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Generating a cylindrical vector beam interferometrically for ellipsometric measurement
Optics Communications 361 (2016) 116–123
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Generating a cylindrical vector beam interferometrically for ellipsometric measurement Jing-Heng Chen a, Ruey-Shyan Chang b, Chien-Yuan Han b,n a b
Department of Photonics, Feng Chia University, Taichung, Taiwan, ROC Department of Electro-Optical Engineering, National United University, Miaoli, Taiwan, ROC
art ic l e i nf o
a b s t r a c t
Article history: Received 29 April 2015 Received in revised form 12 August 2015 Accepted 16 August 2015
Cylindrical vector beams have been widely used in material processing, lithography, optical trapping and manipulating. However, few works discussed their application in polarization metrology. A cylindrical vector beam generated by a concrete interferometer setup is employed to determine the ellipsometric parameters of thin films, which was discussed in this work. A TEM01 mode beam was applied as the light source impinging into a modified Michelson interferometer with contiguous optical elements. The mode of light beam was transformed and the polarization states were coordinated with the optical configuration that made the output beam a doughnut-shaped axially symmetric polarized beam. In addition, the output beam plays the same role as rotating polarization element configuration of an ellipsometer. However, the polarization modulation was in spatial domain instead of temporal domain. By making use of this configuration, ellipsometric parameters of thin films were deduced and the results were very close to theoretical values. & 2015 Published by Elsevier B.V.
Keywords: Ellipsometric measurement Cylindrical vector beams Interferometry
1. Introduction As a widely adopted optical technique, ellipsometry is employed to characterize film thickness for single layer or complex multilayer stacks that ranges from angstroms or tenth of a nanometer to several micrometers in semiconductor industry [1]. Upon analyzing the light polarization change reflected from a sample, the information collected from the sample can determine the complex refractive index or dielectric function, which gives access to fundamental physical parameters and is associated with various sample properties, such as chemical composition, morphology, crystal quality or electrical conductivity. Consequently, ellipsometry has been successfully applied in a number of studies concerning material science, biology and pharmaceuticals as well [2–4]. Ellipsometry determines the material properties by means of analyzing the polarization characteristics of the reflected beam with respect to a known incident polarization, therefore the polarization modulation apparatus has to be introduced into the optical systems. Time domain modulation is the most common polarization modulation technique used in ellipsometry, for instance mechanically rotating a polarizer/analyzer or electrically n
Corresponding author. E-mail address: [email protected] (C.-Y. Han).
http://dx.doi.org/10.1016/j.optcom.2015.08.041 0030-4018/& 2015 Published by Elsevier B.V.
changing the polarization through the application of a liquid crystal or photoelsctic modulator [5–8]. By making use of the subwavelength structured grating, spatial domain modulation has been indicated as an applicable approach to spatially modulate the polarization state in recent time. The grating device consisted of arrays of quarter wave plates and polarizers, which were directly attached to the CCD image sensor for the analysis of incident polarization state and employed in a compact static ellipsometric measurement [9]. Another spatial domain modulation is dealt with a beam that has spatially homogeneous polarization. During the last decade, as the so-called cylindrical vector beam, laser beams with cylindrical symmetry in polarization can be generated by active or passive methods and have obtained much interest in theoretical and experimental investigations [10,11]. A vast majority of applications of cylindrical vector beams concentrated on their propagation and focusing characteristics, such as material processing, optical trapping and manipulating, optical microscopy and lithography [12–14]. However, there has been only a few ideals and experimental results demonstrating in optical metrology and polarimetry [15]. In this work, a solid interferometric configuration was proposed to generate a cylindrical vector beam with the characteristic of high stability and quality, which was applied to ellipsometric measurement, so as to simplify the system configuration and data analysis. As a more practical approach, interferometric
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arrangements have been proposed to generate high quality and high efficient cylindrical vector beams. Nevertheless, a substantial issue of the interferometric configuration is how to keep the phase shift of two beam stably and constantly [16]. Recently, a pentashaped solid state interferometer was demonstrated to stably produce cylindrical vector beams [17]. Although the incident beam was separated by the beam splitter, the separated path was in a solid element minimizing the environmental turbulence and enhancing the phase stability. A modified Michelson interferometer was designed here and one arm of the reflecting mirror was substituted by a right angle prism so as to transform the mode of the incident beam. Two arms of the interferometer (transformed and untransformed mode beams) were superposed by a lateral displacement beam splitter, by which a cylindrical vector beam can be obtained while phase difference of two superimposed beams stably remained a constant. The quality of output polarization is examined with the light beam being evaluated quantitatively by employing the Stokes vectors and overall polarization parameters. Since the output beam is characterized with spatially modulated polarization states, the polarization modulation apparatus is no longer required in the ellipsometric system. Therefore, the output beam served as the light source directly impinge into the sample and the reflected beam was passing through a analyzer set at a fixed angle for ellipsometric measurement. By making use of this configuration, ellipsometric parameters of thin films can be obtained with only one intensity profile captured by a CCD camera.
2. Principles and experiments The generation of a cylindrical polarized beam by means of vector superposition of two orthogonal linearly polarized Hermite–Gaussian TEM01 and TEM10 mode beams was proposed and demonstrated, while their phase difference equals zero [16,18]. A Michelson interferometer with image inversion components was installed to produce the radially polarized beam TEM01*, so that the two modes can be coherently combined. The experimental schematic is shown in Fig. 1. As the light source, a He–Ne Laser (Melles Groit, 05-LPB-670) outputs a TEM01 mode beam with linear polarization into the interferometer. Before entering the
117
interferometer, the light beam passed through a half-wave and a quarter-wave plates. The azimuth of the half-wave plate (HWP) is adjusted to alter the transmitted and reflected intensity of a polarizing beam splitter cube (PBS); the azimuth of the quarter-wave plate (QWP) changes the phase shift of two orthogonal linearly polarized beam in the interferometer. To make the alignment of the optical system easier, the laser beam is expanded by a factor of 5 (the diameter is about 1 cm) after passing through a beam expander and then directing into the Michelson interferometer. The field distribution is shown in Fig. 2(a), which can be expressed as
E01(x, y) =
⎡ ⎛ x2 + y2 ⎞⎤ E0x ⎟⎥(u^ − u^ x) (x − y)exp⎢ − ⎜ ⎢⎣ ⎝ W 2 ⎠⎥⎦ y 4W
(1)
The incident beam is separated into p and s wave components by the PBS and the field distribution along the x- and y-axis is shown in Fig. 2(b) and (c). The field distributions can be written as
E01(x, y) =
⎡ ⎛ x2 + y2 ⎞⎤ E0 (x + y) ^ ⎟⎥ s wave uyexp⎢ − ⎜ ⎢⎣ ⎝ W 2 ⎠⎥⎦ 2 2W
(2)
E01(x, y) =
⎡ ⎛ x2 + y2 ⎞⎤ E0 (x − y) ^ ⎟⎥ p wave uxexp⎢ − ⎜ ⎢⎣ ⎝ W 2 ⎠⎥⎦ 2 2W
(3)
There is a QWP installed in each optical path in both arms of the interferometer. For the sake of maximum transformation efficiency, the azimuth of the QWP was set at 45° for the transmitted beam and the electric field of the beam was converted to linearly polarization perpendicular to x-axis that made the beam be reflected by the PBS. In the other arm of the interferometer, there are totally two internal reflections in the Porro prism, therefore the polarization is changed based on the incident angle and the rotation angle of the prism [19]. The schematic setup in the Fig. 1 made the vector of the light perpendicularly enter the large rectangular face of the prism without the rotation of the prism. Moreover, supposing that the azimuth of the QWP was set at 0°, the output polarization from the QWP was linearly polarized at 45° from the x-axis. After exiting from the PBS, the polarization states of both arms is shown in Fig. 2(d) and (e), which can be expressed as
Fig. 1. Configuration of the solid-state interferometer for generating the inhomogeneous polarized beam. PBS: polarization beam splitter, QWP: quarter-wave plate, HWP: half-wave plate.
