Simultaneous measurement of phase and local orientation of linearly polarized light: implementation and measurement results Sergej Rothau,1,* Christine Kellermann,1 Vanusch Nercissian,1 Andreas Berger,1 Klaus Mantel,2 and Norbert Lindlein1 1
Institute of Optics, Information, and Photonics, Friedrich-Alexander-University Erlangen-Nürnberg (FAU), Staudtstr. 7/B2, 91058 Erlangen, Germany
2
Max Planck Institute for the Science of Light, Günther-Scharowsky-Str. 1/Bldg 24, 91058 Erlangen, Germany *Corresponding author: [email protected] Received 31 January 2014; revised 4 April 2014; accepted 4 April 2014; posted 8 April 2014 (Doc. ID 205784); published 9 May 2014
Optical components manipulating both polarization and phase of wave fields find many applications in today’s optical systems. With modern lithography methods it is possible to fabricate optical elements with nanostructured surfaces from different materials capable of generating spatially varying, locally linearly polarized-light distributions, tailored to the application in question. Since such elements in general also affect the phase of the light field, the characterization of the function of such elements consists in measuring the phase and the polarization of the generated light, preferably at the same time. Here, we will present first results of an interferometric approach for a simultaneous and spatially resolved measurement of both phase and polarization, as long as the local polarization at any point is linear (e.g., for radially or azimuthally polarized light). © 2014 Optical Society of America OCIS codes: (120.3180) Interferometry; (160.1190) Anisotropic optical materials; (120.5050) Phase measurement; (260.5430) Polarization. http://dx.doi.org/10.1364/AO.53.003125
1. Introduction
Nowadays, wave fields with polarization states other than globally linear or circular polarization are in great demand for various applications. Very important polarization states are those with a locally linear state but spatially varying orientation such as radial and azimuthal polarization. Even more exotic polarization states can be created artificially in different ways such as interferometric methods [1], the use of liquid crystals [2], or elements with nanostructured surfaces.
1559-128X/14/143125-06$15.00/0 © 2014 Optical Society of America
Nanostructured elements can be made of metal [3] as well as dielectric semiconductor material [4,5] and can convert either circularly or linearly polarized light into the desired polarization state. Depending upon the materials used, different working principles, desired polarization distribution, and object phase, an extra phase term may appear, e.g., the so-called geometrical or Berry phase [6]. Therefore, polarization states generated by those elements should be characterized with respect to both the phase and the orientation of the polarization. The polarization usually is described using the Stokes parameters [7], which is common practice. This measurement only returns the phase retardation of two orthogonal polarization states with 10 May 2014 / Vol. 53, No. 14 / APPLIED OPTICS
3125
respect to each other rather than the local optical phase of the element under test. The well-known phase shifting interferometry (PSI) is a good method for measuring the phase of a given wavefront [8], severely constraining the polarization state of the element under test (usually to globally linear or circular).There are also some noninterferometric works on modal decomposition of vector light fields using computer generated holograms [9,10]. An interferometric measurement method, in which both the phase and the orientation of polarization of the generated light can be measured simultaneously, entails certain problems (see Section 2.A) and needs a new approach. In [11] a new method for simultaneous phase and orientation measurement of locally linear polarized light was described in theory and different classes of algorithms were presented. In this publication the experimental setup is described and first measurement results are shown. Two different polarization elements (described in Section 4) were used for the measurement. 2. Theoretical Background
For convenience, the theoretical background, which has also been described in [11], is briefly outlined. A.
