1. Introduction

The phase of a paraxial optical beam is used in many applications, including holography, optical metrology, characterization of the optical properties of materials, etc. [1] More recently, the transversal phase of an optical beam has acquired great importance in the field of singular optics [2], where phase singularities of the optical field are used in many applications as micromachining [3], microfluidics [4], data storage [5], new imaging schemes [6, 7] or new astronomical instruments [8]. Also, the transversal spatial structure, both phase and amplitude, of an optical beam defines its orbital angular momentum content [9, 10] which can be transferred to material particles using optical tweezers [11], used to directly control the state of atomic ensembles [12, 13, 14] or in quantum information applications [15].

Nevertheless, the phase of an optical beam is a property which is usually only accessible using interferometric methods, where a reference beam with well defined amplitude and phase is used to retrieve the properties of the desired beam. Interferometric methods usually demand very stable set-ups, which can be difficult to implement in every day applications. As an alternative to those interferometric systems, some self-referenced techniques have been proposed and implemented [16, 17, 18]. Recently one of those techniques, the so called spiral phase interferometry (SPI) [17, 18], is gaining interest in the optical community due to its simple implementation. It has already been successfully used in high resolution microscopy applications [19, 20, 21].

In principle, this technique could be used for characterizing an unknown optical beam, which could be of interest in the above mentioned applications where the transversal structure of an optical beam is used to codify information. In particular it could be very interesting in applications where the orbital angular momentum of an optical beam has to be measured. In this article we expose some of the problems to fulfil this program and a possible way to solve it, by using a slight variation of the usual SPI. We have experimentally implemented such a system and here we present a few examples of the results we obtained with complex beams.

2. Spiral phase contrast revisited

The spiral phase contrast method is based on the convolution of a given image with a spiral filter. To be more specific let us start with an initial paraxial beam with scalar amplitude E_in(x_in,y_in). In an experimental set-up, the input field would be optically Fourier transformed with a 2f system and multiplied by a set of phase masks H_k=_1,2,3 with the following shape:

where θ(x′,y′)=arctan $\frac{y\prime }{x\prime }$ is the azimuthal angle, R is a typically small radius, which separates the two regions of the filter and ${\alpha }_k=\frac{2\pi }{3}k$ is a constant phase which is different for every mask in the set.

After performing another optical Fourier transform, the resulting field is an interference between the two different parts of the beams, traversing the two zones of the mask. As it is nicely explained in ref. [19], when R tends to zero, the output field is the self-referenced interference of the input beam with a plane wave, with a well defined phase. Different masks in the set provide different phases of the plane wave, which in the end allows to reconstruct the initial beam’s intensity and phase.

This method has delivered nice results in the fields of microscopy [19, 21] and for phase modulated constant amplitude input beams [17, 18]. Our aim in this paper is to find a way to extend those results to arbitrary beams. This extension can be problematic in some cases and we will provide some solutions, which we have successfully experimentally tested. Let us start by writing the intensity of the output beam as recorded by the CCD camera:

where (x_in,y_in) represent the transversal coordinates of the input plane, (x_out,y_out) those of the output plane (CCD camera), and (x′,y′) the coordinates of the plane where the filtering takes place (after the first optical Fourier transform). For the sake of simplicity we have normalized the transversal coordinates so that we do not take into account the trivial magnification factors and inversions due to the set of lenses chosen. ℱ{g} represents the optical Fourier transform of the function g and the functions A(x_out,y_out) and B(x_out,y_out) are more easily expressed in cylindrical coordinates:

here, Ẽ_in(r′,θ′)=ℱ{E_in} is the Fourier transform of E_in.

The original decoding method [19] is based on the calculation of these two quantities:

Assuming that B(x_out,y_out) is sufficiently close to a plane wave, one can retrieve the separate information of A(x_out,y_out) and B(x_out,y_out). Unfortunately for general optical beams, this method runs into some problems. First, as already stated in the original paper, there is a limit on how small we can make R. The smaller R is, the smaller the amplitude B is, which reduces the amplitude of I_c. The natural limit for R is then given by the noise in the recording apparatus and also depends on the shape of the input beam. In typical optical beams, one needs R to be a significant fraction of the input beam, to be able to overcome the noise in a standard CCD camera. This invalidates the approximation of B(x,y) as a plane wave. Second, even in the case that B(x,y) is close to a plane wave, the original algorithm to retrieve B(x,y) does not give a single solution when the input beams contain points with zero intensity, as would be the case for beams carrying orbital angular momentum.

