CAPÍTULO III: EL CONTROL DE CONVENCIONALIDAD Y LA POSIBILIDAD
D. SOBRE EL HECHO INTERNACIONALMENTE ILÍCITO Y LA
5.
Optically trapped micro-apertures for phase and coherence
measurements
5.1.Overview
In the previous chapter 3 and 4, the BFP interference patterns from the diffracting light fields of a trapped microsphere is used for position and trap stiffness measurements. The measured and calculated BFP intensity pattern resembles an Airy disk, figure 3.1. An Airy disk pattern is the Fourier transformation of a circular aperture. Hence, the trapped microsphere is acting like a diffracting aperture [1, 2]. If two microspheres are trapped alongside each other in one optical trap, the interference between the diffracting light (Airy pattern) from each microsphere will form a sinusoidal intensity fringes pattern. The two microspheres have in fact formed a microscopic Young’s slit type interferometer. The visibility and arrangements of the interference fringes reflect the correlation of the phase and coherence between the light fields that is emerging from the two microspheres respectively. The sphere can be either trapped by sampling light field or separate optical traps.
In this chapter, I demonstrate two sets of experiments that use optically trapped microparticles as micro-apertures for phase and spatial coherence measurements. The first set of experiments use a LG beam to trap the two or more microspheres within its annular intensity pattern. The microspheres acts like self-aligned apertures and samples the trapping beam. In contrast to the first set of experiments, I next make use of two independent optical traps (dual Gaussian beam optical traps) to manipulate two microparticles individually. The optically controlled micro- apertures (microparticle) are independently used to sample the spatial coherence on a separate sampling light field (LG beam). In the second set of experiments, the independent optical traps that are separated from the sampling light field offer more controllability over the position in which the microparticles are positioned.
5.2.Introduction
Conventional interferometric system makes use of beam splitter cubes, pinhole and slits for measurement of spatial, temporal coherence and phase change (wavefront) within a given light field. In this particular chapter, I shall restrict to interferometer formed with pinholes and slits: Young slits interferometer [3]. Young’s double slits experiment is often used as a map of the spatial coherence of a light field that is well known to result in co-sinusoidal intensity fringes modulated by a sinc function. The lateral shifting and the visibility of the intensity fringes can be used to infer the
phase difference and the spatial coherence between the emerging light fields from the two slits respectively. Traditionally, the Young’s interferometer comprises of two rectangular apertures or slit. With current microlithography techniques, the size of the slits can be reduced to smaller symmetrical apertures or pinhole. The quality of the pinholes (≈10 µm) often relies on good fabrication technique. In addition, precise optical alignment of the aperture within the sample beam is important. This alignment helps to increase the intensity of the transmitted light. Microspheres (polystyrene and silica) are widely available from a large number of companies and they come in a range of size selection (typically from 20 nm to 20 µm). These microspheres serve as the ideal pinholes (soft Gaussian type aperture) for probing the phase and coherence properties of a sample light field. Using optical traps, the controllability of the micro-aperture (microsphere) positions around a sample beam is increased. Next I shall briefly explain the concept of phase and coherence measurements with interference.
Figure 5.1 Spatial (red box) and temporal (blue box) sampling of a point source. The interference between the light fields from points denoted by t1,2,3 and r1,2,3 indicates points is time and space. t1,3 at
r0 indicates to sampling point at different time in z and r1,2 at t2 refers to the sampling points at
different spatial positions (x)
Classical optical interferometry describes the coherent interference of two sets of points, (
E
1,E
2) orE
1,E
3, within a coherent light field. The interference pattern generated from the two sets of points is used to sample the spatial coherence or the temporal coherence of the input light. The interference of the fields results in an intensity pattern at an observation plane given asI
p. The intensity pattern is a way of deducing the state of coherence or phase difference between the twopoints. Depending on the position of the two points (sampling points); it can refer to either the spatial, Ispatial or temporal, Itemporal, interference as described in equation 5.1 or figure 5.1. By spatial
interference, I refer to the sampling of r1 and r2 at the same frame of time t2. On the other hand,
temporal interference refers to the sampling of t1 and t3 at the same position r0 ,.