Two-beam Interferometry
The object wave can be described in the Jones-vector formalism [see Eq. (1)], which carries the information about the specimen under test. Here, we assume a spatially variant but locally linear polarization with orientation angle Ωx; y and a general phase Φx; y, which is the sum of the object phase and the additional geometrical phase term. The reference beam [see Eq. (2)] has a spatially invariant linear polarization with angle ω and constant phase φ. uO and uR are the (real) amplitudes of the waves, which may also be a function of position x; y: cosΩx; y J⃗ O uO x; yeiΦx;y ; (1) sinΩx; y ⃗J R uR x; yeiφ cosω : sinω
I 0 u2O u2R ; 3126
V
2uO uR : u2O u2R
APPLIED OPTICS / Vol. 53, No. 14 / 10 May 2014
If the polarization in the object arm at x; y is orthogonal to the polarization in the reference arm at x; y, a complete loss of contrast results in the intensity pattern (Fig. 1). Such regions with low or even vanishing contrast can make the fringe analysis with common phase-shifting methods impossible. B. Measurement Method
With the new measuring method, the phase is shifted φ → φi [see Eq. (5)] as in the conventional PSI. This allows the recovering of the phase information locally in areas with good contrast, but not for the whole measurement range. In addition, the polarization orientation of the reference wave is rotated ω → ωj [see Eq. (5)]. As a result the areas with the vanishing contrast change their position. Since for every region of the interferogram there is now a measurement, where this region shows good visibility, it is possible to extract the phase information for the whole measurement area. In addition, the polarization information is obtained simultaneously:
(2)
Under the condition that the elements of the setup exert only a negligible influence on polarization and phase of object and reference wave, it is possible to calculate the resulting intensity distribution Ix; y derived from the two-beam interference formula [see Eq. (3)]. I 0 x; y is the sum of the intensities of object and reference wave and Vx; y denotes the visibility [see Eq. (4)]: I I 0 1 V cosΦ − φ cosΩ − ω;
Fig. 1. Interferogram of an object beam J⃗ O with radial polarization and spiral phase and a linearly polarized reference beam J⃗ R with a wedge-shaped phase. In the regions, where the polarizations of J⃗ O and J⃗ R are orthogonal, the visibility of the fringe pattern vanishes.
(3)
(4)
φi i · δφ ;
i ∈ f0; …; N φ − 1g;
ωi i · δω ;
i ∈ f0; …; N ω − 1g:
(5)
The values δφ ; δω and the number N φ ; N ω of phase and polarization shifting steps can be varied depending on the time or the accuracy of the measurement and the object to be measured. The possible measurement algorithms can be combined into two classes, which are essentially different in their measurement strategy. The first class may be called “one phase for each polarization step.” The numbers of steps are identical N N φ N ω ∧ N ≥ 5 ∧ N ≠ 6 and the values are calculated using Eq. (6). For N 5, this will further on
be called the N b5 algorithm. Phase and polarization steps are given by δφ
2π ; N
δω
4π : N
(6)
The second class can be described as “N φ phase steps for each one of N ω polarization steps” and will further be referred to the N N φ ∕N φ ;N ω ∕N ω algorithm. For N φ;ω ≥ 3 the step values δφ;ω can be calculated with the first expression in Eq. (7) and δφ;ω by the second expression in Eq. (7): δφ;ω
2π N ϕ;ω
or
δφ;ω
2π : N ϕ;ω − 1
(7)
The number of steps N φ and N ω may but do not have to be equal. For example, all following measurement algorithms N 7;3 , N 5;6 , N 5 ;4 , and N 5 ;5 are possible. The mathematical analysis and the theoretical background can be found in [11]. 3. Measurement Setup
The measurement setup is based on a modified Mach–Zehnder interferometer. The modifications (Fig. 2) are necessary to minimize the influence of the elements of the measurement setup onto the polarization distribution to be measured. In the reference arm the orientation angle of the linear polarization is controlled by the rotation of a half-wave plate (HWP). In the object arm the mirror (M1) is moveable to allow for phase shifting. An almost normal incidence (5°) on the mirrors is necessary to suppress the influence of the mirror reflections on the polarization states as indicated by the Fresnel coefficients (Fig. 3). A Wollaston prism is used as beam splitter that allows the adjustment of the visibility of the interferograms by controlling the intensity ratio between object and reference wave. The adjustment is done by rotating a HWP in front of the prism. The polarization states of the partial beams behind the prism have well known orientation and are of high quality. As a beam combiner, a specially designed phase grating is used. The influence of the grating parameters such as the period, duty cycle, and etching depth on TE and TM polarization have been investigated through a rigorous diffraction method regarding diffraction efficiency and phase retardation, and
Fig. 2. Sketch of a Mach–Zehnder interferometer adapted to the measurement task. Wollaston prism as a beam splitter, reference arm with an additional HWP, moveable mirror M1 in object arm, phase grating as a beam combiner.