 

Fig. 1. Block diagram of our system. (a) represents the optical processing, OFT mean optical Fourier transform, and (b) represents the numerical post-processing to recover the field, FFT and IFFT are the fast Fourier transform and its inverse, and sqrt{} is the square root operation. More details are given in the text.

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In order to overcome these problems we have devised a slight variation of the original spiral phase contrast method. Our implementation can be seen in Fig. 1. To start with, we perform a pre-processing: we make an image of the Fourier transformed input beam, which allows us to find a proper point (x_R,y_R) where to center our filter.We look for a spot in the beam with a local maximum of intensity.With this simple pre-processing, we obtain several advantages: First, the I_c field will be maximized for a fixed radius R. Second, we also avoid zeros of the input beam which will make I_c close to zero. Finally, the amplitude of the beam within the circle of radius R in the filter is rather constant which will allow us to make some simplifications to retrieve the input beam information. In our implementation we performed this step visually, but it can be easily automatized with proper image processing algorithms. After the spot has been chosen, we proceed in the following way. First, we obtain the three filtered images as in the original spiral phase contrast method and we also record an additional image where the filter has been completely removed, i.e. we take an image of the intensity I ⁽⁰⁾ _out=|E_in(x_out,y_out)|², see Fig. 1(a). This image will be identical to the input beam, except for the trivial rescaling and inversion due to the optical Fourier transforming processes.

The reconstruction of E_in, which is shown in Fig. 1(b) is rather simple and reads:

where the errors in the reconstruction of the intensity are only due to the noise of the imaging and recording systems. On the other hand, the reconstruction of the phase provides a good approximation, as we will show below. When using the proper spot to place our filter, we assure that the function B(x_out,y_out) is very close to diffraction of a circular aperture:

except for trivial rescaling due to the optical Fourier transform. To obtain this expression one has to assume a flat amplitude across the circle R and a linear change in the phase. These approximations are based on the fact that we have chosen the right amplitude spot in the beam and that the radius R is small enough so that we can approximate any changes in the phase to first order. This change in the phase is responsible of the displacement of the diffraction function: (δ_x,δ_y). As we will see, this typically small displacement of the function does not affect our reconstruction. Finally, the added phase (x_Rx+y_Ry) in B(x,y) is due to the displacement of the filter and is corrected in the reconstruction Eq. (5). Note that even under this approximation, the phase of B(x,y) presents some radial phase singularities, i.e. there are π phase jumps at some radial positions, given by the Airy function. Although these singularities could be properly taken into account, usually we do not have to deal with them as they are out of the area of interest. Finally, taking into account Eq. (4) and (5) we observe that the reconstructed phase is actually the result of applying a mean filter of size R to the Fourier transformed beam.

 

Fig. 2. Numerical example of the reconstruction system. (a) Intensity of the input field. (b) Phase of the input field. (c) Amplitude of the Fourier transform of the input beam. The white dot indicates the position of the center of the filter. (d) One of the filters used, the white dot indicates the center of the filter. (e), (f), (g) Output intensities of the system, corresponding to the different filters used. (h) Intensity of the recovered field. (i) Phase of the recovered field.

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In Fig. 2 we present a numerical example of how the reconstruction works. Our input field consists in a beam with some phase singularities. The order of the phase singularities can be easily identified in Fig. 2(b), where we observe that the beam presents one single charged vortex and another second order vortex (where the phase twists twice around the singularity). Both vortices are separated by some distance. Figure 2(c) is the Fourier transform of this beam and the white spot represents the intensity maximum where we position the filters (one of the filters is shown in Fig. 2(d)). The 3 different images obtained after the filtering and the last Fourier transform are displayed in the panels (e), (f) and (g). Finally, in the last panels of the figure, (h) and (i), we present the numerically reconstructed beam with Eqs. (5), which are to be compared with panels (a) and (b).