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Fig. 2. Intensity and field vector distribution in the interferometer: (a) node and polarization of the beam were set at 135° and 45°, respectively, (b) vertical field component of (a), (c) horizontal field component of (a), (d) reflected pattern of (c) from a mirror, (e) reflected pattern of (b) from a porro prism, (f) the result of coherent superposition of (d) and (e), (g) radial polarized beam obtained after passing through a polarization rotator.
E01(x, y) =
⎡ ⎛ x2 + y2 ⎞⎤ E0 (x + y) ^ ⎟⎥ uyexp⎢ − ⎜ ⎢⎣ ⎝ W 2 ⎠⎥⎦ 2 2W
(4)
E01(x, y) =
⎡ ⎛ x2 + y2 ⎞⎤ E0 (x − y) ^ ⎟⎥ uxexp⎢ − ⎜ ⎢⎣ ⎝ W 2 ⎠⎥⎦ 2 2W
(5)
Eqs. (4) and (5) respectively represent the field distribution of the transmitted and reflected waves form the PBS. In practice, it can be observed that the polarization states from both reflected arms did not match with the transmission axis of the PBS, which makes the intensity of both arms are not equal. Therefore, the azimuth angle of the input polarization state impinging into the PBS was set at about 54.73°, which makes intensity ratio for transmitted arm and reflected arm be 1:1.414 after the first exit of the PBS. Before the entrance of the interferometer, this work was achieved by means of adjusting the azimuth angle of the HWP. Finally, the intensity ratio of both beams was balanced before impinging into succeeding elements. At the output channel, the lateral shift of light beams was corrected with the help of a lateral displacement polarizing beam splitter (Edmund optics, 47-191) and two arms of light beams were reflected three and four times, which made the output pattern complement each other as a donut shape. Optical elements such as PBS, QWP, mirror, Porro prism and lateral displacement PBS, were attached to each other with a very small amount of immersion oil, whose refractive index matched with optical elements exactly and firmly held together through cohesive force to establish a solid configuration. The field distribution at the exit of the lateral displacement polarizing beam splitter is expressed as
ET (x, y) =
⎡ ⎛ x2 + y2 ⎞⎤ ⎟⎥⎡ (x + y)eiδ u^ x + (y − x)u^ y⎤⎦ exp⎢ − ⎜ ⎢⎣ ⎝ W 2 ⎠⎥⎦⎣ 2 2W E0
(6)
δ in Eq. (6) represents the phase difference between p and s waves of the two beams. While the phase difference of two superimposed beams was stably remained at zero or a multiple of 2π, the field distribution was spiral polarization, as shown in Fig. 2 (f). This work was achieved by changing the azimuth of the QWP at the entrance of the interferometer. The objective of this configuration was to obtain a radially polarized beam for ellipsometric
measurement, and a polarization rotator was installed at the exit channel of the lateral displacement PBS that changes the azimuth of the linear polarization to be a doughnut-shaped TEM01* mode beam with radial polarization whose field distributions can be written as
ET (x, y) =
⎡ ⎛ x2 + y2 ⎞⎤ E0 ⎟⎥⎡ xu^ + yu^ y⎤⎦ exp⎢ − ⎜ ⎢⎣ ⎝ W 2 ⎠⎥⎦⎣ x W
(7)
Two qualities, Stokes parameters and overall parameters, were adopted to examine the polarization characteristic of the output beam from the interferometer. A classical measurement method which composed of a linear polarizer and a QWP at wavelength of 632.8 nm was used to determine the Stokes parameters in the experiment. To obtain the S0, S1, and S2 was quite straightforward. In practice, they were easily determined by removing the QWP from the optical train and set the azimuth angle of the polarizer at 0°, 45°, and 90° respectively. However, the measurement of S3 was more difficult, the QWP had to be reinserted into the optical train with its azimuth set at 0° and the polarizer set at 45°. Since the QWP absorbed some optical energy, the absorption factor of the QWP had to be introduced that an accurate measurement of the Stokes parameters especially S3 can be evaluated. The data analysis was also followed by the protocol that was being used in the Ref. [20]. However, the only difference between our approach and the approach in Ref. [20] was that we used a 16-bit CCD camera (MX516, Starlight-Xpress) rather than a photodiode to observe the intensity profile of the output beam to deduce the two dimensional values of the Stokes parameters. The overall parameters ρ˜R and σ˜R are determined from the values of the Stokes parameters S0 S1 and S2 [21, 22], where ρ˜R represented the radial polarization contents of an inhomogeneous polarized beam with the local value of irradiance as a density function, while σ˜R gave the information concerning the uniformity of polarization contents across the beam profile. According to the definition of ρ˜R and σ˜R , they can be written in the form
ρ˜R =
1 1 + 2 2P
∞
∫0 ∫0
2π
⎡⎣ cos( 2θ )s ( r , θ ) + sin( 2θ )s ( r , θ )⎤⎦r dr dθ 1 2
(8)
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Fig. 3. The schematic setup of a cylindrical vector beam for the ellipsometric measurement. The upper right: measured locations of 36 points on the image plane.