Fig. 3. Fresnel coefficients as a function of the angle of incidence.
the grating is designed to conserve any polarization patterns present in the incident beams. The polarization state of light can be chosen to be linear or circular (with a help of a quarter wave plate), depending on the polarization element being measured. Currently, the following average values for repeatability and reproducibility are reached: • repeatability: Φ 0.01 rad, Ω 0.004 rad; • reproducibility: Φ 0.02 rad, Ω 0.006 rad. 4. Measurement Results
We present results of three different measurement algorithms. N b5 from the first class and two from the second: N 3;3 and N 5;4 . Two different polarization elements were measured. The simulations of the robustness of individual algorithms [11] indicate that small step errors by the algorithm N b5 of the first class lead to relatively large deviations in the measurement results. The algorithms of the second class should deliver even better results with respect to robustness against phase and polarization errors. With an increasing number of intensity images the measurement results should become more robust. A. Objects to be Measured
The first device under test was a radial polarizer (radPOL) made from a subwavelength aluminum radial grating on a glass substrate for the wavelength λ 633 nm [3]. The circular input polarization is converted into linearly polarized light parallel to the grating vector Fig. 4 left]: This optical element generates an additional vortex phase term in form of a 2π spiral together with the desired polarization distribution [3,6]. This phase vortex can be removed in the measured data by subtracting it via software. A more elegant solution is to repeat the measurement with an illumination of reverse handed circular polarization. By reversing the sense of rotation of the polarization, the sign of the geometric phase changes. By adding these two measured phases, the geometric term cancels and we obtain the pure object phase. The second element is made of the semiconductor material Si3 N4 . The nanostructured Si3 N4 layer on a glass substrate generates a phase shift of π, so the structure can be considered as a HWP with a locally variant orientation [12],radial HWP (radHWP) (Fig. 5). 10 May 2014 / Vol. 53, No. 14 / APPLIED OPTICS
3127
Fig. 4. Schematic representation of the effect of a radPOL. RadPOL with an additional phase vortex is generated from right and left circular input polarizations, respectively.
Therefore, the incident linear polarization is rotated by twice the angle α, where α is the angle between the polarization direction of light and the local grating vector of the element structure. The local orientation of the output polarization can be changed by the rotation of the local subwavelength grating with respect to the input polarization. This element allows the generation of radial or azimuthal polarization or an “intermediate state” by just changing the orientation of polarization of the incident wave. B.
N b5 Algorithm
The N b5 algorithm with only five pictures is the algorithm with the smallest number of measurements in order to measure both phase and polarization. The used steps are δφ
2π 5
δω
4π : 5
and
The measured phase distribution and the orientation of the polarization are demonstrated in Fig. 6 for radPOL and in Fig. 7 for radHWP. This phase vortex can be removed in the measured data by subtracting it via software to get the deviations of the distribution from the ideal spiral form.
Fig. 6. Orientation of polarization (left) and relative phase (right) measured with N b5 algorithm for the element radPOL.
C.
N 5 ;4 Algorithm
By the N 5 ;4 algorithm five phase images with δφ
π 2
are taken each of four polarization positions with π δω : 2 This method averages over the phase range of 2π so the first and the last phase image by each polarization are identical. From a total of 20 images the phase and the orientation of the polarization was calculated, the deviations from the ideal 2π spiral are shown in Fig. 8 for radPOL and for radHWP in Fig. 9. The 2π spirals in polarization distributions behind the radPOL and the radHWP were removed numerically by the software. The phase distribution behind the radPOL is the result of the sum of two measured phase results with left and right circularly polarized incident light. As described in Section 4.A, the additional 2π geometric phase terms of both results have opposite signs and can be eliminated by the sum. D.