3. Experimental set-up

 

Fig. 3. Sketch of the experimental setup. A computer generated phase hologram is illuminated with a collimated diode laser light to produce a Laguerre-Gaussian-like beam in the object plane, using the lens L1 and an iris (to select the first order of diffraction). Then L2 makes the Fourier transform of the object and puts it on the SLM surface, where the filters are displayed. After the filtering, we make the Fourier transform again with L3 and we rescale the image to fit the CCD chip with the imaging system 1. With the imaging system 2 we make an image of the SLM on the CCD to find a proper point where to center the filters.

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The experimental setup of our system is sketched in Fig. 3. Our source of light was a 810nm diode laser which was coupled to a single mode fiber to obtain a pure Gaussian spatial mode. The light from the output of the optical fiber was collimated and illuminated a computer generated phase hologram. As the holograms we use are custom made, we can produce the appropriate hologram to modulate the optical beam in the desired way. In the example in Fig. 3 we present a simple fork-like dislocation [22]. This kind of holograms are well known to produce superpositions of Laguerre-Gaussian beams [23, 24]. A numerical example of the intensity shape of the resulting beam can be seen in Fig. 3 in the object plane. In this case, the hologram dislocation is shifted from the center of the illuminating Gaussian beam. Then the output from the hologram is a superposition of a Gaussian beam and a Laguerre-Gaussian beam [23]. We use then lens L1 and an iris to select the first order of diffraction from the hologram.

With lens L2 we Fourier transform the object onto the surface of an spatial light modulator (SLM). The SLM allows us to display on real time the different filters needed for the protocol. Our SLM was set to work in phase mode (only affecting the phase of the incoming beam). L1 and L2 were chosen to magnify the beam to take advantage of the SLM surface.

Finally, with two flip mirrors (dashed lines in the figure) we could choose to direct the light from the SLM to a CCD camera either through an imaging system (imaging system 2) or with a Fourier transforming system (lens L3) rescaled (with lens L4) so that the resulting image fits the CCD chip.

First, we scanned with the imaging set-up the shape of the beam in the SLM, looking for maxima of intensity. Once we found a suitable zone, we switched to the Fourier set-up and took the four images needed for the protocol. The first one was taken with a blank filter in the SLM, thus we just retrieved the intensity pattern of the object plane. The three other images were taken with three different filters in the SLM as explained previously. Each filter consisted on a fork-like pattern (similar to that in the hologram of Fig. 3), but the position of the dislocation was covered with a circle of variable radius (depending on the visibility conditions). Every filter had a different relative phase in the circle.

 

Fig. 4. Characterization of a Gaussian beam with four embedded phase singularities. Upper row, simulation of the expected field. Lower row, experimental reconstruction. (a) and (c) Intensity pattern with the four zeroes associated with the four phase dislocations. (b) and (d) Phase pattern.

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In Fig. 4 we present an example of the experimental results we obtained with our system. The initial vortex beam traversed a hologram consisting on four phase dislocations, forming a square. In the upper row of the figure we present, for the sake of comparison, a numerical calculation of the beam we expected. In the calculation we used our knowledge of both the incoming beam and the hologram we used. Observe that the reconstruction phase follows remarkably well the expected features. The reconstructed phase is rather noisy far from the center of the beam, where the method is prone to give worse results as the noise of the camera is of the same order as the recovered signal. Note also that from the phase measurements we can observe that the beam has a small divergence, which can be observed from the curvature of the iso-phase lines. This is an indication that the laser beam was not perfectly collimated in the object plane. Finally, from the intensity measurements a small ellipticity in the beam can be observed. This is probably due to some inhomogeneities of the SLM and is in agreement with other series of measurements not shown here.

 

Fig. 5. Characterization of a Gaussian beam with a phase jump. Upper row, simulation of the expected field using a 0.8π phase jump. Lower row, experimental reconstruction. (a) and (c) Intensity pattern. (b) and (d) Phase pattern.

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Another example of the reconstruction process can be found in Fig. 5. Here, the hologram that we used consisted in a simple phase jump. The reconstruction shows that the jump was actually of approximately 0.8π, a value consistent with the design of our hologram. Note again the small curvature of the beam as in the previous case.

In conclusion, we have presented here a method to measure the amplitude and phase of Laguerre-Gaussian-like beams. This method is based on a small variation of the spiral phase interferometry technique. Our method avoids some technical problems that can be found in the reconstruction of finite sized beams and beams with phase singularities. We have shown a few examples of the use of our technique for the characterization of complex beams.