1. Introduction

The ability to sculpt the transverse optical intensity profile of a light field leads to many interesting insights into fundamental and applied photonics. In this area, the last decades have seen interest in so termed “non-diffracting light fields”. The Bessel mode [1, 2] has been a prime example of such a field but others exist, most notably the Airy beam that not only is immune to diffraction, but additionally its intensity exhibits a transversal acceleration as it propagates [3]. These “non-diffracting” beams may theoretically propagate for an infinite distance without spreading, assuming the presence of infinite energy. Experimentally, it is possible to propagate such a beam over a finite distance without appreciable spreading.

The existence of the Airy wavepacket was predicted 30 years ago in the context of a solution to the Schrödinger equation [4]. It is the equivalence between the Schrödinger equation and the paraxial wave equation that allowed their realization in the optical domain in 2007 [3]. The generation of this beam has sparked interest due to its unusual “non-diffracting” properties and its parabolic trajectory which has recently been employed for particle clearing using optical forces at the microscopic scale [5]. The existence of a transverse component in the acceleration of the wavepacket is the origin of this parabolic propagation. The Poynting vector follows a line that is tangent to the propagation direction of the beam [6]. Similar to the Bessel beam [7], the Airy beam reconstructs around obstructions placed in the beam path [8]. Experimentally, the Airy beam has finite energy and is best represented by an exponentially apertured perfect Airy beam [9].

The Airy beam can be thought of as an interference pattern, similar to the Bessel beam [1, 2] which itself is generated using an axicon to interfere conical waves. In the case of the Airy beam, this interference pattern is based on a cubic phase mask that corresponds to a gradual tilt of the phase front. As a result the constituent beams constructively interfere on a parabola. Since interference requires the presence of spatial coherence, it is interesting to investigate the effects of partial coherence on the propagation and generation of Airy beams. To study the wavelength dependence of the Airy beam parameters in the context of spatial coherence we use a white light supercontinuum source.

In this paper, we generate a “white light” Airy beam using a spatial light modulator (SLM) in conjunction with a supercontinuum light source. This permits a detailed investigation of the Airy beam propagation parameters. More precisely, we determine the wavelength and spatial coherence dependence of the deflection coefficient, b ₀, the characteristic length, x ₀, and the aperture coefficient, a ₀, for this intriguing optical field. We first present an extended Gaussian-Schell model where we use Huygen’s integral to propagate the Airy beam through general ABCD optical elements. After describing the experimental setup, we present, discuss and verify the results.

2. Theory

2.1. Propagation of the coherent Airy beam

The Airy beam, the exponentially apertured Airy beam [9] and Gaussian beams are solutions of the paraxial equation

where z is the direction of propagation, k=2πn/λ the wavevector and exp(i(ωt-kz)) the carrier plane wave considered here. The electric, E, and magnetic, H, fields associated with the scalar field u₀(x,y, z) are determined via the magnetic vector potential A and the Lorenz gauge conditions,

where x̂ represents the x-axis unit vector and ε ₀ and µ ₀ are the permittivity and permeability of free space. For clarity, we have omitted the time dependence exp(iωt).

In the case of the three beams mentioned above, the scalar field u₀(x,y,z)=u_x(x,z)u_y(y,z) can be decomposed into two independent parts, u_x(x,z) and u_y(y,z), each solutions of:

In the following, without any loss of generality, we consider only the transverse dimension x. A symmetric 2D solution can be obtained by replacing the coordinate x with y and multiplying the two scalar fields together.

The scalar field, in the apex plane of the parabola (plane z=0 for the parabola defined by z=b ₀ x ²), of an exponentially apertured Airy beam is

where Ai is the Airy function, x ₀ its characteristic length and a0 the aperture coefficient which determines the beam propagation distance. The associated spatial Fourier transform, û_x, of this beam is

which describes the scalar field at the SLM plane (z=zSLM); the first exponential term represents the incident Gaussian beam and the second represents the cubic phase mask generated on the SLM. Here k_x represents the transversal K-space wavevector in the x direction and corresponds to the x coordinate in the SLM plane. Note that changing the cubic phase term in the SLM plane changes directly the characteristic length x ₀ of the Airy beam, while the aperture coefficient is given by a ₀=ω ² ₀/(4x ² ₀), where ω ₀ is the beam waist of the incident Gaussian beam. This equation links the incident beam parameters to the Airy beam parameters.

We now solve the paraxial equation for the initial conditions using the free-space Huygens-Fresnel integral [10]

where x_m(z)=z ²/(4k ² x ³ ₀) defines the lateral position of the Airy beam. This integral is fully equivalent with the paraxial equation, Eq. 5, and determines the scalar field u_x at a distance z as a function of the field at z=0.