N 3;3 Algorithm
The N 3;3 -algorithm with δφ;ω
2π 3
is the minimal measurement algorithm of the second class, for the calculation only nine images are used. The calculated orientation of polarization
Fig. 5. Functional sketch of the generation of radial (top), azimuthal (bottom) polarization and an “intermediate state” (middle) from linearly polarized light. 3128
APPLIED OPTICS / Vol. 53, No. 14 / 10 May 2014
Fig. 7. Orientation of polarization (left) and relative phase (right) measured with N b5 algorithm for the element radHWP.
Fig. 8. Orientation of polarization after addition of reverse spiral term (left) and phase distribution after addition of a measured phase with an illumination of reverse circular polarization (right) measured with N 5 ;4 algorithm of radPOL.
Fig. 11. Orientation of polarization (left) without the spiral term and relative phase (right) measured with N 3;3 algorithm.
In this paper three different algorithms from both classes were presented. All presented measurement results (see Figs. 6–11) calculated from different numbers of polarization orientations and phase steps correspond to the expectations and are consistent with the theory. The polarization has the form of a 2π spiral, the small deviations from the ideal radial orientation can be explained by small structural defects. The visible stripes in both distributions are the artifacts caused by lithographic fabrication of the grid used for beam combination.
The residual polarizations in Figs. 9 and 11 show a systematic structure. It seems plausible to attribute this residual systematic to a small ellipticity still present in the polarization field. The effect of such small deviations from the assumption of local linear polarization is currently under investigation. The object phase of radPOL is superimposed with the geometrical term (see Fig. 6). This additional phase vortex can be eliminated in the two ways described in Section 4. If we compare the pure object phases after applying these two methods, then it becomes clear that the results are very similar. By forming the difference of two measurement results of different algorithms, it can be seen whether the data coincide or not. Figure 12 shows the difference of results calculated with N 3;3 and N 6;6 algorithms. The rms values of both data are in the same order of magnitude as the reproducibility and do not have any areas with significant mismatch. Figure 12 shows the difference of phase and polarization results calculated with N 3;3 and N 6;6 algorithms. The robustness against stepping errors was analyzed by extra measurements with misaligned phase and polarization steps. The robustness increased with an increasing number of measurement steps, as predicted in the simulation [11]. Especially for the N b5 algorithm, the steps must be adjusted very precisely in order to get good results. On the other hand, it turned out that a relatively large number of measuring steps in both the phase as well as the polarization do not necessarily lead to an improvement of the measurement accuracy. The longer the measurement time, the greater the influence of external disturbances on the measurement result such as air turbulence and vibrations. The values
Fig. 10. Orientation of polarization (left) and relative phase (right) measured with N 3;3 algorithm of radPOL after addition of reverse spiral term.
Fig. 12. Difference of results measured with N 3;3 and N 6;6 algorithms showing polarization (left) and phase (right), rms values are 0.005 and 0.01, respectively.
Fig. 9. Orientation of polarization after addition of reverse spiral term (left) and phase (right) measured with N 5 ;4 algorithm of radHWP.
and relative phase measured with N 3;3 algorithm of radPOL and radHWP, after addition of reverse spiral term, are shown in Figs. 10 and 11, respectively. The spiral phase term behind the radPOL and of the polarization spirals behind both elements were subtracted by the software. 5. Discussion
10 May 2014 / Vol. 53, No. 14 / APPLIED OPTICS
3129
of repeatability and reproducibility do not become better by a larger number of measuring steps and stay approximately constant.
5.
6. Conclusion
The first measurements show very good results. It was experimentally shown that just five or nine shots, respectively, are sufficient for the simultaneous measurement of the phase and the local orientation of the linear polarization. The next step is the extension of the measurement algorithm to arbitrary states of local elliptical polarization.
6. 7. 8. 9.
References 1. 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). 2. J. A. Davis, D. E. McNamara, D. M. Cottrell, and T. Sonehara, “Two-dimensional polarization encoding with a phase-only liquid-crystal spatial light modulator,” Appl. Opt. 39, 1549– 1554 (2000). 3. Z. Ghadyani, I. Vartiainen, I. Harder, W. Iff, A. Berger, N. Lindlein, and M. Kuittinen, “Concentric ring metal grating for generating radially polarized light,” Appl. Opt. 50, 2451–2457 (2011). 4. Z. Ghadyani, S. Dmitriev, N. Lindlein, G. Leuchs, O. Rusina, and I. Harder, “Discontinuous space variant sub-wavelength
3130
APPLIED OPTICS / Vol. 53, No. 14 / 10 May 2014
10.
