Chapter 2 Principle and setup
2.3 Summery
(a) (b) Fig. 2-8 (a) Solver SNOM. (b) Inverted optical microscope Olympus IX81.
2.3 Summery
Consideration of the better synchronization in the dynamic control, the circular approach of interference method was chose. Further, experiment setup was introduced.
In the following chapter, we will use these instruments to accomplish the synthesis system, and the simulation software will be utilized to verify our system.
Chapter 3
Experiment and verification
The experiment of synthesizing the cylindrical vector beams and interference will be discussed in this chapter. First LG beam has the optical vortex phase which leads to wire patterns. The performance and the feasibility of our experiment setup would be obtained by checking these interference pattern .
3.1 Optical vortex
Optical vortices are characterized by a dark core of destructive interference in a coherent beam. The last decade has seen a resurgence of interest in optical vortices, owing to new potential applications. Optical vortices have been used to enhance laser trapping of low index particles, and laser tweezing of biological samples. In nonlinear optical systems, optical vortices may induce a waveguide, which may be useful as an optical switching technique. Optical vortices have also sparked interest in quantum computing due to their unique wavefront topology.
An optical vortex is essentially a helical phase and can be described by a phase profile given by:
Φ , , (3-1) where (r, ,z) are the cylindrical coordinates centered on the vortex, and m is a signed integer known as the topological charge or strength of optical vortex. The topological charge of a defect may be found from the line integral:
m π ds (3-2)
where is the gradient of the phase of the field and ds is a line enclosing the defect.
Around the vortex center the helical phase increases by an integer multiple of 2π.
Except the vortex centet(r=0) where the phase exhibits singularity, the phase distribution is continuous for all paths along direction. However, this singularity is physically acceptable due to the zeor intensity at the beam center.
A monochromatic light beam traveling along a given axis z can transport angular momentum oriented in two different forms. The first form is caused by circular polarization, and the photon carry of angular momentum depending on the handedness of the polarization. Another form is associated with optical phase distribution in the transversal plane. From the equation E r, E r exp , the photon propagate in the free space with the of angular momentum.
A single optical vortex in the center of a scalar monochromatic beam propagating in the z directio man y be written in cylindrical coordinates ( , , ):
E , , , E , e e (3-3) where E(r, z) is a circularly symmetric amplitude function, 2 / is the wave number of a monochromatic field of wavelength λ, w is the angular frequency, and m is the topological charge. The amplitude and phase of a typical vortex beam are shown in Fig. 3.1. The vortex nature of the field is governed by the phase factor, e . At a fixed instant of time helical surfaces of constant phase given by mf kz const are produced for integer values of m. Along the helix axis (r 0) the phase is undefined and thus this point is known as a phase singularity. The amplitude also vanishes along the helix axis (r=0) owing to destructive interference in the vicinity of the vortex core i.e. E 0, z 0. The Laguerre-Gaussian modes are an orthonormal set of solutions to the wave equation in cylindrical coordinates and have the property of the optical vortices.
Fig. 3-1 Wavefront and phase distribution with different vortex beam number m.
3.2 Interference pattern
In general, when two or more scalar waves with the same polarization interfere in space, complete destructive and constructive interference occurs on lines called nodal lines, and on points called phase singularity, wave dislocation, or optical vortices. The standard Mach–Zehnder interferometer can merge two beams at the output of the interferometer. If one beam be tilted from the optical axis, irradiance of interference pattern can be obtained.
When using circular approach for the synthesis of cylindrical vector beams, the opposite topological charge was adopted in the interferometer. Consideration of tilting one beam leads to the interference pattern by passing through a polarizer. From difference of interference pattern, we can easily distinguish between the radially-polarized beams and azimuthally-polarized beams and check the precision of this synthesis.
3.2.1 Analytical description of interference
The adopted method of synthesizing the inhomogeneous beams is circular polarization approach, so the two coherent beams of interference are with particular circular polarization. One of the incident beams, i.e. E , is a wave propagating from a direction in the x-z plane making a small angle with respect to the z axis. The mathematical properties of two coherent beams are described as following:
E , , E e e y (3-4)
E , , E e e y (3-5)
E e e e cos sin ̂ y .
When the two waves are e d, the field E The total field could be decomposed into three components depending on the state of polarization. The intensity profile as seen through a polarizer in the x-transmission
d as ollowing:
S bstit ting cos into (3-6) due to consistency of the coordinate, we obtain
I E 1 cos 2 cos cos cos sin cos 1
which can be abbreviated into
I 1 cos cos sin 2cos cos sin .
