Chapter 3 Experiment and verification
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.
When taking radial-like polarization as an example, we can use the phase information with respective circular polarization as illustrated in Fig. 4-1 (b), (c). By using the software Diffract, we can simulate the propagation of radial-like beam passing
through the polarizer. Fig. 4-2 describes the simulation results depending on the transmission axis of polarizer. The result was brought into harmony with our prediction.
(a) (b) (c) (d) (e) (f)
Fig. 4-1 Two examples of synthesizing the inhomogeneous beams. (a) Radial-like polarization with specific phase (b), (c) and (d) azimuthal-like beams polarization with specific phase (e), (f).
(a) (b) (c) (d) Fig. 4-2 The simulation of passing the radial-like beam through the transmission axis at the angle (a) 0°, (b) 45°, (c) 90°, (d) -45° of polarizer.
4.2 Synthesis of the inhomogeneous beams
In the demonstration, the SLM was utilized to generate localized phase information based on the relationship between retardance and polarization as mentioned in section 4.1.2. Therefore, in order to demonstrate correctly, it is important to make the optical paths of interference the same due to our derivation without any different path, so it needs more patients to adjust the position of beamsplitter or mirror. Another way of adjust the optical path is to put the glass plate
on the one arm of interferometer so the tilted angle of glass plate is used to compensate the difference of optical paths.
The following is the calibration procedure:
(1) The experimental setup gets ready and the corresponding phase pattern on the SLM can result in radially polarized beam.
(2) We can observe the irradiance of the emerge beam passing through the polarizer and adjust the tilted angle of two optical paths until the wire shape was transformed to the fan shape.
(3) If the angle of fan shape doesn’t match with the angle of transmission axis of polarizer, we adjust the tilted angle of glass plate until they coincide.
(4) Finally we can put the pattern on SLM to synthesize the inhomogeneous beam we want.
Fig. 4-3 shows the experimental result of passing the radial-like beam through the transmission axis at the angle of polarizer.
(a) (b) (c) (d) Fig. 4-3 The experiment of passing the radial-like beam through a polarizer at the angle (a) 0°, (b) 45°, (c) 90°, (d) -45° of the transmission axis.
Another way to check the performance of the inhomogeneous beam is using the SNOM to observe the focal spot. The focal intensity distribution of the radial-like
beam and azimuthal-like beam is shown in Fig. 4-4.
(a) (b) Fig. 4-4 The simulation of focusing (a)the radial-like beam and (b)azimuthal-like beam.
4.3 Optical encryption by polarization
In this section, we present an idea that the optical information can be carried by the beam and we can extract the information by using the polarizer, i.e. data encryption by polarization. Basically it is based on our circular approach of synthesizing the inhomogeneous beam. First, the information was edited by computer and stored to SLM, and then the beam was superposed by interferometer. At a receiver set, we can use the polarizer to choose the channel which is decided by the angle of transmission axis of a polarizer. Fig. 4-5 show output pattern by of a receiver with different transmission angle. We can see that the C-shape appear singly only in specific angle and disappeared in orthogonal angle. The choice of transmission axis leads to the particular pattern and the carried information can be extracted by the transmission control protocol at the receiver.
There are some issues to limit the development of this method. First, the uniformity of output pattern is not desirable, because it is easy to find the remaining intensity in dark region as showed in Fig. 4-5(a). If the variation of phase on SLM is
too large, the edge of the phase will result in the dark or bright line which spoiled the uniformity and caused the failure of reading the data. Finally, the optical component such as the diaphragm will lead to the airy disk due to diffraction effect ,and we will discuss more on the diffraction effect in next section.
(a) (b) (c) Fig. 4-5 The output of a receiver with different transmission angle. (a)0°, (b)45°, (c)90°.
4.4 Discussion of optimizing the performance
The performance of the emergent beam was diminished by the imperfect or physical property of the optical component which causes the aberration or diffraction effect.