σ˜R =
1 1 − 2 2P
∞
∫0 ∫0
where P =
∞
2π
⎡⎣ cos( 2θ )s ( r , θ ) + sin( 2θ )s ( r , θ )⎤⎦r dr dθ 1 2
I = k 0 + k1 cos 2P + k2 sin 2P (9)
2π
∫0 ∫0 s0(r , θ ) r dr dθ denotes the total power of the
beam, and S0, S1, and S2 are the Stokes parameters determined at each point in the polar coordinates system (r, θ). The experimental setup and measurement procedure to evaluate the both overall parameters of a radial polarized beam were the same as the configuration to determine the Stokes parameters. After determining the Stokes parameters across the beam profile, the overall parameters can be evaluated from Stokes parameters at each point with its irradiance according to the Ref. [22]. In general, ellipsometry enable to measure optical characteristics of thin films and the simplest scheme is the polarizer-sample-analyzer (PSA) configuration with a rotating optical element in the system, a rotating analyzer, a rotating polarizer or a compensator. At various azimuthal angles of the rotating optical element (e.g. analyzer or compensator), the intensity distributions of the reflected beam from the sample were recorded. Besides, through the application of Fourier coefficients analysis or photometric approach, ellipsometric parameters were deduced based on intensity profiles. Fig. 3 shows the measurement configuration, since the polarization states of a radially polarized beam were linear polarization in the manner of axially symmetrical distribution, the intensity distribution was not measured in the way of spatial variation instead of temporal variation, in an analogous to the rotating polarizer ellipsometry approach, so as to investigate the thickness of thin films. Ellipsometric parameters Ψ and Δ are generally defined as
tan2ψ⋅eiΔ =
rp rs
(10)
where rp and rs are respectively the reflection coefficients in the planes parallel (p) and perpendicular (s) to the incident plane. The intensity measured can be written as 2
2
2
2
2
2
I = I0(sin P⋅sin A + tan Ψ ⋅cos A + 0.5tan Ψ ⋅cos Δ⋅ sin 2P⋅ sin 2 A)
(11)
I0, P, and A represent the incident flux, azimuth angle of the polarizer, and analyzer respectively. Rotating analyzer or polarizer configuration is the most practical design for rotating optical component ellipsometry. In the rotating polarizer ellipsometer, the analyzer is fixed and the output signal varied temporally along with the rotation of the polarizer. The intensity emerging from the analyzer can be further expressed as:
(12)
where
k 0 = I0(sin2A + tan2Ψ cos2A),
(13)
k1 = I0(tan2Ψ cos2A − sin2A),
(14)
k2 = I0 tan Ψ sin 2A cos Δ,
(15)
while employing k0, k1 and k2, ellipsometric parameters can be written as
k 0 + k1
tan Ψ =
k 0 − k1
tan A (16)
k2
cos Δ =
k 02 − k12
(17)
It is found to optimize precision for numerous substrate materials while the polarizer usually set at 30° [23]. Therefore, the azimuth of the analyzer A also set at 30° in Eq. (13)–(16) in our approach. With the help of a radial polarized beam as light source, the time-dependent flux due to the rotation of the polarization element is substituted for the spatial distributed flux by the radial polarized beam emerging from the fixed angle analyzer, and then a number of points on the image plane with varied azimuth angles of input linear polarizations are required for deducing the k0, k1 and k2. The coefficients of three components can be calculated by Fourier transformation:
k0 =
1 n
n
∑ Ii,
k1 =
i=1
k2 =
2 n
2 n
n
∑ Iicos 2Pθi,
n
∑ Iisin 2Pθi i=1
and
i=1
θi =
2π ni
(18)
The component Ii is the measured intensity from n points, taken at equal intervals over a cycle of the intensity, and Pθi is the polarization orientation angle of the each measurement. In practice, the intensity distribution of impinging light to the sample was doughnut-shaped with an intensity of zero in the middle and average high intensity in an annular region around the central position. In the different radial directions of the annular region, the orientation of the linear polarization varied with cylindrical symmetry. However, being reflected from the reference wafer, the intensity in the annular region was not identical any more. In
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addition, the intensity pattern varied in the annular region delivering the information of optical characteristics of thin films on the wafer. With various angles in the radial direction, the intensity pattern was examined in the manner of measuring maximum intensities of the annular region. In the experiment, a broadband silver mirror was installed as test sample without insertion of the analyzer in the optical path for the calibration because the reflection coefficient between the p and s waves was with small difference. Therefore, 36 points of maximum intensity in the annular region of the radially polarized TEM01* mode beam were located by a CCD camera through rotating the analyzer at the interval of 10°, as shown in the upper right of Fig. 3. After replacing the sample by the reference wafer and inserting the analyzer, intensities of the 36 points in the same locations on the image plane were then examined by taking a snapshot, so that k0, k1 and k2 can be estimated by 36 intensities from Eq. (18). Consequently, ellipsometric parameters of the examined sample can be obtained from Eqs. (16) and (17).
3. Results and discussion Easy methods to examine output polarization states were to set an analyzer with various angles at the exit of the solid state interferometer. On the condition that the radially polarized beam passed through the analyzer, a two-lobe pattern could be observed. The light beam was expanded into a large diameter and the analyzer was rotated to record the intensity profiles with the help of a CCD camera. When the analyzer was rotated in a counterclockwise direction, the two-lobe pattern was rotated in the same direction. Fig. 4 reveals the recorded mode pattern when the azimuth of analyzer were respectively set at 0°, 45°, 90° and 135°. This result indicates that all field vectors are of nearly linear polarization with a radial distribution. As the output beam was desired for ellipsometric measurement, a more detailed examination
for its polarization properties was then demonstrated. However, this approach makes sense only when the beam is assumed with uniform polarization properties over its cross-sectional area. For examining the polarization quality of an inhomogeneous polarized beam quantitatively, the spatial distributions of Stokes parameters through the application of a CCD camera were demonstrated in Fig. 5. Stokes parameter S0 described the total intensity of the optical beam, and S0 profile of the output beam was demonstrated in Fig. 5(a) which was close to ideal intensity profile of the TEM01* mode. S1 is the intensity difference between horizontal and vertical polarization; it can be observed that the maximum (equal to 1) and minimum values (equal to 1) are respectively along the xand y-axis, and S1 values on two diagonals were equal to 0, which was shown in Fig. 5(b). Moreover, S2 represents the intensity difference between linear polarization oriented at þ45° and 45°, and the spatial distribution is displayed in Fig. 5(c). The distribution is similar to the profile of S1 after being rotated by 45°. The intensity difference between right and left circular polarization can be denoted by S3. Since the values of S3 are much smaller than S0, S1, and S2, the scale of S3 is set between 0.5 and 0.5, as shown in Fig. 5(d), and it can be observed that there are some small variations of the S3, minimum and maximum around 0.25 and 0.78, located in the inner region of the beam. Imperfection of optical elements in the interferometric configuration and inaccuracy in the initial adjustment of the rotating wave plate are concerns to introduce small amount of S3. However, the appearance of small amount of S3 has a small influence on the evaluation of overall parameters because the S3 are not taken account into the evaluation, as shown in Eqs. (8) and (9). Also, it has very slight effect on deducing the ellipsometric parameters because the intensities are examined in the annular region of TEM01* beam, where values of S3 are close to zero. The results are close to the theoretical values, which confirmed that the radial polarization purity of the light beam was desirable for ellipsometric measurement. In addition to the measurement of the spatial distributions
Fig. 4. Intensity distributions with various azimuth angles of the analyzer: (a) observed without polarizer, and observed with a polarizer with transmitting axis of (b) 0°, (c) 45°, (d) 90°, and (e) 135°.
