Eq. (2) without introducing unnecessary terms. We describe two criteria that sets limits to the number of Zernike polynomials relevant to the expansion or, in other words, upper bounds for n and m.

5.1. Radial limit

The first constraint concerns the radial part of the Zernike polynomial. An important property of these polynomials is that the normalization has been chosen so that, for all permisible values of n and m, Rnm(1)=1. We will therefore express the radial coordinate normalized by the aperture radius of our measurements R. Then, since the normalized minimum resolvable spatial feature is Δρ/R the associated angular spatial frequency is
kmax=RΔρ.
(6)

Consequently we will neglect Zernike polynomials which spectral components fall out of the range [0, kmax]. The radial part of the Fourier transform of the Zernike polynomials (𝔕n(k)) is a function that depends on the nth-order Bessel function of the first kind [32

32. G. M. Dai, “Zernike aberration coefficients transformed to and from Fourier series coefficients for wavefront representation,” Opt. Lett. 31, 501–503 (2006). [CrossRef] [PubMed]

]. This spectral density have the shape of a bandpass spatial filter which central frequency increases with n. Therefore high order Zernike polynomials will contribute to carry high frequency noise. Our criterion is based on including only the terms necessary to describe the data.

At this point we define a ratio that provides a normalized power density of |𝔕n(k)| in the mentioned range as a function of the radial degree:
β=0kmax|𝔕n(k)|2dk0|𝔕n(k)|2dk.
(7)

For a given n, β is a measure of the amount of power spectral density that lies within the range, see Fig. 4. Consider a particular kmax, β decreases with increasing n, because the central frequency of the Fourier transform lies further. Alternatively, as kmax increases, the spectral density of more polynomials lay within the range and therefore the power ratio β increases. Four cases are plotted to illustrate this behavior.

Fig. 4 Evolution of β as a function of the radial polynomial degree n (evaluated within the range 0 < k < kmax). For the experimental conditions described in this paper kmax = 25.6. The β parameter is above 99% until n = 21.

We will discard polynomials for which the power ratio β falls 1% (a −0.45 dB change). To some extent, this threshold is taken arbitrarily: due to the high spatial resolution, kmax is large, and the curves fall slowly. Since our measurement system is slow no information is carried by high spatial frequencies. Therefore including high order polynomials in the expansion becomes less and less relevant. More importantly, if we employ too few polynomials we are filtering the signal: smoothing of the maps and loss of contrast will result.

Zernike polynomials are typically used to expand the wavefront aberration of optical systems. Consequently, low order zernike polynomials are often sufficient to describe aberrations such as tilt, defocus, astigmatism or coma. However, in other applications, where more complicated surfaces are to be described, such as the topography of mechanical samples [33

33. S. V. Pinhasi, R. Alimi, S. Eliezer, and L. Perelmutter “Fast optical computerized topography,” Phys. Lett. A 374, 2798–2800 (2006). [CrossRef]

], a larger set of polynomials may be required. In our case the characteristics of the irradiance maps naturally requires higher order polynomials. According to Eq. (6) our experimental conditions lead to kmax = 25.6. In this case β is above the predefined threshold (β ≥ 0.99), for nmax = 21 (see Fig. 4). Therefore polynomials to the radial order 21 will be employed in the expansion. That accounts for the first 253 polynomials in the Noll notation [25

25. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976). [CrossRef]

].

5.2. Azimuthal limit

Therefore |m|max = m′, i.e. the maximum azimuthal index that should be employed in the expansion corresponds with the number of knife-edge cuts in our measurements.

Another way of seeing this relation follows from the back-projection process. The inverse Radon transform is based on the Fourier slice theorem [30

30. R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using MATLAB, (Gatesmark Publishing2008).

]. This states that the complete back-projected image, resulting from a set of parallel-beam projections at Δθ angular variations, is computed from the discrete 1D Fourier transform of each projection. In the 2D spatial frequency domain that implies that these 1D transforms are arranged in lines at Δθ incremental angles. The final image is obtained by integrating all the inverse 1D transforms. By considering polynomials with angular frequencies higher than the ones present in the 2D transform we may be introducing signals that were not there to begin with.

