2.1. Analytic Solutions
Following the discussion from Berry [
23] for the propagation of the vortex, the transverse wavevectors follow the Poynting vector as it spirals around the azimuth. The wave can be written as a superposition of plane waves with the transverse wavevectors for a number of phase steps all the way around the azimuth. The wave is then written as a summation over the transverse wavevectors, expressed as
As the number of phase steps approaches infinity, or in the limit as
around the azimuth, the summation can be replaced as an integral over all transverse wavevectors,
KConsidering the orthogonality relation between the basis functions, the amplitude Am can be determined as a plane wave with the added vortex phase from the near field,
Multiplying both sides of Equation (2) by
, and integrating over
R, at
z = 0
The expression from Equation (3) is then solved for the amplitude, and the remaining integral can be solved in terms of Bessel functions,
where
is a Bessel function of the first kind, implemented as
, where
m is the vortex charge and
θ is the azimuth angle. Equation (4) represents the amplitude of the wave, which can then be substituted into the propagating paraxial wave, where
, using the approximation for a paraxial wave. The plane wave can then be rewritten as
where
, is the combination of half-integer Bessel functions.
Then, the superposition of a pair of vortices leads to the intensity contributions from each vortex
and
, as well as the interference term
that governs the interference pattern, with
. The latter term can be determined by Equation (5), where the phase difference arises from the difference in the Bessel functions. The phase difference will then have a dependence on the charge of the vortex in addition to the length of propagation and initial phase. The interference of two vortices is solved analytically by writing
and
, as
and
Then, the intensity of their superposition would be expressed as
Since the electric field , the intensity term , and the expected interference pattern would present 2m fringes around the azimuthal angle. When the two vortices interact, the resulting interference depends on the charge of the vortices, as presented in the numerical simulations.
2.2. Numerical Simulations
In order to determine the far-field amplitude and phase distributions of vortex beams, a computer program was used to simulate the propagation of an incident beam with an arbitrary amplitude and phase distribution through a focusing lens. The code, written in Wolfram Mathematica, sums the Huygens wavelet contributions using the Fresnel approximation from the near field to points in the far field [
22]. In this paper, we model a flattop beam passing through a spiral phase plate and a perfect focusing lens, however, the computer code can be used to calculate the effect of a wide range of beam profiles, phase distributions and optical aberrations. Since we use an SLM to imprint phase patterns onto an incident beam, the phase distribution is written modulo 2π, allowing for the full use of the dynamic range of the SLM in the zero to 2π range, while still allowing for the equivalent of arbitrarily high phases to be expressed. For extremely short pulses, where the number of optical cycles in the pulse is comparable to the value of the charge
m, using the full phase pattern would result in a temporal “walk-off” of the different phase-shifted portions of the beam within the focal region. Using the modulo 2π phase has the added benefit of limiting these ultrashort pulse width effects. In
Figure 1, we present an example of the code and output of the simulations done in Wolfram Mathematica.
The program from
Figure 1a computes the radial intensity plot of a vortex through paraxial approximation using Equation (5), and computing
.
Figure 1b,c correspond to the 2D and 3D plots of the radial intensity of a vortex with a topological charge
, computed with respect to a horizontal rate
, respectively. The superposition of a pair of vortices is simulated by implementing code that allows the visualization of the intensity profile of various charge vortices limited by a Gaussian beam profile and then computes their interference, based on Equations (6)–(8). The simulated intensity patterns are presented in
Figure 2 and
Figure 3, for equally and opposite charged coplanar and non-coplanar vortices, respectively. In these simulations, no free parameters are used, in order to compare the experimental results directly with the simulations presented in this section.
For two equally charged coplanar vortices, there should only be constructive interference since both vortices are identical, as presented in
Figure 2a. When both coplanar vortices have opposite charges, there should be destructive interference producing a number of fringes around the vortex equal to twice the charge, as presented in
Figure 2b.
A phase tilt can be added to the numerical results to better simulate what occurs in the laboratory, where individual vortices intersect at a small angle. For equally charged vortices, the phase tilt causes interference fringes to appear in the vortex. The number of fringes increases as the phase tilt does. These fringes run along one direction, as the typical fringes seen when two coherent and in-phase laser beams intersect. When two oppositely charged vortices are superimposed with a phase tilt, the result is an interference pattern that has varying spacing between the interference fringes around the vortex. This pattern is known as the firebird pattern, becoming more pronounced as the phase tilt increases.
Figure 3 shows the described interference patterns.
It is also of interest to obtain the intensity patterns of the superposition of multiple wavelength vortices. Through the use of a blazed fork pattern to diffract vortices into the first order, a wide range of wavelengths can be used to generate quality vortices without the need for multiple spatial light modulators. Therefore, the full spectrum of the supercontinuum light source can be used to either selectively pick the desired wavelength of a vortex or to generate multiple vortices. If the full supercontinuum spectrum of wavelengths is incident on the SLM, many vortices are generated and diffracted into the first order. While each vortex of different wavelength would have a slightly different size and position, all vortices would overlap to some extent.
Figure 4 presents a simulated supercontinuum vortex generated by the superposition of individual charge ten vortices with wavelengths λ = 520 nm, 570 nm, 600 nm and 650 nm. This image was generated with a custom computer code written in the Interactive Data Language (IDL), previously used to demonstrate excellent comparisons with experiment for extremely detailed patterns [
27].