J.-H. Chen et al. / Optics Communications 361 (2016) 116–123
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Fig. 5. Spatial distributions of Stokes vectors of a radially polarized beam: (a) S0 (b) S1 (c) S2, and (d) S3.
of Stokes parameters, two overall parameters ρ˜R and σ˜R adopted for characterizing the radial and azimuthal polarization contents of inhomogeneous polarized beams were introduced [21,22]. As a result, it was obtained that the output beam ρ˜R equals to 0.983, and the average values of the dispersions σ˜R is equal to 0.01147 0.0001. What’s more, this presents the agreement with the values ρ˜R = 1 and σ˜R = 0 for a pure azimuthally polarized beam. A 5-step SiO2/Si thin film reference wafer (Ocean optics) was set for ellipsometric measurement, where the beam from the interferometer was impinged into the wafer at the angle of 70° and the reflected light distribution from a SiO2/Si reference wafer whose silicon dioxide thicknesses were respectively 124.8 nm, 220.3 nm and 312.4 nm was examined by a CCD camera equipped with an analyzer. Besides, the normalized light intensity reflected from the wafer was a function of linear polarization’s azimuth angle. When the azimuth angle of analyzer was set at 30°, measured values and theoretical curves of SiO2 films are demonstrated
in Fig. 6, which respectively have a thickness of 124.8 nm, 220.3 nm and 312.4 nm. As it can be observed, there is a substantial change in the intensity profiles with different SiO2 film thicknesses in theory. For each thickness of SiO2 film, 36 discretely measured intensities with different azimuth angles are demonstrated; It can be found that the majority of measured values were pretty close to theoretical curves, which verifies sufficient accuracy for the application of radially polarized beam as the light source for an ellipsometric measurement system. The data analysis for the ellipsometric parameters Ψ and Δ are based on the Eq. (12)–(17) and the comparison between the deduced film thicknesses and references values are listed in Table 1, which shows a good agreement with reference values by utilizing this new approach. 4. Conclusions With the help of a modified Michelson interferometer, the
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Table 1 Results of ellipsometric measurements with radial polarized beam for SiO2 film on a Si substrate. SiO2 thickness (nm)
124.8 220.3 312.4
Theoretical values
Measured values
Ψ (deg)
Δ (deg)
Ψ (deg)
Δ (deg)
62.28 26.65 15.96
86.13 272.05 117.41
67.65 27.81 16.81
56.58 272.1 117.01
Deduced film thickness (nm)
128.0 217.4 312.2
temporal stability of this configuration be better than other interferometric methods. Moreover, the output polarization with continuously varied azimuth angles resemble the manner of rotating polarization configuration but in spatial domain. Both features make the output beam be a practical light source for polarization metrology. As a result, the ellipsometric information can be obtained with only one intensity profile captured by a CCD camera. Therefore, the duration of one measurement mainly depended on the exposure time and data transfer speed of the CCD camera. In our measurement with the reference wafer, the exposure time was in the time scale of tens millisecond. With a higher data transfer equipment and processor, the speed of measurement can be expected to possibly reduce down to less than 0.1 s in the future.
Acknowledgment The authors thank the Ministry of Science and Technology, Taiwan, for financially supporting this research under contract no. MOST 103-2221-E-035-035.
Reference
Fig. 6. Intensity distributions with different film thickness (a) 124.8 nm, (b) 220.3 nm, and (c) 312.4 nm for various azimuthal angles of the linear polarization states.
stable generation of a radially polarized laser beam is demonstrated in this paper. The merits of the generation of an cylindrical vector beam by means of this solid-state interferometric approach are wavelength switchable and obtaining a high-quality radially polarized beam simply by means of installing commercial optical elements. As light beams undergo in a solid configuration, vibrations and other disturbance effects can be minimized to make the
[1] H. Tompkins, E.A. Irene, Handbook of Ellipsometry, William Andrew, 2005. [2] J.D. Yeager, M. Dubey, M.J. Wolverton, M.S. Jablin, J. Majewski, D.F. Bahr, D. E. Hooks, Examining chemical structure at the interface between a polymer binder and a pharmaceutical crystal with neutron reflectometry, Polymer 52 (2011) 3762–3768. [3] I.R. Hooper, M. Rooth, J.R. Sambles, Dual-channel differential surface plasmon ellipsometry for bio-chemical sensing, Biosens. Bioelectron. 25 (2009) 411–417. [4] R.S. Moirangthem, Y.C. Chang, P.K. Wei, Investigation of surface plasmon biosensing using gold nanoparticles enhanced ellipsometry, Opt. Lett. 36 (2011) 775–777. [5] Y.F. Chao, W.C. Lee, C.S. Hung, J.J. Lin, A three-intensity technique for polarizersample-analyser photometric ellipsometry and polarimetry, J. Phys. D. Appl. Phys. 31 (1998) 1968–1974. [6] J.N.A. Moritani, Y. Okuda, H. Kubo, High-speed retardation modulation ellipsometer, Appl. Opt. 22 (1983) 2429–2436. [7] M.W. Wang, F.H. Tsai, Y.F. Chao, In situ calibration technique for photoelastic modulator in ellipsometry, Thin Solid Films. 455–456 (2004) 78–83. [8] I.R. Hooper, J.R. Sambles, Differential ellipsometric surface plasmon resonance sensors with liquid crystal polarization modulators, Appl. Phys. Lett. 85 (2004) 3017–3019. [9] T. Sato, T. Araki, Y. Sasaki, T. Tsuru, T. Tadokoro, S. Kawakami, Compact ellipsometer employing a static polarimeter module with arrayed polarizer and wave-plate elements, Appl. Opt. 46 (2007) 4963–4967. [10] Q. Zhan, Cylindrical vector beams: from mathematical concepts to applications, Adv. Opt. photon- 1 (2009) 1–57. [11] A.M. Beckley, T.G. Brown, Pupil polarimetry using stress-engineered optical elements, Proc. SPIE 7570 (2010) 757011. [12] A.F. Abouraddy, K.C. Toussaint, Three-dimensional polarization control in microscopy, Phys. Rev. Lett. 96 (2006) 1–4. [13] M. Meier, V. Romano, T. Feurer, Material processing with pulsed radially and azimuthally polarized laser radiation, Appl. Phys. A Mater. Sci. Process. 86 (2007) 329–334. [14] H. Kawauchi, K. Yonezawa, Y. Kozawa, S. Sato, Calculation of optical trapping forces on a dielectric sphere in the ray optics regime produced by a radially polarized laser beam, Opt. Lett. 32 (2007) 1839–1841. [15] S. Tripathi, K.C. Toussaint, Rapid Mueller matrix polarimetry based on parallelized polarization state generation and detection, Opt. Express. 17 (2009)
J.-H. Chen et al. / Optics Communications 361 (2016) 116–123
21396–21407. [16] N. Passilly, R. de Saint Denis, K. Aït-Ameur, F. Treussart, R. Hierle, J.F. Roch, Simple interferometric technique for generation of a radially polarized light beam, J. Opt. Soc. Am. A. Opt. Image Sci. Vis. 22 (2005) 984–991. [17] H.F. Chen, C.Y. Han, R.S. Chang, Solid-state interferometry of a pentaprism for generating cylindrical vector beam, Opt. Rev. 20 (2013) 189–192. [18] X. Chen, C. Jin, L. Li, B. Lu, Z. Ren, J. Bai, Generation of various vector beams based on vector superposition of two orthogonal linearly polarized TEM01 beams, Opt. Commun. 316 (2014) 140–145. [19] Y.D. Liu, C. Gao, X. Qi, Field rotation and polarization properties of the Porro
123
prism, J. Opt. Soc. Am. A. Opt. Image Sci. Vis. 26 (2009) 1157–1160. [20] D. Goldstein, Polarized Light, Second edition, 2003. [21] P.M. Mejı, G. Piquero, R. Martı, Overall parameters for the characterization of non-uniformly totally polarized beams, Opt. Commun. 265 (2006) 6–10. [22] P.M. Mejı, G. Piquero, V. Ramı, Global parameters for characterizing the radial and azimuthal polarization content of totally polarized beams, Opt. Commun. 281 (2008) 1976–1980. [23] L.Y. Chen, D.W. Lynch, Scanning ellipsometer by rotating polarizer and analyzer, Appl. Opt. 26 (1987) 5221–5228.