By merging together the radial and azimuthal criteria we can select a set of Zernike polynomials that belongs to the accesible expansion in a reliable manner. Those polynomials having a larger value of their radial or azimuthal order fall beyond the measurement capabilities of the experimental set-up.

We show an example of the consequences of applying both conditions. Consider a set of measurements that lead to nmax = 4 and |m|max = 2. If we arrange the Zernike polynomials in a triangle, as illustrated in Fig. 5, the vertical position sets its n order and the horizontal its m order. The radial constrain sets limits to the height of this triangle and the azimuthal condition establishes the maximum width. Only the functions within these limits, drawn in a gray box, should be called upon in the expansion. If we consider the radial constraint we would use a set of polynomials with n ≤ 4, a total number of N = 15. However taking into account the azimuthal limit we would have to remove four of them: Z3±3 and Z4±4 and therefore use a new set of N′ = 11 polynomials.

Fig. 5 Representation of Zernikes arranged according to their radial and azimuthal parameters ( Znm). As an example, if experimental conditions are defined with nmax = 4 and |m|max = 2, only the polynomials within the gray area should be employed.

Analogously, in our case, we would have to remove from our list of polynomials with n ≤ 21 (N = 253) the ones with |m| > 9. Leaving the series expansion with N′ = 169 terms.

6. Phase retrieval results and discussion

Fig. 6 (a) Irradiance contour maps reconstructed using the tomographic technique. The planes are separated Δz = 108 μm. (b) Five phase maps can be retrieved from seven irradiance maps.

To check the consistency of experimentally measured irradiance data we have evaluated the energy conservation law by integrating δzI within the aperture (of radius R = 200 μm). Loss of intensity in a region arises through energy flow across the boundary of the region. For a phase solution to exist this value must be zero. δzI is computed from non-consecutive planes (IP+1 and IP−1), where P goes from 2 to 6. Integration of δzI yields a value below 10−6 for any P and it has a minimum, below 10−8, at P = 4. We will consider that negligible.

Then Eq. (2) was employed to retrieve the wavefront associated to this intensity field. We used the subset of N′ Zernike polynomials calculated previously and the parameter Nf = 110. From each group of three consecutive irradiance planes (IP+1, IP, IP−1), a phase map could be generated. Hence a total number of five phase maps were recovered and they are plotted in Fig. 6(b). For instance, to retrieve the phase map at plane 2, the irradiance maps from planes 1, 2 and 3 were employed. Astigmatism clearly governs its propagation and change of curvature from positive to negative can also be seen as the wavefront crosses plane 4. Fig. 7(a) shows the phase map recovered at this plane.

Fig. 7 (a) Phase map reconstructed at the focal plane using N′ Zernike coefficients. (b) Difference between the phase map reconstructed with N′ and one with N (magnitude in radians).

To evaluate the impact of a particular Zernike expansion in the retrieval algorithm we have compared the wavefront reconstructed with a N′ expansion set of polynomials, plotted in Fig. 7(a), and the one produced with the N set (at the same plane). Fig. 7(b) shows the difference between both wavefronts. Clearly the extra number of polynomials has an impact in the phase retrieval. The highest differences are in the order of 10%, and they are particularly located at the edges of the aperture. This supports the idea that, as we explained in section 5, the extra Zernike polynomials with |m| > |m|max, (polynomials with strong angular changes), introduce more uncertainty in the final recovered wavefront.

Finally, it should be emphasized that expanding this approach to high-numerical apertures systems is possible. Indeed, vectorial effects could be describe within this experimental framework [35

35. J. M. Braat, S. Van Haver, A. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point spread functions,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), 349–468. [CrossRef]

].

7. Conclusions

Characterization of light beams, both in amplitude an phase, is required to test nanophotonic devices such as optical antennas. This process becomes an issue when the width of the beam is of the same order than the pixel size of an imaging system, which happens differently depending on the available detection technology at the corresponding spectral range. Moreover, a reliable recovery of the wavefront typically reveals new problems when setting interferometric measurements. As a consequence, alternative methods have been developed.