2.3. Experimental Setup
The experimental setup for generating white-light vortices consists of a femtosecond oscillator, a microstructured photonic crystal fiber, a two-dimensional SLM and various detectors, as depicted in
Figure 5. These devices are arranged in a Fourier lens system with the SLM in the Fourier plane to generate optical vortices in the image plane. The laser oscillator is a Nd:YAG pumped Ti:Sapphire oscillator emitting at a center wavelength of 800 nm. This laser generates ultrashort pulses when mode-locked with a 20 fs duration. Immediately after the oscillator, the beam passes through a pulse-shaping compressor, essential for optimizing the super-continuum output by pre-chirping the input pulse. Before the beam is coupled into the microstructured fiber, steering mirrors and a waveplate are used to ensure a level and horizontally polarized beam. A 20X microscope objective lens is mounted onto an XYZ translation stage to couple the beam into the nonlinear photonic crystal fiber (NKT Photonics FemtoWhite 800), mounted on a similar stage. The fiber consists of a honeycomb structure with a central core surrounded by air gaps. The core of the fiber is 1.8 ± 0.3 µm, allowing a single transverse mode and a nonlinear interaction through the length of the fiber, which produces a broad spectral bandwidth from 500 nm to over 1000 nm. The Femtowhite 800 fiber is designed to work optimally with a Ti:Sapphire oscillator.
The coherent supercontinuum beam exiting the fiber then travels through a 4f Fourier lens system. The output light expanding from the fiber is collimated with a 70 cm focal length lens, and is incident onto a Hamamatsu X8267-14DB two-dimensional SLM, located at 70 cm from the lens. This SLM has a pixel array of 768 × 768 active pixels, with a pixel size of 26 µm
2 and a diffraction efficiency above 70% for a blazed grating conformed by a four-pixel sawtooth phase lattice. Its 2π phase modulation depends upon the wavelength, it requires a gray level scale from 0 to 97 for 540 nm, increasing to 228 levels for 900 nm [
28]. Since the SLM has a large aperture (2 cm × 2 cm), various colored filters can be placed in front of it to segment the SLM by color, or the entire supercontinuum spectrum can be used to generate white light vortices. The modulated light is reflected back through the f = 70 cm lens and onto a pick-off mirror that reflects the light to the CCD detector in the imaging plane. This well-known technique creates a far-field pattern at the CCD that is a Fourier transform of the field at the SLM, as presented in
Figure 6.
Various detectors are used to collect the white light vortex data. The supercontinuum spectrum is collected using an Ocean Optics spectrum analyzer (USB2000 + XR), the vortex intensity image (in false color) is captured using a DataRay WinCamD-UHR and the colored optical vortices are recorded with a 24.7 × 15.6 mm CCD sensor in a Nikon D70s DSLR camera.
The optimized supercontinuum spectrum ranges from 500 nm to over 1000 nm in the near IR. Various experiments call for the use of filters to narrow down the spectrum incident on the SLM. A variable wavelength filter placed directly after the fiber may be used to select a narrow bandwidth, approximately 20 nm wide, that can be centered anywhere over the entire range of the supercontinuum. Red and green filters (Edmund Optics NT46-139 and NT30-634) that transmit a larger region of the spectrum are employed as well. These filters can be placed side-by-side directly in front of the SLM to generate independently controllable vortices of different color.
The supercontinuum spectrum can be altered by changing the power or pulse shape of the input beam into the fiber. By lowering the power of the input beam, a simple method is provided to switch between a broadband supercontinuum light source from the fiber and the narrowband (~40 nm) femtosecond source. In the low-power case, the fiber acts merely as a spatial filter for the mode locked laser. This is used as a quick reference to view the difference between SCG vortex interference and narrowband interference, and it is also possible to un-modelock the laser to compare with the CW case. When the SCG vortices are not filtered to a narrow bandwidth, the various colors will diffract at different angles, giving the vortex image a spatial chirp. With a CW beam, there is no spatial chirp.
Optical vortices are generated by encoded phase patterns onto the SLM resulting in phase modulation of the SCG beam. To generate a single vortex, a forked grating pattern is displayed on the 768 × 768 pixel active region of the SLM, generated by a LabView VI that allows for real time control of the blazed grating and spiral phase (vortex charge). Multiple vortices can also be displayed on the SLM, allowing for independent control of different vortices.
Figure 7 shows the VI with a sample vortex pattern that is displayed onto the SLM.
The LabView VI generates a vortex pattern through the combination of a blazed grating with a spiral phase. The blazed grating is set from 0 to 2π through the grayscale of the SLM, and this allows for the vortex to be diffracted away from the zeroth order. The SLM has been calibrated for various wavelengths in order to display the correct gray level scale for the wavelength incident on the vortex pattern. The size of the blazed grating in both the x and y-direction, the charge of the spiral phase plate, the gray level scale, the quantity of vortex patterns displayed onto the SLM and their location can all be controlled through the LabView VI. This provides ultimate flexibility in generating multiple vortices of arbitrary size, charge and location.
The broad spectrum of the supercontinuum light allows for the use of optical filters for selecting the color of the vortex. Multiple filters can be used, each covering a small section of the SLM that matches a specific vortex pattern. Since the Hamamatsu SLM has a large active area and high-resolution, it is possible to separate different regions of the SLM as effectively independent modulators. This is used to demonstrate vortex interference, whereby two vortices are created on each half of the SLM, then made to interfere in the image plane. Since the two sections of the SLM are independently controllable by the LabView code, vortex generation of equal or unequal charge is possible. Additionally, arbitrary blazed gratings can be added to each vortex controlling their location in the image plane. This setup can produce multiple vortices, each at different wavelengths with the use of only a single light source. With this experimental setup we have been able to produce up to six independent vortices with full control. For clarity, we present results here for only a pair of independent vortices.