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Generating a cylindrical vector beam interferometrically for ellipsometric measurement Jing-Heng Chen a, Ruey-Shyan Chang b, Chien-Yuan Han b,n a b
Department of Photonics, Feng Chia University, Taichung, Taiwan, ROC Department of Electro-Optical Engineering, National United University, Miaoli, Taiwan, ROC
art ic l e i nf o
a b s t r a c t
Article history: Received 29 April 2015 Received in revised form 12 August 2015 Accepted 16 August 2015
Cylindrical vector beams have been widely used in material processing, lithography, optical trapping and manipulating. However, few works discussed their application in polarization metrology. A cylindrical vector beam generated by a concrete interferometer setup is employed to determine the ellipsometric parameters of thin films, which was discussed in this work. A TEM01 mode beam was applied as the light source impinging into a modified Michelson interferometer with contiguous optical elements. The mode of light beam was transformed and the polarization states were coordinated with the optical configuration that made the output beam a doughnut-shaped axially symmetric polarized beam. In addition, the output beam plays the same role as rotating polarization element configuration of an ellipsometer. However, the polarization modulation was in spatial domain instead of temporal domain. By making use of this configuration, ellipsometric parameters of thin films were deduced and the results were very close to theoretical values. & 2015 Published by Elsevier B.V.
Keywords: Ellipsometric measurement Cylindrical vector beams Interferometry
1. Introduction As a widely adopted optical technique, ellipsometry is employed to characterize film thickness for single layer or complex multilayer stacks that ranges from angstroms or tenth of a nanometer to several micrometers in semiconductor industry [1]. Upon analyzing the light polarization change reflected from a sample, the information collected from the sample can determine the complex refractive index or dielectric function, which gives access to fundamental physical parameters and is associated with various sample properties, such as chemical composition, morphology, crystal quality or electrical conductivity. Consequently, ellipsometry has been successfully applied in a number of studies concerning material science, biology and pharmaceuticals as well [2–4]. Ellipsometry determines the material properties by means of analyzing the polarization characteristics of the reflected beam with respect to a known incident polarization, therefore the polarization modulation apparatus has to be introduced into the optical systems. Time domain modulation is the most common polarization modulation technique used in ellipsometry, for instance mechanically rotating a polarizer/analyzer or electrically n
Corresponding author. E-mail address: [email protected] (C.-Y. Han).
http://dx.doi.org/10.1016/j.optcom.2015.08.041 0030-4018/& 2015 Published by Elsevier B.V.
changing the polarization through the application of a liquid crystal or photoelsctic modulator [5–8]. By making use of the subwavelength structured grating, spatial domain modulation has been indicated as an applicable approach to spatially modulate the polarization state in recent time. The grating device consisted of arrays of quarter wave plates and polarizers, which were directly attached to the CCD image sensor for the analysis of incident polarization state and employed in a compact static ellipsometric measurement [9]. Another spatial domain modulation is dealt with a beam that has spatially homogeneous polarization. During the last decade, as the so-called cylindrical vector beam, laser beams with cylindrical symmetry in polarization can be generated by active or passive methods and have obtained much interest in theoretical and experimental investigations [10,11]. A vast majority of applications of cylindrical vector beams concentrated on their propagation and focusing characteristics, such as material processing, optical trapping and manipulating, optical microscopy and lithography [12–14]. However, there has been only a few ideals and experimental results demonstrating in optical metrology and polarimetry [15]. In this work, a solid interferometric configuration was proposed to generate a cylindrical vector beam with the characteristic of high stability and quality, which was applied to ellipsometric measurement, so as to simplify the system configuration and data analysis. As a more practical approach, interferometric
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arrangements have been proposed to generate high quality and high efficient cylindrical vector beams. Nevertheless, a substantial issue of the interferometric configuration is how to keep the phase shift of two beam stably and constantly [16]. Recently, a pentashaped solid state interferometer was demonstrated to stably produce cylindrical vector beams [17]. Although the incident beam was separated by the beam splitter, the separated path was in a solid element minimizing the environmental turbulence and enhancing the phase stability. A modified Michelson interferometer was designed here and one arm of the reflecting mirror was substituted by a right angle prism so as to transform the mode of the incident beam. Two arms of the interferometer (transformed and untransformed mode beams) were superposed by a lateral displacement beam splitter, by which a cylindrical vector beam can be obtained while phase difference of two superimposed beams stably remained a constant. The quality of output polarization is examined with the light beam being evaluated quantitatively by employing the Stokes vectors and overall polarization parameters. Since the output beam is characterized with spatially modulated polarization states, the polarization modulation apparatus is no longer required in the ellipsometric system. Therefore, the output beam served as the light source directly impinge into the sample and the reflected beam was passing through a analyzer set at a fixed angle for ellipsometric measurement. By making use of this configuration, ellipsometric parameters of thin films can be obtained with only one intensity profile captured by a CCD camera.
2. Principles and experiments The generation of a cylindrical polarized beam by means of vector superposition of two orthogonal linearly polarized Hermite–Gaussian TEM01 and TEM10 mode beams was proposed and demonstrated, while their phase difference equals zero [16,18]. A Michelson interferometer with image inversion components was installed to produce the radially polarized beam TEM01*, so that the two modes can be coherently combined. The experimental schematic is shown in Fig. 1. As the light source, a He–Ne Laser (Melles Groit, 05-LPB-670) outputs a TEM01 mode beam with linear polarization into the interferometer. Before entering the
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interferometer, the light beam passed through a half-wave and a quarter-wave plates. The azimuth of the half-wave plate (HWP) is adjusted to alter the transmitted and reflected intensity of a polarizing beam splitter cube (PBS); the azimuth of the quarter-wave plate (QWP) changes the phase shift of two orthogonal linearly polarized beam in the interferometer. To make the alignment of the optical system easier, the laser beam is expanded by a factor of 5 (the diameter is about 1 cm) after passing through a beam expander and then directing into the Michelson interferometer. The field distribution is shown in Fig. 2(a), which can be expressed as
E01(x, y) =
⎡ ⎛ x2 + y2 ⎞⎤ E0x ⎟⎥(u^ − u^ x) (x − y)exp⎢ − ⎜ ⎢⎣ ⎝ W 2 ⎠⎥⎦ y 4W
(1)
The incident beam is separated into p and s wave components by the PBS and the field distribution along the x- and y-axis is shown in Fig. 2(b) and (c). The field distributions can be written as
E01(x, y) =
⎡ ⎛ x2 + y2 ⎞⎤ E0 (x + y) ^ ⎟⎥ s wave uyexp⎢ − ⎜ ⎢⎣ ⎝ W 2 ⎠⎥⎦ 2 2W
(2)
E01(x, y) =
⎡ ⎛ x2 + y2 ⎞⎤ E0 (x − y) ^ ⎟⎥ p wave uxexp⎢ − ⎜ ⎢⎣ ⎝ W 2 ⎠⎥⎦ 2 2W
(3)
There is a QWP installed in each optical path in both arms of the interferometer. For the sake of maximum transformation efficiency, the azimuth of the QWP was set at 45° for the transmitted beam and the electric field of the beam was converted to linearly polarization perpendicular to x-axis that made the beam be reflected by the PBS. In the other arm of the interferometer, there are totally two internal reflections in the Porro prism, therefore the polarization is changed based on the incident angle and the rotation angle of the prism [19]. The schematic setup in the Fig. 1 made the vector of the light perpendicularly enter the large rectangular face of the prism without the rotation of the prism. Moreover, supposing that the azimuth of the QWP was set at 0°, the output polarization from the QWP was linearly polarized at 45° from the x-axis. After exiting from the PBS, the polarization states of both arms is shown in Fig. 2(d) and (e), which can be expressed as
Fig. 1. Configuration of the solid-state interferometer for generating the inhomogeneous polarized beam. PBS: polarization beam splitter, QWP: quarter-wave plate, HWP: half-wave plate.