In this contribution we have addressed measurements in the IR range and a tomographic method has been used to reconstruct the irradiance map. The tomography is obtained from multi-angle knife-edge scans across the beam. Then, the phase has been retrieved using a solution of the TIE, which is based on a truncated expansion of both the irradiance and the phase information in a basis of Zernike polynomials.

The results show how it can be recovered both phase and modulus of a laser beam at the focal region of a lens. We have illustrated the method using a lens that has been tilted to produce an orthogonal astigmatic beam. The orthogonal beam waists appear at different planes along the propagation axis. The obtained phase and irradiance maps are in accordance with this situation and a complete characterization of the electric field at this region was presented.

Although we have worked this approach out under paraxial conditions, further extension to high numerical aperture systems are feasible, hence putting forward an experimental method to ease the characterization of antenna-coupled detectors and, in general, nanophotonic devices.

Acknowledgments

This work was supported by the Ministerio de Ciencia e Innovacin through the grant ENE2009-14340.

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R. L. Olmon, P. M. Krenz, B. A. Lail, L. V. Saraf, G. D. Boreman, and M. B. Raschke, “Determination of electric-field, magnetic-field, and electric-current distributions of infrared optical antennas: A near-field optical vector network analyzer,” Phys. Rev. Lett. 105, 167403 (2010). [CrossRef]

9.

C. Fumeaux, G. D. Boreman, W. Herrmann, F. K. Kneubhl, and H. Rothuizen, “Spatial impulse response of lithographic infrared antennas,” Appl. Opt. 38, 37–46 (1999). [CrossRef]

10.

J. Alda, C. Fumeaux, L. Codreanu, J. A. Schaefer, and G. D. Boreman, “Deconvolution method for two dimensional spatial response mapping of lithographic infrared antennas,” Appl. Opt. 383993–4000 (1999). [CrossRef]

11.

J. M. López-Alonso, B. Monacelli, J. Alda, and G. D. Boreman, “Uncertainty analysis in the measurement of the spatial responsivity of infrared antennas,” Appl. Opt. 44, 4557–4568 (2005). [CrossRef] [PubMed]

12.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

13.

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]

14.

J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A 15, 1662–1669 (1998). [CrossRef]

15.

W. Chen and X. Chen, “Quantitative phase retrieval of complex-valued specimens based on noninterferometric imaging,” Appl. Opt. 50, 2008–2015 (2011). [CrossRef] [PubMed]

16.

H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. H. Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A 23, 1179–1200(2006). [CrossRef]

17.

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nature Phys. 2, 101–104 (2006). [CrossRef]

18.

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

19.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001). [CrossRef]

20.

J. M. Rico-García, L. M. Sánchez-Brea, and J. Alda, “Application of tomographic techniques to the spatial-response mapping of antenna-coupled detectors in the visible,” Appl. Opt. 47, 768–775 (2008). [CrossRef] [PubMed]

21.

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, 7244–7249 (2011). [CrossRef] [PubMed]

22.

M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983). [CrossRef]

23.

F. Roddier, “Wavefront sensing and the irradiance transport equation,” Appl. Opt. 29, 1402–1403 (1990). [CrossRef] [PubMed]

24.

T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A 13(8), 1670–1682 (1996). [CrossRef]

25.

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976). [CrossRef]

26.

J. Alda, J. Alonso, and E. Bernabéu, “Characterization of aberrated laser beams,” J. Opt. Soc. Am. A 14, 2737–2747 (1997). [CrossRef]

27.

S. Barbero and L. N. Thibos, “Error analysis and correction in wavefront reconstruction from the transport-of-intensity equation,” Opt. Eng. 45, 094001 (2006). [CrossRef]

28.

M. Soto, E. Acosta, and S. Rios “Performance analysis of curvature sensors: optimum positioning of measurement planes,” Opt. Express 11, 2577–2588 (2003). [CrossRef] [PubMed]

29.