11.
12.
structures for generating radially polarized light in visible region,” J. Eur. Opt. Soc. 6, 11041 (2011). Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Spacevariant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. 27, 1141–1143 (2002). G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Formation of helical beams by use of Pancharatnam-Berry phase optical elements,” Opt. Lett. 27, 1875–1877 (2002). D. Goldstein and D. H. Goldstein, Polarized Light (Dekker, 2011). H. Schreiber and J. H. Bruning, Phase Shifting Interferometry (Wiley, 2006). D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, and M. Duparré, “Measuring the spatial polarization distribution of multimode beams emerging from passive step-index large-mode-area fibers,” Opt. Lett. 35, 3429–3431 (2010). C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013). A. Berger, V. Nercissian, K. Mantel, and I. Harder, “Evaluation algorithms for multistep measurement of spatially varying linear polarization and phase,” Opt. Lett. 37, 4140–4142 (2012). S. Dmitriev, I. Harder, and N. Lindlein, “Artificial wave plates made from sub-wavelength structures,” in DGaO Proceedings (2012), http://www.dgao‑proceedings.de/download/113/ 113_p16.pd.
Institute of Optics, Information, and Photonics, Friedrich-Alexander-University Erlangen-Nürnberg (FAU), Staudtstr. 7/B2, 91058 Erlangen, Germany
2
Max Planck Institute for the Science of Light, Günther-Scharowsky-Str. 1/Bldg 24, 91058 Erlangen, Germany *Corresponding author: [email protected] Received 31 January 2014; revised 4 April 2014; accepted 4 April 2014; posted 8 April 2014 (Doc. ID 205784); published 9 May 2014
Optical components manipulating both polarization and phase of wave fields find many applications in today’s optical systems. With modern lithography methods it is possible to fabricate optical elements with nanostructured surfaces from different materials capable of generating spatially varying, locally linearly polarized-light distributions, tailored to the application in question. Since such elements in general also affect the phase of the light field, the characterization of the function of such elements consists in measuring the phase and the polarization of the generated light, preferably at the same time. Here, we will present first results of an interferometric approach for a simultaneous and spatially resolved measurement of both phase and polarization, as long as the local polarization at any point is linear (e.g., for radially or azimuthally polarized light). © 2014 Optical Society of America OCIS codes: (120.3180) Interferometry; (160.1190) Anisotropic optical materials; (120.5050) Phase measurement; (260.5430) Polarization. http://dx.doi.org/10.1364/AO.53.003125
1. Introduction
Nowadays, wave fields with polarization states other than globally linear or circular polarization are in great demand for various applications. Very important polarization states are those with a locally linear state but spatially varying orientation such as radial and azimuthal polarization. Even more exotic polarization states can be created artificially in different ways such as interferometric methods [1], the use of liquid crystals [2], or elements with nanostructured surfaces.
1559-128X/14/143125-06$15.00/0 © 2014 Optical Society of America
Nanostructured elements can be made of metal [3] as well as dielectric semiconductor material [4,5] and can convert either circularly or linearly polarized light into the desired polarization state. Depending upon the materials used, different working principles, desired polarization distribution, and object phase, an extra phase term may appear, e.g., the so-called geometrical or Berry phase [6]. Therefore, polarization states generated by those elements should be characterized with respect to both the phase and the orientation of the polarization. The polarization usually is described using the Stokes parameters [7], which is common practice. This measurement only returns the phase retardation of two orthogonal polarization states with 10 May 2014 / Vol. 53, No. 14 / APPLIED OPTICS
3125
respect to each other rather than the local optical phase of the element under test. The well-known phase shifting interferometry (PSI) is a good method for measuring the phase of a given wavefront [8], severely constraining the polarization state of the element under test (usually to globally linear or circular).There are also some noninterferometric works on modal decomposition of vector light fields using computer generated holograms [9,10]. An interferometric measurement method, in which both the phase and the orientation of polarization of the generated light can be measured simultaneously, entails certain problems (see Section 2.A) and needs a new approach. In [11] a new method for simultaneous phase and orientation measurement of locally linear polarized light was described in theory and different classes of algorithms were presented. In this publication the experimental setup is described and first measurement results are shown. Two different polarization elements (described in Section 4) were used for the measurement. 2. Theoretical Background
For convenience, the theoretical background, which has also been described in [11], is briefly outlined. A.