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Fig. 2. Intensity and field vector distribution in the interferometer: (a) node and polarization of the beam were set at 135° and 45°, respectively, (b) vertical field component of (a), (c) horizontal field component of (a), (d) reflected pattern of (c) from a mirror, (e) reflected pattern of (b) from a porro prism, (f) the result of coherent superposition of (d) and (e), (g) radial polarized beam obtained after passing through a polarization rotator.
E01(x, y) =
⎡ ⎛ x2 + y2 ⎞⎤ E0 (x + y) ^ ⎟⎥ uyexp⎢ − ⎜ ⎢⎣ ⎝ W 2 ⎠⎥⎦ 2 2W
(4)
E01(x, y) =
⎡ ⎛ x2 + y2 ⎞⎤ E0 (x − y) ^ ⎟⎥ uxexp⎢ − ⎜ ⎢⎣ ⎝ W 2 ⎠⎥⎦ 2 2W
(5)
Eqs. (4) and (5) respectively represent the field distribution of the transmitted and reflected waves form the PBS. In practice, it can be observed that the polarization states from both reflected arms did not match with the transmission axis of the PBS, which makes the intensity of both arms are not equal. Therefore, the azimuth angle of the input polarization state impinging into the PBS was set at about 54.73°, which makes intensity ratio for transmitted arm and reflected arm be 1:1.414 after the first exit of the PBS. Before the entrance of the interferometer, this work was achieved by means of adjusting the azimuth angle of the HWP. Finally, the intensity ratio of both beams was balanced before impinging into succeeding elements. At the output channel, the lateral shift of light beams was corrected with the help of a lateral displacement polarizing beam splitter (Edmund optics, 47-191) and two arms of light beams were reflected three and four times, which made the output pattern complement each other as a donut shape. Optical elements such as PBS, QWP, mirror, Porro prism and lateral displacement PBS, were attached to each other with a very small amount of immersion oil, whose refractive index matched with optical elements exactly and firmly held together through cohesive force to establish a solid configuration. The field distribution at the exit of the lateral displacement polarizing beam splitter is expressed as
ET (x, y) =
⎡ ⎛ x2 + y2 ⎞⎤ ⎟⎥⎡ (x + y)eiδ u^ x + (y − x)u^ y⎤⎦ exp⎢ − ⎜ ⎢⎣ ⎝ W 2 ⎠⎥⎦⎣ 2 2W E0
(6)
δ in Eq. (6) represents the phase difference between p and s waves of the two beams. While the phase difference of two superimposed beams was stably remained at zero or a multiple of 2π, the field distribution was spiral polarization, as shown in Fig. 2 (f). This work was achieved by changing the azimuth of the QWP at the entrance of the interferometer. The objective of this configuration was to obtain a radially polarized beam for ellipsometric
measurement, and a polarization rotator was installed at the exit channel of the lateral displacement PBS that changes the azimuth of the linear polarization to be a doughnut-shaped TEM01* mode beam with radial polarization whose field distributions can be written as
ET (x, y) =
⎡ ⎛ x2 + y2 ⎞⎤ E0 ⎟⎥⎡ xu^ + yu^ y⎤⎦ exp⎢ − ⎜ ⎢⎣ ⎝ W 2 ⎠⎥⎦⎣ x W
(7)
Two qualities, Stokes parameters and overall parameters, were adopted to examine the polarization characteristic of the output beam from the interferometer. A classical measurement method which composed of a linear polarizer and a QWP at wavelength of 632.8 nm was used to determine the Stokes parameters in the experiment. To obtain the S0, S1, and S2 was quite straightforward. In practice, they were easily determined by removing the QWP from the optical train and set the azimuth angle of the polarizer at 0°, 45°, and 90° respectively. However, the measurement of S3 was more difficult, the QWP had to be reinserted into the optical train with its azimuth set at 0° and the polarizer set at 45°. Since the QWP absorbed some optical energy, the absorption factor of the QWP had to be introduced that an accurate measurement of the Stokes parameters especially S3 can be evaluated. The data analysis was also followed by the protocol that was being used in the Ref. [20]. However, the only difference between our approach and the approach in Ref. [20] was that we used a 16-bit CCD camera (MX516, Starlight-Xpress) rather than a photodiode to observe the intensity profile of the output beam to deduce the two dimensional values of the Stokes parameters. The overall parameters ρ˜R and σ˜R are determined from the values of the Stokes parameters S0 S1 and S2 [21, 22], where ρ˜R represented the radial polarization contents of an inhomogeneous polarized beam with the local value of irradiance as a density function, while σ˜R gave the information concerning the uniformity of polarization contents across the beam profile. According to the definition of ρ˜R and σ˜R , they can be written in the form
ρ˜R =
1 1 + 2 2P
∞
∫0 ∫0
2π
⎡⎣ cos( 2θ )s ( r , θ ) + sin( 2θ )s ( r , θ )⎤⎦r dr dθ 1 2
(8)
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Fig. 3. The schematic setup of a cylindrical vector beam for the ellipsometric measurement. The upper right: measured locations of 36 points on the image plane.