P. Toft, “The Radon transform: theory and implementation,” Ph. D. dissertation (Technical University of Denmark, 1996), http://petertoft.dk/PhD.

30.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using MATLAB, (Gatesmark Publishing2008).

31.

M. Born and E. Wolf, Principles of Optics, (Pergamon Press, 1980).

32.

G. M. Dai, “Zernike aberration coefficients transformed to and from Fourier series coefficients for wavefront representation,” Opt. Lett. 31, 501–503 (2006). [CrossRef] [PubMed]

33.

S. V. Pinhasi, R. Alimi, S. Eliezer, and L. Perelmutter “Fast optical computerized topography,” Phys. Lett. A 374, 2798–2800 (2006). [CrossRef]

34.

J. Alda, “Laser and Gaussian beam propagation and transformation,” in Encyclopedia of Optical Engineering (Marcel Dekker Inc. 2003), 999–1013.

35.

J. M. Braat, S. Van Haver, A. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point spread functions,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), 349–468. [CrossRef]

OCIS Codes
(010.7350) Atmospheric and oceanic optics : Wave-front sensing
(100.5070) Image processing : Phase retrieval
(100.6950) Image processing : Tomographic image processing
(140.3295) Lasers and laser optics : Laser beam characterization

ToC Category:
Image Processing

History
Original Manuscript: June 20, 2012
Revised Manuscript: July 23, 2012
Manuscript Accepted: July 23, 2012
Published: October 3, 2012

Citation
Manuel Silva-López, José María Rico-García, and Javier Alda, "Measurement limitations in knife-edge tomographic phase retrieval of focused IR laser beams," Opt. Express 20, 23875-23886 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-21-23875