Similarly, the propagation of a beam through a paraxial optical system defined by the general ABCD matrix can be described through a generalized Huygens-Fresnel integral of the form:

where u ₁(x ₁) and u ₂(x ₂) are respectively the fields in the initial and final transverse planes. The coefficients A, B, C and D are obtained through the matrix product of all associated matrices of the optical elements involved [10]. Considering again the exponentially apertured Airy beam, $u_1\left(x_1\right)=\mathrm{Ai}(x_1⁄x_0)e^{a_0{x}_1⁄x_0}$, as the initial field we can now determine the field profile after the optical system

where we used the general property of the ABCD matrix, AD-BC=1. We remark that the propagation through an optical system does conserve the intensity profile of the exponentially apertured Airy beam in the same way as the free space propagation does. This result is similar to [11].

Using the free space Huygens-Fresnel integral, Eq. 9, for a distance z after the ABCD element we can determine the path of the beam profile. This path is no longer parabolic and changes to a rational polynomial of the form

This is the most general transverse motion of the Airy beam. A parabolic path occurs when the coefficient C=0, this is the case for the experiments presented here.

In the general case of Eq. 12, we note the singular behaviour of the focussed Airy beam at the position A+Cz=0 where the lateral shift diverges and the beam seems to disappear at +∞ to reappear at -∞. This divergent behaviour is accompanied by the “wavelength” of the Airy oscillations becoming infinitely small rendering the paraxial approximation invalid. It is interesting to remark that this behaviour is also present in the case of Schrödinger’s equation where the equation does maintain its validity. In this case, it is the size of the initial wavefunction that determines the minimal wavelength probed.

2.2. Partially coherent Airy beams from Gaussian Schell-model sources

The second order cross spectral density function describes partially coherent beams. It corresponds to the measure of the degree of coherence between two transverse points, r ₁ and r ₂, in an optical field [12]:

where µ(r ₁–r ₂) is the spectral degree of coherence which has a value of 1 for perfect coherence. Within the Gaussian Schell-model this spectral coherence is given by

where σ_µ is the coherence length. The average intensity of the partially coherent beam at a propagation distance z is given by I(x,y,z)=W(r,r,z).

Using the second order cross spectral density function together with the Huygens integral we can study the propagation of a partially coherent beam through general ABCD optical elements. Here, we are interested in the initial scalar field of the form:

where the first product corresponds to the incident Gaussian beam, while the second to the cubic phase imposed on the beam by the SLM. Practically, $c_0=\sqrt[3]{6\varphi }⁄l$ where l is the hologram side length and ϕ the maximal phase shift across the hologram.

After the SLM, we propagate through a single lens to form the far field in its focal plane. The associated ABCD matrix is given by

$\left(\begin{array}{cc}\multicolumn{1}{c}{-z⁄f}& \multicolumn{1}{c}{f}\\ \multicolumn{1}{c}{-1⁄f}& \multicolumn{1}{c}{0}\end{array}\right)=\left(\begin{array}{cc}\multicolumn{1}{c}{1}& \multicolumn{1}{c}{f+z}\\ \multicolumn{1}{c}{0}& \multicolumn{1}{c}{1}\end{array}\right)\left(\begin{array}{cc}\multicolumn{1}{c}{1}& \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{-1⁄f}& \multicolumn{1}{c}{1}\end{array}\right)\left(\begin{array}{cc}\multicolumn{1}{c}{1}& \multicolumn{1}{c}{f}\\ \multicolumn{1}{c}{0}& \multicolumn{1}{c}{1}\end{array}\right)$

Integrating the Huygens integral we find that the intensity of the apertured Airy beam is given by:

$I(x,y,z)\propto \exp \left(2\genfrac{}{}{0.1ex}{}{a_0}{x_0}\left(x-{b-}_0{z}^2\right)\right)\mathrm{Ai}\left(\genfrac{}{}{0.1ex}{}{x-b_0{z}^2}{x_0}+\genfrac{}{}{0.1ex}{}{{\mathrm{ia}}_0}{{\mathrm{kx}}_0^2}z+a_0^2\right)\mathrm{Ai}\left(j^2\left(\genfrac{}{}{0.1ex}{}{x}{x_0}-b_0{z}^2-\genfrac{}{}{0.1ex}{}{{\mathrm{ia}}_0}{{\mathrm{kx}}_0^2}z+a_0^2\right)\right)$$\exp \left(2\genfrac{}{}{0.1ex}{}{a_0}{x_0}\left(y-b_0{z}^2\right)\right)\mathrm{Ai}\left(\genfrac{}{}{0.1ex}{}{y-b_0{z}^2}{x_0}+\genfrac{}{}{0.1ex}{}{{\mathrm{ia}}_0}{{\mathrm{kx}}_0^2}z+a_0^2\right)\mathrm{Ai}\left(j^2\left(\genfrac{}{}{0.1ex}{}{y}{x_0}-b_0{z}^2-\genfrac{}{}{0.1ex}{}{{\mathrm{ia}}_0}{{\mathrm{kx}}_0^2}z+a_0^2\right)\right)$