(
) (
)
(
) (
)
2 2 1 2 2 1 1 2 2 2 2 12 1 1 2 2 2 2 12 2 2 0 1 3 1 0 1 3 0 3 13 1 0 1 3 0 3 13 ( , , ) ( , ) ( , ) 2. ( , ) . ( , ) .cos( ) ( , , ) ( , ) ( , ) 2. ( , ) . ( , ) .cos( ) spatial p temporal I r r t E r t E r t E r t E r t I I r t t E r t E r t E r t E r tγ
φ
γ
φ
⎧ = + + Δ ⎪ ⎨ = + + Δ ⎪⎩ (5.1) From the interference equation (for temporal and spatial) given in equation 4.1, the cosine and gamma terms determines the phase difference and degree of the coherence between the two electric fields, E1 and E2, at sampled points respectively. A difference in the phase between thesampling points would result in a spatial shift in the intensity patterns and the degree of the coherence between the points change in the visibility of the interference fringes.
A simple means of measuring the azimuthal phase variation is to use a Young’s double slits experiment to sample the phase difference at the opposing sides of the beam (spatial sampling)[4]. In figure 5.2 A, the interference pattern after a spatially coherent Gaussian beam incident passes through two pinholes is numerically calculated. In figure 5.2 B, the interference fringe pattern (shifted by π as compared to figure 5.2A) of an LG beam that is incident on the same pinholes. Due to the phase difference (π) between the two points, the resulting interference pattern is shifted by a single interference fringe.
Figure 5.2 Young’s slit interference with two pinholes. The cross-sectional plot of interference pattern between two coherent points in a monochromatic Gaussian beam (A) and a 1
0
LG beam (B). Left inset shows the interference pattern. Right inset shows the incident beam. The two pinholes are separated by one beam waist and centred to beam central axis.
Figure 5.3 Temporal interference of LG beam and Gaussian beam. The numerically calculated interference between a LG06 and a plane wave at an angle [5, 6]. The number of bifurcations, 6, (red dot) in the fork intensity patterns shows the azimuthal phase variation of the LG06 beam.
On the other hand, by temporally interfering the LG beam with a plane wave (Mach Zehnder or Michelson interferometer), the intensity pattern (fork like pattern) can measure the degree of azimuthal phase variation, l [5-7] based on the number of bifurcations. Figure 5.3 shows the numerical simulation of a 6
0
LG beam interfering with a plane wave. A fork facing upwards indicates a positive azimuthal phase variation while the reverse indicates a negative azimuthal phase variation. The level of azimuthal phase variation (l) is signified by the number of bifurcations in the fork pattern. This particular temporal interference method was first used to measure the azimuthal phase variation in the LG beam modes [5]. The limitation lies in stability of the interference pattern and difficult to use to measure polychromatic LG sources (short coherence length ≈10µm). Aperture based wavefront sampling techniques (i.e. Young’s slits type experiment [4] or Shack-Hartmann wavefront sensor [8] ) are suited to diagnose the wavefront curvature of a monochromatic and polychromatic sources. I aim to further develop the aperture-based interferometry technique using optically controlled microapertures.
In the next two sections, I use the trapped microspheres to perform two phase measurements. The first experiment is to perform discrete phase measurement about a 1
0 LG beam with two optically trapped microspheres (Young’s slits). This demonstrates that the microspheres can be optically controlled onto a chosen position on the 1
0
LG beam. The second experiment is to use multi-point interferometry with several trapped microspheres. Using several optically trapped spheres (all of which resides on the circumference of the same beam), I analysed the far-field interference pattern that elucidates the sign of the azimuthal phase variations of the trapping 3
0 LG
Hartmann wavefront measurement. Using the optical forces, it is possible to “tune” the number of optically trapped microspheres within the 1
0
LG trapping beam.