(3-7) In another case of using a polarizer with y transmission axis, the following formula was derived :
I 2sin cos sin . (3-8)
When combining the Eqs. (3-7) and (3-8), the total field irradiance could be observed by the CCD at exit of interferometer.
I Ix I cos2 1 2 In the experiment of synthesizing the cylindrical vector beams with titled angle, the irradiance of emergent beam is Ix or Iy depending on the transmission axis of a polarizer. The experiment result will be shown in the Chapter 3 where it is well match with theoretical derivation.
3.2.1 Simulation of interference pattern
In this section we will discuss three cases which are distinguished with different topological charge and tiled angle. For the case 1, m1=m2=1, it can be linked to our synthesis of cylindrical vector beams and the interference pattern could be obtained by experiment which is introduce in the chapter 3. From Eq. (3-9) the total irradiance of interference can be can be abbreviated into
I Ix I cos2 1
2 cos sin sin cos sin . (3-10)
We can use mathematical software such as matlab to simulate intensity distribution of the emergent beam by using the formula derived in previous section. Fig. 3-2 (b - d) shows the interference pattern. If the tiled angle does not
equal to zero, the wires of the interference pattern become broader due to the decreasing tiled angle. The pattern of the first row is in the condition of x-transmission axis of polarizer. Similarly, the second row is in the condition of y-transmission axis of polarizer. We can find that they are spatially complementary between the first and second rows. From the relationship between pattern and tilted angle it can be applied to judge the precision of interferometer. As we decrease the titled angle carefully, the density of pipe wires decrease simultaneously. Till the tilted angle approaches zero, the fan-shaped pattern is observed as shown as Fig. 3-2 (a).
(a) (b) (c) (d) Fig. 3-2 Simulated interference pattern of 1 in different angle (a) 0°,(b) 0.02°, (c) 0.05° ,(d) 0.1°. The first row was represented in the x-transmission axis of polarizer. The second row was represented in the y-transmission axis of polarizer.
Note that m=m1 and m=-m2. The density of pipe wires decrease when the titled angle is decreased.
For the case 2, m1=m2, we consider that the beam with the same unsigned topological charge was interfered when the tilted angle is 0.02°. The formula which describes the intensity of emergent beam from the polarizer with x-transmission axis is
I cos cos sin . (3-11)
Each of the interference patterns shown as Fig. 3-3 is with the same unsigned topological charge. The extra folk of interference pattern can be calculated by compare the wires of the upper plane of pattern with ones of the bottom plane. For instance, the mtotal equal to 2 in the Fig. 3-3(a). In general, mtotal can be expressed as
. (3-12) According to this equation, we can know how many topological charges the emergent beam has by calculate the extra forks. Moreover, it is observed that all patterns have good symmetry in spatial 2D plane. This phenomenon of symmetry will be discussed in the case 3 in comparison with the different topological charge. Another observed phenomenon is that the interference of the odd topological charge leads to the dark wire of centerline in y-axis. In contrast, the interference of the even topological charge leads to the bright wire of midline in y-axis.
Fig. 3-3 Simulated interference pattern of spiral phase with (a) 1, (b) 2, (c) 3, (d) 4. The tilted angle is 0.02° and the transmission axis of the polarizer is x-axis.
For the case 3, , the beam with the different unsigned topological charge was interfered when the tilted angle is 0.02°. The formula which describes the intensity of the emergent beam from the polarizer with x transmission axis is-
I cos cos sin .
The interference pattern with different topological charge was shown as Fig. 3-4. The well symmetry of interference pattern in x axis is obtained due to the same unsigned topological charge, i.e. m1=m2. Interference of different unsigned topological charge leads to spatial asymmetry in x axis, as showed as Fig. 3-4(b)(d)(f)(h). The general rule to describe the symmetry of pattern is to check parity of the sum of topological charge. If the sum of m1 and m2 is even, the irradiance of good symmetry is observed.
Differently, the odd number of summing m1 and m2 result in the spatial asymmetry in the bottom plane. Note that the general rule to judge the extra wires of pattern is the same with Eq. (3-12) derived in the case 2.
During the synthesis procedures as creating cylindrical vector beam, the precision of optical path of our setup can be checked by these interference patterns.
Only the perfect supposition of two beams in the interferometer could lead to the perfect cylindrical vector beams. When one optical axis coincide with the other in the interferometer, we observe the fan shape pattern in the exit pupil plane due to the perfect overlap of high precision, as showed in the Fig. 3-5.