4.4.1 The diffraction at boundaries
Diffraction can be understood by considering the wave nature of light. Huygen’s principle states that each point on a propagating wavefront is an emitter of secondary wavelets. The combined focus of these expanding wavelets forms the propagating wave. Interference between the secondary wavelets gives rise to a fringe pattern that
rapidly decreases in intensity with increasing angle from the initial direction of propagation. Huygen’s principle nicely describes diffraction, but rigorous explanation demands a detailed study of wave theory.
We consider the effects of that diffract screen introduced in the plane z 0.
Define the amplitude transmittance function of the aperture as the ratio of the transmitted field , ; 0 to the incident field amplitude , ; 0 at each (x, y) in the z 0 plane [19],
, , ;, ; . (4-3)
Then the angular spectrum ⁄ , ⁄ of the incident field and the angular spectrum ⁄ , ⁄ of the transmitted field are related by the convolution theorem,
, , , (4-4)
where , , exp 2 , and is the symbol
for convolution. For example, if the incident plane-wave illuminates the diffracting structure normally, i.e.
, δ , , (4-5)
and , , , , . (4-6)
In this case, the transmitted angular spectrum is found directly by Fourier transforming the amplitude transmittance function of the aperture. Note that, if the diffracting structure is an aperture that limits the extent of the field distribution, this is a broadening of the angular spectrum of the disturbance, from smaller the aperture, the broader the angular spectrum behind the aperture.
In our experimental setup, the beam was cut by the diaphragm producing the dark rings. The diaphragm is the circular aperture so the transform of circle function is written
, , ,
1 z exp 2 . (4-7)
exp j
where and , and the bandwidth limitation associated with evanescent waves was explicitly introduced through the use of a circ. function. Within the circular bandwidth, the modulus of the transfer function is unity but frequency-dependent phase shifts are introduced. The phase dispersion of the system is most significant at high spatial frequency as showed in Fig. 4-6 and vanishes as both and approach zero. In other words, the aperture is the smaller, and the angular spectrum behind the aperture is the broader. For any fixed spatial frequency pair the phase dispersion increases as the distance of propagation z increases.
Fig. 4-6 The diffraction at boundaries
Based on the previous discussion our experimental setup could be redesigned as depicted as Fig. 4-7. The diaphragm was moved behind the final beamsplitter due to increasing of phase dispersion as the distance of propagation z increases. Nevertheless, the diffraction remains because the beam was cut by the boundaries of diaphragm, so
we add spatial filter to diminish the dark rings.
Fig. 4-7 The schematic of optimized setup. BS: beamsplitters, M: mirror, and D: diaphragm.
4.5 Summary
In closing we presented and demonstrated the idea of the synthesis of inhomogeneous beam and the data encryption by polarization which allows us to conveniently edit the arbitrary local phase delay by SLM. In addition, some of their obvious limit have been found and been diminished by adding the spatial filter and reset the position of diaphragm.
Chapter 5 Con
clusions and Future work
5.1 Conclusions
In this thesis, we demonstrated a circularly polarization interference method to generate the spatially inhomogeneous polarized beams which include cylindrical vector beams and the 8-zone radial-like and azmuthal-like polarized beams. Based on the numerical and experimental studies, different polarization states can be switched by loading different phase arrangement subject to the SLM. The focal irradiance of cylindrical vector beams is nearly equal to the size of their vector diffraction limited spot size.
From a set of systematic studies, we found that the interference patterns can be used to not only make a distinction between radially and azimuthally polarized beams but also ensure the experimental precision. And from interference patterns caused by optical vortex, the topological charges lead to the extra pipe-line patterns and fan-shaped pattern. As the tilted angle of two optical paths approaches to zero, the density of pipe line will decrease and yield a fan-shaped pattern. Moreover, we redesigned the experimental setup in order to diminish the diffraction at the boundary and improve the performance of emergent beam.
The flexibility of SLM yields an opportunity to synthesize arbitrary inhomogeneous polarization for other application. The present work unambiguously demonstrates the beam could locally have different states of polarization in 2D plane, and it will trigger future studies addressing near field fabrication with a resolution
beyond the diffraction limit of light as well as fundamental studies and theories of photo-induced nano movements in polymers.