where j ³=−1, Ai(x) is the Airy function and

Here we observe that only the aperture coefficient depends on the coherence length σ_µ. This shows that the deflection coefficient b ₀ and the characteristic length x ₀ are not affected by the spatial coherence of the generating beam. This means that the general beam properties remain the same regardless of the degree of spatial coherence. In contrast, the aperture coefficient a ₀, which determines the distance over which the Airy beam propagates, is affected by the degree of spatial coherence.

The beam quality factor of the incident Gaussian beam is defined by its waist ω ₀ and Rayleigh range z_r and is given by

The spatial coherence length is given by [12] σ ² _µ=w ² ₀/((M ²)²−1. Using this relation we can determine the aperture coefficient a ₀

as a function of the beam waist ω_slm of the Gaussian beam incident on the SLM. We base our experimental data upon this equation as well as Eq. 16–20.

3. Experimental setup

We used a supercontinuum source (Fianium Ltd. 4ps, 10MHz) to generate a “white light” Airy beam. Appropriate filtering was used to explore the relative propagation of the Airy beam for four of the source wavelength components, obviating the need to change sources and system to explore the propagation characteristics of the Airy light field. The experimental set up is shown in Fig. 1 and is similar to that used by Siviloglou et al. [13]. The supercontinuum source has a spectral bandwidth of 464–1750nm and an output power of 5.5W. The visible wavelength range was selected for the experiment using a dielectric mirror and the supercontinuum spectrum after this mirror is shown in Fig. 1. We used a Labview program to generate the Airy beam hologram which was displayed on a spatial light modulator (SLM, Holoeye LC-R 2500). The hologram consisted of the superposition of a 2D cubic phase mask and a diffraction grating. This imposed a cubic phase profile on the incident Gaussian beam. The first order of diffraction was then Fourier transformed using a lens to generate the Airy beam. This diffraction of the SLM introduces a wavelength dependent dispersion. This was compensated by adjusting the grating spacing on the SLM hologram and imaging it onto a 10° prism oriented to induce an opposite sensed dispersion. A CCD camera (Basler PLA640-210gc) was used to view the resultant beam after lens L5 (see Fig. 1).

 

Fig. 1. On the left is the experimental set up. A visible mirror M (Comar 25MF02) was used to select the visible wavelength range from the source. On the right is the spectrum taken after L5. Wavelength components were selected using interference filters F (Comar, 3nm bandwidth). L1 and L2 are achromatic lenses used to expand the beam with focal lengths of 50mm and 500mm respectively. The grating was imaged onto a 10° prism using achromatic lenses L3 (f=500mm) and L4 (f=200mm). An aperture was used to select the first order diffracted beam. The dispersion compensated Airy beam was viewed on a CCD camera via achromatic lens L5 (f=250mm).

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For the wavelength dependent studies, an interference filter (3nm bandwidth) was used to pass each of the chosen wavelengths and a CCD was positioned after lens L3. Airy beams were generated at 515nm, 546nm, 578nm and 633nm wavelengths. Each wavelength diffracted at a different angle from the SLM hologram, it was therefore necessary to maintain the beam path through the center of the optics by adjusting the grating spacing on the SLM for each wavelength. For a two dimensional Airy beam with a wavelength λ and a characteristic length x ₀, the deflection coefficient of the beam from a straight path is given by [3]:

where the √2 factor arises due to the Airy beam being two-dimensional.

The deflection coefficient, b ₀, was found in each case by determining the transverse displacement of the beam as it propagated. Images of the beam oriented at 90° and 270° were taken at 2cm intervals along the propagation axis using a CCD that was placed on rails parallel to that axis. This resulted in around 15 data points for each case. Correction for any mismatch between the axis of the camera rails and the beam propagation axis was achieved by taking data for Airy beams with opposite deflections (90° and 270° beams), the data points were then averaged. A parabola was fitted to this combined data and the deflection coefficient was determined to be the z ² coefficient of the fit.

Using the images of our beams at the focus, the values of x ₀ and the aperture coefficient, a ₀, were found for each beam. The experimental x ₀ values were determined by fitting the Airy function to cross sections of the two beams (oriented at 90° and 270°) at z=0, the average value was then taken to be the experimentally measured x ₀. The a ₀ values were determined by fitting the Airy function to a cross section of the beams.