Fig. 3-4. Comparison of simulated interference pattern with different topological charge. The tilted angle is 0.02° and the transmission axis of the polarizer is x-axis.
Fig. 3-5. Simulated patterns of superposition of m1 + m2= (a) 2, (b) 3, (c) 4, (d) 5, (e)6 with no tilted angle.
3.3 Experiment
3.3.1 Synthesis of cylindrical vector beams
In this section we will show the experiment result of cylindrical vector beams.
Based on the experiment setup discussed in previous section, the beam was combined with particular phase arrangement with specific circular polarization. First the correct phase arrangement was displayed in the SLM, and then the beam was reflected from SLM and was split into two beams by beamsplitter. In the interferometer each beam passed through a polarizer and a quarter wave plate and was transform from linear polarization to right handed or left handed circular polarization, as described in Fig.
2-2. The beam was superposed by the final beamsplitter and its irradiance was obtained from CCD.
A polarizer placed in front of CCD could be used to check the accuracy of state of polarization. Fig. 3-6 shows the irradiance of the synthesized beam with different transmission axis of a polarizer. In comparing the bright region of irradiance with the transmission axis of the polarizer, the derivation of angle between them can be adjusted. There are three methods to overcome the derivation due to the different length of the two arms of the interferometer. First, we can adjust the position of the
mirror or final beamsplitter to change the length of optical path. When the correct irradiance was obtained, the correct position of optical elements could be decided.
However, it is difficult to simultaneously adjust the element and keep the tilted angle which will lead to the change of irradiance. Another way is to change the phase arrangement on the SLM because the synthesis of cylindrical vector is based on the superposition of the opposite optical vortex. For instance, we control the rotation of the phase on the SLM to compensate the difference of optical paths of interferometer. Fig. 3-7 shows the respective rotation angle of particular phase on the SLM. We can adjust the angle φ until the correct irradiance is observed.
Finally, the inclined glass plate was put into one arm of interferometer to adjust the phase delay of interferometer. When inclined angle of glass plate is changing , we can find the best angle to correct the derivation.
(a) (b) (c) Fig. 3-6 The irradiance of the synthesized beam with different transmission axis of a polarizer. (a)0°, (b) 45°, (c) 90°.
(a) (b)
Fig. 3-7 The phase arrangement on the SLM. (a) Before adjustment. (b) After adjustment.
Cylindrical vector beams have special property in the focal region. The radially polarized beam have strong longitudinal polarization component which leads to generate extra small spot size at the focal point as simulated in Fig. 3-8(a).
Furthermore, the azimuthally polarized beam will lead to spot of the donut-like shape due to the week longitudinal polarization component.
(a) (b) Fig. 3-8 (a) The focal spot of (a) the radially polarized beam and (b) the azimuthally polarized beam.
3.3.2 Interference pattern
The previous section mentions the synthesis of cylindrical vector beams with no tilted angle. If the two optical paths of interferometer don’t coincide perfectly, the interference pattern was obtained by CCD. Therefore, we can adjust the tilted angle of mirror according to the patterns, as showed in Fig. 3-9. It’s useful for us to do fine tuning. This experimental result is well matched with our simulation in Fig. 3-2.
Fig. 3-9 Interference patterns with different tilted angle.
The optical phase is the dominate issue of producing the interference pattern which is easy to calculate the summation of unsigned topological charge. We can set the corresponding phase pattern on the SLM to obtain the interference pattern that we want. The results of the experiment were assembled in the table. From the table as the tilted angle is equal to zero, the wire shape of interference pattern was transformed to the fan shape.
Fig. 3-10 Experimental interference pattern of spiral phase with (a) 1, (b) 2, (c) 3, (d) 4.
(a) (b) (c) (d) (e) Fig. 3-11 Experimental interference patterns of superposition of m1 + m2= (a) 2, (b) 3, (c) 4, (d) 5, (e)6 with no tilted angle.
3.4 Summary
The experimental setup for synthesis process and interference has been completed by fine tuning. When the tilted angle approaches to 0° and the SLM provide the corresponding spiral phase, the fan shape intensity distribution depending on the transmission axis of a polarizer is most similar to the simulated one. By the results, the synthesis of cylindrical vector beam has been verified. Moreover, interference pattern of different topological charge shows the different extra wires and it’s easy to check the number of topological charge.
Chapter 4
The spatially inhomogeneous polarized beam
The method was proposed to synthesize the inhomogeneous beam. Furthermore, the optimized setup has been utilized to enhance performance and reduce the diffraction effect. In order to verify this method, a demonstration based on the circular polarization approach will be completed.