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References

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  2. G. Volpe, S. Cherukulappurath, R. J. Parramon, G. Molina-Terriza, and R. Quidant, “Controlling the Optical Near Field of Nanoantennas with Spatial Phase-Shaped Beams,” Nano Lett.93608–3611 (2009). [CrossRef] [PubMed]
  3. E. C. Kinzel, J. C. Ginn, R. L. Olmon, D. J. Shelton, B. A. Lail, I. Brener, M. B. Sinclair, M. B. Raschke, and G. D. Boreman, “Phase resolved near-field mode imaging for the design of frequency-selective surfaces,” Opt. Express20, 11986–11993 (2012). [CrossRef] [PubMed]
  4. E. Cubukcu, N. Yu, E. J. Smythe, L. Diehl, K. B. Crozier, and F. Capasso, “Plasmonic Laser Antennas and Related Devices,” J. Sel. T. Q. Elec.14, 1448–1461 (2008). [CrossRef]
  5. L. Novotny and N. Van Hulst, “Antennas for light,” Nature Phot.5, 83–90 (2011). [CrossRef]
  6. J. Alda, J. M. Rico-García, J. M. López-Alonso, and G. D. Boreman, “Optical antennas for nanophotonics applications,” Nanotechnology16, S230, (2005). [CrossRef]
  7. M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antenna,” Science332, 702–704, (2011). [CrossRef] [PubMed]
  8. R. L. Olmon, P. M. Krenz, B. A. Lail, L. V. Saraf, G. D. Boreman, and M. B. Raschke, “Determination of electric-field, magnetic-field, and electric-current distributions of infrared optical antennas: A near-field optical vector network analyzer,” Phys. Rev. Lett.105, 167403 (2010). [CrossRef]
  9. C. Fumeaux, G. D. Boreman, W. Herrmann, F. K. Kneubhl, and H. Rothuizen, “Spatial impulse response of lithographic infrared antennas,” Appl. Opt.38, 37–46 (1999). [CrossRef]
  10. J. Alda, C. Fumeaux, L. Codreanu, J. A. Schaefer, and G. D. Boreman, “Deconvolution method for two dimensional spatial response mapping of lithographic infrared antennas,” Appl. Opt.383993–4000 (1999). [CrossRef]
  11. J. M. López-Alonso, B. Monacelli, J. Alda, and G. D. Boreman, “Uncertainty analysis in the measurement of the spatial responsivity of infrared antennas,” Appl. Opt.44, 4557–4568 (2005). [CrossRef] [PubMed]
  12. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik35, 237–246 (1972).
  13. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt.21, 2758–2769 (1982). [CrossRef] [PubMed]
  14. J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects,” J. Opt. Soc. Am. A15, 1662–1669 (1998). [CrossRef]
  15. W. Chen and X. Chen, “Quantitative phase retrieval of complex-valued specimens based on noninterferometric imaging,” Appl. Opt.50, 2008–2015 (2011). [CrossRef] [PubMed]
  16. H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. H. Riege, C. Cui, M. R. Howells, R. Rosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, “High resolution ab initio three-dimensional x-ray diffraction microscopy,” J. Opt. Soc. Am. A23, 1179–1200(2006). [CrossRef]
  17. H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused x-ray fields,” Nature Phys.2, 101–104 (2006). [CrossRef]
  18. A. H. Firester, M. E. Heller, and P. Sheng, “Knife-edge scanning measurements of subwavelength focused light beams,” Appl. Opt.16, 1971–1974 (1977). [CrossRef] [PubMed]
  19. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs “The focus of light - theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B72, 109–113 (2001). [CrossRef]
  20. J. M. Rico-García, L. M. Sánchez-Brea, and J. Alda, “Application of tomographic techniques to the spatial-response mapping of antenna-coupled detectors in the visible,” Appl. Opt.47, 768–775 (2008). [CrossRef] [PubMed]
  21. 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. Express19, 7244–7249 (2011). [CrossRef] [PubMed]
  22. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am.73, 1434–1441 (1983). [CrossRef]
  23. F. Roddier, “Wavefront sensing and the irradiance transport equation,” Appl. Opt.29, 1402–1403 (1990). [CrossRef] [PubMed]
  24. T. E. Gureyev and K. A. Nugent, “Phase retrieval with the transport-of-intensity equation. II. Orthogonal series solution for nonuniform illumination,” J. Opt. Soc. Am. A13(8), 1670–1682 (1996). [CrossRef]
  25. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am.66, 207–211 (1976). [CrossRef]
  26. J. Alda, J. Alonso, and E. Bernabéu, “Characterization of aberrated laser beams,” J. Opt. Soc. Am. A14, 2737–2747 (1997). [CrossRef]
  27. S. Barbero and L. N. Thibos, “Error analysis and correction in wavefront reconstruction from the transport-of-intensity equation,” Opt. Eng.45, 094001 (2006). [CrossRef]
  28. M. Soto, E. Acosta, and S. Rios “Performance analysis of curvature sensors: optimum positioning of measurement planes,” Opt. Express11, 2577–2588 (2003). [CrossRef] [PubMed]
  29. P. Toft, “The Radon transform: theory and implementation,” Ph. D. dissertation (Technical University of Denmark, 1996), http://petertoft.dk/PhD .
  30. R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using MATLAB, (Gatesmark Publishing2008).
  31. M. Born and E. Wolf, Principles of Optics, (Pergamon Press, 1980).
  32. G. M. Dai, “Zernike aberration coefficients transformed to and from Fourier series coefficients for wavefront representation,” Opt. Lett.31, 501–503 (2006). [CrossRef] [PubMed]
  33. S. V. Pinhasi, R. Alimi, S. Eliezer, and L. Perelmutter “Fast optical computerized topography,” Phys. Lett. A374, 2798–2800 (2006). [CrossRef]
  34. J. Alda, “Laser and Gaussian beam propagation and transformation,” in Encyclopedia of Optical Engineering (Marcel Dekker Inc. 2003), 999–1013.
  35. J. M. Braat, S. Van Haver, A. E. M. Janssen, and P. Dirksen, “Assessment of optical systems by means of point spread functions,” in Progress in Optics, E. Wolf, ed. (Elsevier, 2008), 349–468. [CrossRef]

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