σ˜R =
1 1 − 2 2P
∞
∫0 ∫0
where P =
∞
2π
⎡⎣ cos( 2θ )s ( r , θ ) + sin( 2θ )s ( r , θ )⎤⎦r dr dθ 1 2
I = k 0 + k1 cos 2P + k2 sin 2P (9)
2π
∫0 ∫0 s0(r , θ ) r dr dθ denotes the total power of the
beam, and S0, S1, and S2 are the Stokes parameters determined at each point in the polar coordinates system (r, θ). The experimental setup and measurement procedure to evaluate the both overall parameters of a radial polarized beam were the same as the configuration to determine the Stokes parameters. After determining the Stokes parameters across the beam profile, the overall parameters can be evaluated from Stokes parameters at each point with its irradiance according to the Ref. [22]. In general, ellipsometry enable to measure optical characteristics of thin films and the simplest scheme is the polarizer-sample-analyzer (PSA) configuration with a rotating optical element in the system, a rotating analyzer, a rotating polarizer or a compensator. At various azimuthal angles of the rotating optical element (e.g. analyzer or compensator), the intensity distributions of the reflected beam from the sample were recorded. Besides, through the application of Fourier coefficients analysis or photometric approach, ellipsometric parameters were deduced based on intensity profiles. Fig. 3 shows the measurement configuration, since the polarization states of a radially polarized beam were linear polarization in the manner of axially symmetrical distribution, the intensity distribution was not measured in the way of spatial variation instead of temporal variation, in an analogous to the rotating polarizer ellipsometry approach, so as to investigate the thickness of thin films. Ellipsometric parameters Ψ and Δ are generally defined as
tan2ψ⋅eiΔ =
rp rs
(10)
where rp and rs are respectively the reflection coefficients in the planes parallel (p) and perpendicular (s) to the incident plane. The intensity measured can be written as 2
2
2
2
2
2
I = I0(sin P⋅sin A + tan Ψ ⋅cos A + 0.5tan Ψ ⋅cos Δ⋅ sin 2P⋅ sin 2 A)
(11)
I0, P, and A represent the incident flux, azimuth angle of the polarizer, and analyzer respectively. Rotating analyzer or polarizer configuration is the most practical design for rotating optical component ellipsometry. In the rotating polarizer ellipsometer, the analyzer is fixed and the output signal varied temporally along with the rotation of the polarizer. The intensity emerging from the analyzer can be further expressed as:
(12)
where
k 0 = I0(sin2A + tan2Ψ cos2A),
(13)
k1 = I0(tan2Ψ cos2A − sin2A),
(14)
k2 = I0 tan Ψ sin 2A cos Δ,
(15)
while employing k0, k1 and k2, ellipsometric parameters can be written as
k 0 + k1
tan Ψ =
k 0 − k1
tan A (16)
k2
cos Δ =
k 02 − k12
(17)
It is found to optimize precision for numerous substrate materials while the polarizer usually set at 30° [23]. Therefore, the azimuth of the analyzer A also set at 30° in Eq. (13)–(16) in our approach. With the help of a radial polarized beam as light source, the time-dependent flux due to the rotation of the polarization element is substituted for the spatial distributed flux by the radial polarized beam emerging from the fixed angle analyzer, and then a number of points on the image plane with varied azimuth angles of input linear polarizations are required for deducing the k0, k1 and k2. The coefficients of three components can be calculated by Fourier transformation:
k0 =
1 n
n
∑ Ii,
k1 =
i=1
k2 =
2 n
2 n
n
∑ Iicos 2Pθi,
n
∑ Iisin 2Pθi i=1
and
i=1
θi =
2π ni
(18)
The component Ii is the measured intensity from n points, taken at equal intervals over a cycle of the intensity, and Pθi is the polarization orientation angle of the each measurement. In practice, the intensity distribution of impinging light to the sample was doughnut-shaped with an intensity of zero in the middle and average high intensity in an annular region around the central position. In the different radial directions of the annular region, the orientation of the linear polarization varied with cylindrical symmetry. However, being reflected from the reference wafer, the intensity in the annular region was not identical any more. In
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addition, the intensity pattern varied in the annular region delivering the information of optical characteristics of thin films on the wafer. With various angles in the radial direction, the intensity pattern was examined in the manner of measuring maximum intensities of the annular region. In the experiment, a broadband silver mirror was installed as test sample without insertion of the analyzer in the optical path for the calibration because the reflection coefficient between the p and s waves was with small difference. Therefore, 36 points of maximum intensity in the annular region of the radially polarized TEM01* mode beam were located by a CCD camera through rotating the analyzer at the interval of 10°, as shown in the upper right of Fig. 3. After replacing the sample by the reference wafer and inserting the analyzer, intensities of the 36 points in the same locations on the image plane were then examined by taking a snapshot, so that k0, k1 and k2 can be estimated by 36 intensities from Eq. (18). Consequently, ellipsometric parameters of the examined sample can be obtained from Eqs. (16) and (17).
3. Results and discussion Easy methods to examine output polarization states were to set an analyzer with various angles at the exit of the solid state interferometer. On the condition that the radially polarized beam passed through the analyzer, a two-lobe pattern could be observed. The light beam was expanded into a large diameter and the analyzer was rotated to record the intensity profiles with the help of a CCD camera. When the analyzer was rotated in a counterclockwise direction, the two-lobe pattern was rotated in the same direction. Fig. 4 reveals the recorded mode pattern when the azimuth of analyzer were respectively set at 0°, 45°, 90° and 135°. This result indicates that all field vectors are of nearly linear polarization with a radial distribution. As the output beam was desired for ellipsometric measurement, a more detailed examination
for its polarization properties was then demonstrated. However, this approach makes sense only when the beam is assumed with uniform polarization properties over its cross-sectional area. For examining the polarization quality of an inhomogeneous polarized beam quantitatively, the spatial distributions of Stokes parameters through the application of a CCD camera were demonstrated in Fig. 5. Stokes parameter S0 described the total intensity of the optical beam, and S0 profile of the output beam was demonstrated in Fig. 5(a) which was close to ideal intensity profile of the TEM01* mode. S1 is the intensity difference between horizontal and vertical polarization; it can be observed that the maximum (equal to 1) and minimum values (equal to 1) are respectively along the xand y-axis, and S1 values on two diagonals were equal to 0, which was shown in Fig. 5(b). Moreover, S2 represents the intensity difference between linear polarization oriented at þ45° and 45°, and the spatial distribution is displayed in Fig. 5(c). The distribution is similar to the profile of S1 after being rotated by 45°. The intensity difference between right and left circular polarization can be denoted by S3. Since the values of S3 are much smaller than S0, S1, and S2, the scale of S3 is set between 0.5 and 0.5, as shown in Fig. 5(d), and it can be observed that there are some small variations of the S3, minimum and maximum around 0.25 and 0.78, located in the inner region of the beam. Imperfection of optical elements in the interferometric configuration and inaccuracy in the initial adjustment of the rotating wave plate are concerns to introduce small amount of S3. However, the appearance of small amount of S3 has a small influence on the evaluation of overall parameters because the S3 are not taken account into the evaluation, as shown in Eqs. (8) and (9). Also, it has very slight effect on deducing the ellipsometric parameters because the intensities are examined in the annular region of TEM01* beam, where values of S3 are close to zero. The results are close to the theoretical values, which confirmed that the radial polarization purity of the light beam was desirable for ellipsometric measurement. In addition to the measurement of the spatial distributions
Fig. 4. Intensity distributions with various azimuth angles of the analyzer: (a) observed without polarizer, and observed with a polarizer with transmitting axis of (b) 0°, (c) 45°, (d) 90°, and (e) 135°.