4. Results and discussion

A white light Airy beam was generated using a spatial light modulator to impose a phase shift of 40×2π over the beam using a hologram of side length 0.0146m, similar to Siviloglou et al. [13]. In Fig. 2 is a cross-section profile of the beam, the insert shows an image of the beam at z=0. The cross-section profile of the Airy beam shows the lobes indicating that dispersion is well compensated. It is interesting to note that changing the grating spacing of the SLM hologram allows direct control over the spreading of the wavelengths in the Airy beam. This may be useful for enhancing the difference in the beam trajectories between wavelengths for purposes of optical micromanipulation and even optical sorting using such beams [5].

 

Fig. 2. Shown is the profile of a dispersion compensated “white light” Airy beam, the main lobe is shown by the dashed line and is taken from the cross section of an unsaturated picture. The insert shows an image of the beam at z=0 with the red line indicating where the cross-section was taken.

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Deflections of the Airy beam in the wavelength study are shown as a function of propagation distance in Fig. 3; on the right are images of each Airy beam at z=0. The 515nm Airy beam had the largest deflection coefficient and as the wavelength was increased the deflection coefficient decreased, see Fig. 3. Error bars of ±10% are used for the deflection coefficient values. The parabolas have been arranged for their minima to overlap at the origin for clear comparison of the beam deflections. It is interesting to note that the x0 value also varies with wavelength (see Table 1), therefore the side lobes of the Airy beam are more closely spaced for shorter wavelength beams. If this was not the case then the deflection coefficient would vary only as λ ² (see Eq. 21), resulting in the opposite trend of the deflection coefficient increasing with increasing wavelength. It is therefore extremely important to take into account the variation of x0 with wavelength. This wavelength dependence of the lobe spacing compared to that seen in a “white light” Bessel beam [14], where the rings of shorter wavelength beams are closer together than those of longer wavelength beams.

 

Fig. 3. Plotted are the data points showing how the position of the main beam lobe varied along the propagation direction, the lines indicate the parabolic fit to these points. The inserted table displays the values of the deflection coefficients b ₀. On the right are pictures of the beams taken at z=0, (a) 515nm, (b) 546nm, (c) 578nm and (d) 633nm.

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Table 1. Beam parameter x₀ determined using different methods.

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Table 1 shows the experimental (direct fit) x ₀ values and two calculated x ₀ values. The column labelled “Eq. 21” has x ₀ values determined by Eq. 21 using the deflection values shown in Fig. 3. The column labelled “Eq. 16” has x ₀ values determined using Eq. 16.

The aperture coefficient a0 was also determined for each beam, the results are shown in Fig. 4. The blue (diamond) points show the values obtained by fitting an airy function to horizontal cross sections of our beams. The red (square) points indicate the values determined using Eq. 20 which predicts the expected a ₀ values based on the M² value, the beam spot size incident on the SLM, ω_slm and parameter c ₀ (described in section 2.2). The value of c ₀ for our system was calculated to be 785.9m⁻¹. The experimentally determined a ₀ values agree within 10% of the calculated values determined using Eq. 20. Table 2 shows the beam waists incident on the SLM ω_slm, the M² values and the experimental (direct fit) and calculated a ₀ values.

 

Fig. 4. A graph of how the aperture coefficient a ₀ for the horizontal lobes varies with the different wavelength beams, a dashed blue line is shown to guide the eye. Error bars of 10% are shown. The red (square) points show the values calculated using Eq. 20.

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Table 2. Beam parameter a₀ determined using different methods.

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The M ² values were determined for the horizontal axis of each beam after the SLM and Fourier lens. The M ² values were found using a beam profiler (Thorlabs BP104-vis) to determine how the beam waist varied with distance. The Rayleigh range was then used to calculate the M² values using Eq. 19. Where ω ₀ is the beam waist, λ is the wavelength and z_r is the Rayleigh range. The M² variability with wavelength probably originates from the different filters used to select the wavelengths.

5. Conclusion

We have modelled and generated a spatially partial coherent Airy beam showing clear cross-sectional profile of the beam’s lobes at a variety of wavelengths. As predicted by the extended Gaussian-Schell model presented here, we observe a wavelength and coherence dependence of the different Airy beam parameters. More precisely, we show that the deflection coefficient of an Airy beam decreases with increasing wavelength as a result of the wavelength dependence of the lobe spacing. The characteristic length of the Airy beam has been shown to increase linearly with wavelength. The aperture coefficient is not expected to be wavelength dependent, but due to varying M² beam quality values of the generating Gaussian beam, we observe a variation of this coefficient with wavelength. Lastly, the Airy beam shape (determined by x ₀ and b ₀) is not affected by the degree of spatial coherence but its propagation range (determined by a ₀) increases with increasing coherence.