5.2 Future Work
The uniformity of output irradiance and the purity of polarization is the big issue of our experiment. In our thesis we have tried to improve the performance of our synthesized beams. Moreover it is not good enough so these beams should be improved by advanced method.
The paper presents a fast and accurate method for the surface correction of spatial light modulators [20]. The method uses one single image of a focused doughnut beam, created by the SLM, to find the corresponding phase hologram with the Gerchberg Saxton (GS) algorithm. They have demonstrated, both experimentally and in computer simulations, that the GS algorithm is able to reconstruct hardware phase errors on the order of one or two wavelengths. The presented method promises applicability in research areas where the light phase has to be modified by flexible devices like SLMs or micro mirror arrays with high accuracy. The method could also be used to optimize the entire optical setup, since the determined correction function also corrects distortions which are introduced by other optical elements.
In closing, we will combine the phase correction and our experimental setup to improve the uniformity and the purity of output irradiance and develop the desired spatially inhomogeneous polarized beam for other application.
Reference
[1] S. Quabis, R. Dorn, M. Eberler, O. Glo¨ckl, G. Leuchs, “Focusing light to a tighter spo,” Opt. Commun. 179, 1–7 (2000).
[2] E. A. Saleh, M. C. TEICH, Fundamental of Photonics, (1991).
[3]E. Wolf, Progress in Optics, pp. 296-317 (1999).
[4] M.W. Beijersbergen, L. Allen, H.E.L.O. van der Veen, and J.P. Woerdman,
“Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt.
Comm. 96, 123-132 (1996).
[5] L. Allen, M.W. Beijersbergen, R.J.C. Spreeuw, and J.P. Woerdman, “Orbital angular momentum and the transformation of Laguerre-Gaussian modes,” Phys.
Rev. A 45, 8185-8189 (1992).
[6] A. V. Nesterov, V. G. Niziev, and V. P. Yakunin, "Generation of high-power radially polarized beam," J. Phys. D Appl. Phys. 32, 2871-2875 (1999).
[7] G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, ” Efficient extracavity generation of radially an azimuthally polarized beams” Opt. Lett. 32, No.
11, 1468-1470 (2007).
[8] A. A. Tovar, "Production and propagation of cylindrically polarized Laguerre Gaussian laser beams," J. Opt. Soc. Am. A, 15, 27052711 (1998).
[9] Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, "Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings," Opt.
Lett. 27, 285-287 (2002).
[10] B. Jia, X. Gan, and Min Gu, “Direct measurement of a radially polarized focused evanescent field facilitated by a single LCD”, Opt. Express 13, No18, September (2005)
[11] S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29, 2234-2239 (1990).
[12] R. Oron, S. Blit, N. Davidson, and A. A. Friesem, etc, “The formation of laser beams with pure azimuhtal or radial polarization,” Appl. Phys. Lett. 77, pp. 3322-3324 (2000).
[13] MErd´elyi and Zs Bor, “Radial and azimuthal polarizers,” J. Opt. A: Pure Appl.
Opt. 8 737–742 (2006).
[14] Tidwell S. C., Ford D. H. and Kimura W. D., “Transporting and focusing radially polarized laser beams,” Opt. Eng. 31 1527–31(1992).
[15] D. P. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express 9, No. 10, 490-497 (2001).
[16] Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt.
Express 12, 3377 (2004).
[17] S. C. Tidwell, G. H. Kim, and W. D. Kimura, “Efficient radially polarized laser beam generation with a double interferometer,” Appl. Opt. 32, 5222-5229 (1993).
[18] Lars Egil Helseth, “Roles of polarization, phase and amplitude in solid immersion lens systems,” Opt. Commun. 191, 161-172 (2001).
[19] Joseph. W. Goodman , Introduction to Fourier Optics, 3rd edition.
[20] A. Jesacher, A. Schwaighofer, S. F¨urhapter, C. Maurer, S. Bernet, and M.
Ritsch-Marte, “Wavefront correction of spatial light modulators using an optical vortex image, ” Opt. Express 15, 5801-5808 (2007).