4.1 Synthesis of 2-dimensional inhomogeneous polarized beams
Based on the interferometrical setup, an idea is proposed that arbitrary state of polarization can be obtained in spatial plane. The following sections include background information of mathematics and the analytical description of synthesis process. The first section focuses on the Jones calculus which is a useful tool to analyze the state of polarization.
4.1.1 Jones calculus
Jones Calculus which includes Jones matrix and Jones vector is a quantitatively mathematical description of polarized light. The operation of optical systems can be described by a cascade multiplication based on 2 x 2 matrices. Each component of the system has associated Jones matrix, and the analysis of the system as a whole is performed by multiplication of the 2 x 2 component matrices. Moreover, from the final vector of multiplication the state of polarization is clearly known at each stage of the calculation. The polarization information can be reduced to a vector, called the Jones vector. The field was described by the Jones vector whose vector components
are including Ex and Ey. From Eqs. Ex Ae and Ey Be ∆ , the Jones vector of the electric field is
E E
E e A
Be ∆ where A and B are amplitude, and ∆ is the phase retardation of the field. For convenience of calculation, the time-dependent term is omitted and then we denote
tors in their normalized forms. For instance, the linearly polarized beam the Jones vec
Ae
0 can be normalized to 10 .
When putting an optical component as a converter of the state of polarization from E E into E E , the function of the optical component is represented by Jones matrix.
where J is the incident beam, J’ is the emergent beam, and M is the Jones matrix.
The Jones calculus also describes the optical functionality of components. This method is useful to describe the propagation of polarized light through optical elements such as polarizer and wave-plates. A polarizer is an optical element that passes a state of polarization by absorbing or reflecting the orthogonal mode of polarization. A wave-plate is an optical element that consists of an anisotropic material, which changes the phase of transmitted light depending on its polarization state. The following table summarizes the Jones vectors for common optical components.
Optical element Jones matrix Linear polarized with axis of
transmission at angle
Cos Cos sin
Cos sin Sin Half-wave plate with fast axis in
x-direction
0 0 Quarter-wave plate with fast axis in
x-direction E 1 0
0 Quarter-wave plate with fast axis at
angle 45°
1
√2
1 j
j 1 Quarter-wave plate with fast axis at
angle -45°
1
√2
1 j
j 1
Table 4-1. The reference between optical element and Jones matrix.
Common optical elements are described by a 2×2 Jones matrix, enabling the description of the change of polarization states by multiplication of the Jones vector with the Jones matrix. Table 4-2 shows some examples of Jones vectors and Jones matrices of the optical elements.
Polarization Jones ector v Illustration
Linear in x-direction 1
Left circular 1
Table 4-2 The reference between optical element and Jones vector.
4.1.2 Mathematics of synthesis
The synthesis of the inhomogeneous beams is based on the interferometrical circular approach. The key issue is to find the relationship between the phase retardation and polarization. Moreover, the derived formula can be used to modulate the spatial light modulator in our experiment. The associated instrument will be introduced in the chapter 3.
Consideration of letting the beam pass through the polarizer and a quarter wave plate can result in turning the x-polarized beam to left-handed circularly polarized beams. Jones mat x ri em gent from quarter-waveplate with fast axiser at angle -45° is
√
Similarly, the x-polarized beam can be turned to right-handed circularly polarized beam. Jones matrix emergent from quarter-waveplate w h fast axis at angle 45° is it
√
The terms of phase retardation of light emergent from reflected SLM are e ∆ and e∆ respectively. As combining these beam, we can get the final vector which describes the state of polarization.
where Jl is Jones vector of the emergent beam at local point and the relative phase difference is the retardance ∆. Additionally, it is easy to control the state of polarization from SLM by referring to Eq. (4-2). When ∆ is 0°, the polarization is linear x-polarized beam. We put the retardance and Jones vector in order, as showed in Table 4-3. From this table, it is convenient for us to design the pattern on the SLM.
Table 4-3 The relationship of retardance and Jones vector.
In order to check feasibility of this method, we create a eight-zones polarization pupil to simulate the property of the inhomogeneous beams, as shown as showed in Fig. 4-1. Two cases of examples are radial-like and azimuthal-like beams.
In order to check feasibility of this method, we create a eight-zones polarization pupil to simulate the property of the inhomogeneous beams, as shown as showed in Fig. 4-1. Two cases of examples are radial-like and azimuthal-like beams.