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Fig. 5. Spatial distributions of Stokes vectors of a radially polarized beam: (a) S0 (b) S1 (c) S2, and (d) S3.
of Stokes parameters, two overall parameters ρ˜R and σ˜R adopted for characterizing the radial and azimuthal polarization contents of inhomogeneous polarized beams were introduced [21,22]. As a result, it was obtained that the output beam ρ˜R equals to 0.983, and the average values of the dispersions σ˜R is equal to 0.01147 0.0001. What’s more, this presents the agreement with the values ρ˜R = 1 and σ˜R = 0 for a pure azimuthally polarized beam. A 5-step SiO2/Si thin film reference wafer (Ocean optics) was set for ellipsometric measurement, where the beam from the interferometer was impinged into the wafer at the angle of 70° and the reflected light distribution from a SiO2/Si reference wafer whose silicon dioxide thicknesses were respectively 124.8 nm, 220.3 nm and 312.4 nm was examined by a CCD camera equipped with an analyzer. Besides, the normalized light intensity reflected from the wafer was a function of linear polarization’s azimuth angle. When the azimuth angle of analyzer was set at 30°, measured values and theoretical curves of SiO2 films are demonstrated
in Fig. 6, which respectively have a thickness of 124.8 nm, 220.3 nm and 312.4 nm. As it can be observed, there is a substantial change in the intensity profiles with different SiO2 film thicknesses in theory. For each thickness of SiO2 film, 36 discretely measured intensities with different azimuth angles are demonstrated; It can be found that the majority of measured values were pretty close to theoretical curves, which verifies sufficient accuracy for the application of radially polarized beam as the light source for an ellipsometric measurement system. The data analysis for the ellipsometric parameters Ψ and Δ are based on the Eq. (12)–(17) and the comparison between the deduced film thicknesses and references values are listed in Table 1, which shows a good agreement with reference values by utilizing this new approach. 4. Conclusions With the help of a modified Michelson interferometer, the
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Table 1 Results of ellipsometric measurements with radial polarized beam for SiO2 film on a Si substrate. SiO2 thickness (nm)
124.8 220.3 312.4
Theoretical values
Measured values
Ψ (deg)
Δ (deg)
Ψ (deg)
Δ (deg)
62.28 26.65 15.96
86.13 272.05 117.41
67.65 27.81 16.81
56.58 272.1 117.01
Deduced film thickness (nm)
128.0 217.4 312.2
temporal stability of this configuration be better than other interferometric methods. Moreover, the output polarization with continuously varied azimuth angles resemble the manner of rotating polarization configuration but in spatial domain. Both features make the output beam be a practical light source for polarization metrology. As a result, the ellipsometric information can be obtained with only one intensity profile captured by a CCD camera. Therefore, the duration of one measurement mainly depended on the exposure time and data transfer speed of the CCD camera. In our measurement with the reference wafer, the exposure time was in the time scale of tens millisecond. With a higher data transfer equipment and processor, the speed of measurement can be expected to possibly reduce down to less than 0.1 s in the future.
Acknowledgment The authors thank the Ministry of Science and Technology, Taiwan, for financially supporting this research under contract no. MOST 103-2221-E-035-035.
Reference
Fig. 6. Intensity distributions with different film thickness (a) 124.8 nm, (b) 220.3 nm, and (c) 312.4 nm for various azimuthal angles of the linear polarization states.
stable generation of a radially polarized laser beam is demonstrated in this paper. The merits of the generation of an cylindrical vector beam by means of this solid-state interferometric approach are wavelength switchable and obtaining a high-quality radially polarized beam simply by means of installing commercial optical elements. As light beams undergo in a solid configuration, vibrations and other disturbance effects can be minimized to make the
[1] H. Tompkins, E.A. Irene, Handbook of Ellipsometry, William Andrew, 2005. [2] J.D. Yeager, M. Dubey, M.J. Wolverton, M.S. Jablin, J. Majewski, D.F. Bahr, D. E. Hooks, Examining chemical structure at the interface between a polymer binder and a pharmaceutical crystal with neutron reflectometry, Polymer 52 (2011) 3762–3768. [3] I.R. Hooper, M. Rooth, J.R. Sambles, Dual-channel differential surface plasmon ellipsometry for bio-chemical sensing, Biosens. Bioelectron. 25 (2009) 411–417. [4] R.S. Moirangthem, Y.C. Chang, P.K. Wei, Investigation of surface plasmon biosensing using gold nanoparticles enhanced ellipsometry, Opt. Lett. 36 (2011) 775–777. [5] Y.F. Chao, W.C. Lee, C.S. Hung, J.J. Lin, A three-intensity technique for polarizersample-analyser photometric ellipsometry and polarimetry, J. Phys. D. Appl. Phys. 31 (1998) 1968–1974. [6] J.N.A. Moritani, Y. Okuda, H. Kubo, High-speed retardation modulation ellipsometer, Appl. Opt. 22 (1983) 2429–2436. [7] M.W. Wang, F.H. Tsai, Y.F. Chao, In situ calibration technique for photoelastic modulator in ellipsometry, Thin Solid Films. 455–456 (2004) 78–83. [8] I.R. Hooper, J.R. Sambles, Differential ellipsometric surface plasmon resonance sensors with liquid crystal polarization modulators, Appl. Phys. Lett. 85 (2004) 3017–3019. [9] T. Sato, T. Araki, Y. Sasaki, T. Tsuru, T. Tadokoro, S. Kawakami, Compact ellipsometer employing a static polarimeter module with arrayed polarizer and wave-plate elements, Appl. Opt. 46 (2007) 4963–4967. [10] Q. Zhan, Cylindrical vector beams: from mathematical concepts to applications, Adv. Opt. photon- 1 (2009) 1–57. [11] A.M. Beckley, T.G. Brown, Pupil polarimetry using stress-engineered optical elements, Proc. SPIE 7570 (2010) 757011. [12] A.F. Abouraddy, K.C. Toussaint, Three-dimensional polarization control in microscopy, Phys. Rev. Lett. 96 (2006) 1–4. [13] M. Meier, V. Romano, T. Feurer, Material processing with pulsed radially and azimuthally polarized laser radiation, Appl. Phys. A Mater. Sci. Process. 86 (2007) 329–334. [14] H. Kawauchi, K. Yonezawa, Y. Kozawa, S. Sato, Calculation of optical trapping forces on a dielectric sphere in the ray optics regime produced by a radially polarized laser beam, Opt. Lett. 32 (2007) 1839–1841. [15] S. Tripathi, K.C. Toussaint, Rapid Mueller matrix polarimetry based on parallelized polarization state generation and detection, Opt. Express. 17 (2009)
J.-H. Chen et al. / Optics Communications 361 (2016) 116–123
21396–21407. [16] N. Passilly, R. de Saint Denis, K. Aït-Ameur, F. Treussart, R. Hierle, J.F. Roch, Simple interferometric technique for generation of a radially polarized light beam, J. Opt. Soc. Am. A. Opt. Image Sci. Vis. 22 (2005) 984–991. [17] H.F. Chen, C.Y. Han, R.S. Chang, Solid-state interferometry of a pentaprism for generating cylindrical vector beam, Opt. Rev. 20 (2013) 189–192. [18] X. Chen, C. Jin, L. Li, B. Lu, Z. Ren, J. Bai, Generation of various vector beams based on vector superposition of two orthogonal linearly polarized TEM01 beams, Opt. Commun. 316 (2014) 140–145. [19] Y.D. Liu, C. Gao, X. Qi, Field rotation and polarization properties of the Porro
123
prism, J. Opt. Soc. Am. A. Opt. Image Sci. Vis. 26 (2009) 1157–1160. [20] D. Goldstein, Polarized Light, Second edition, 2003. [21] P.M. Mejı, G. Piquero, R. Martı, Overall parameters for the characterization of non-uniformly totally polarized beams, Opt. Commun. 265 (2006) 6–10. [22] P.M. Mejı, G. Piquero, V. Ramı, Global parameters for characterizing the radial and azimuthal polarization content of totally polarized beams, Opt. Commun. 281 (2008) 1976–1980. [23] L.Y. Chen, D.W. Lynch, Scanning ellipsometer by rotating polarizer and analyzer, Appl. Opt. 26 (1987) 5221–5228.