Two-beam Interferometry
The object wave can be described in the Jones-vector formalism [see Eq. (1)], which carries the information about the specimen under test. Here, we assume a spatially variant but locally linear polarization with orientation angle Ωx; y and a general phase Φx; y, which is the sum of the object phase and the additional geometrical phase term. The reference beam [see Eq. (2)] has a spatially invariant linear polarization with angle ω and constant phase φ. uO and uR are the (real) amplitudes of the waves, which may also be a function of position x; y: cosΩx; y J⃗ O uO x; yeiΦx;y ; (1) sinΩx; y ⃗J R uR x; yeiφ cosω : sinω
I 0 u2O u2R ; 3126
V
2uO uR : u2O u2R
APPLIED OPTICS / Vol. 53, No. 14 / 10 May 2014
If the polarization in the object arm at x; y is orthogonal to the polarization in the reference arm at x; y, a complete loss of contrast results in the intensity pattern (Fig. 1). Such regions with low or even vanishing contrast can make the fringe analysis with common phase-shifting methods impossible. B. Measurement Method
With the new measuring method, the phase is shifted φ → φi [see Eq. (5)] as in the conventional PSI. This allows the recovering of the phase information locally in areas with good contrast, but not for the whole measurement range. In addition, the polarization orientation of the reference wave is rotated ω → ωj [see Eq. (5)]. As a result the areas with the vanishing contrast change their position. Since for every region of the interferogram there is now a measurement, where this region shows good visibility, it is possible to extract the phase information for the whole measurement area. In addition, the polarization information is obtained simultaneously:
(2)
Under the condition that the elements of the setup exert only a negligible influence on polarization and phase of object and reference wave, it is possible to calculate the resulting intensity distribution Ix; y derived from the two-beam interference formula [see Eq. (3)]. I 0 x; y is the sum of the intensities of object and reference wave and Vx; y denotes the visibility [see Eq. (4)]: I I 0 1 V cosΦ − φ cosΩ − ω;
Fig. 1. Interferogram of an object beam J⃗ O with radial polarization and spiral phase and a linearly polarized reference beam J⃗ R with a wedge-shaped phase. In the regions, where the polarizations of J⃗ O and J⃗ R are orthogonal, the visibility of the fringe pattern vanishes.
(3)
(4)
φi i · δφ ;
i ∈ f0; …; N φ − 1g;
ωi i · δω ;
i ∈ f0; …; N ω − 1g:
(5)
The values δφ ; δω and the number N φ ; N ω of phase and polarization shifting steps can be varied depending on the time or the accuracy of the measurement and the object to be measured. The possible measurement algorithms can be combined into two classes, which are essentially different in their measurement strategy. The first class may be called “one phase for each polarization step.” The numbers of steps are identical N N φ N ω ∧ N ≥ 5 ∧ N ≠ 6 and the values are calculated using Eq. (6). For N 5, this will further on
be called the N b5 algorithm. Phase and polarization steps are given by δφ
2π ; N
δω
4π : N
(6)
The second class can be described as “N φ phase steps for each one of N ω polarization steps” and will further be referred to the N N φ ∕N φ ;N ω ∕N ω algorithm. For N φ;ω ≥ 3 the step values δφ;ω can be calculated with the first expression in Eq. (7) and δφ;ω by the second expression in Eq. (7): δφ;ω
2π N ϕ;ω
or
δφ;ω
2π : N ϕ;ω − 1
(7)
The number of steps N φ and N ω may but do not have to be equal. For example, all following measurement algorithms N 7;3 , N 5;6 , N 5 ;4 , and N 5 ;5 are possible. The mathematical analysis and the theoretical background can be found in [11]. 3. Measurement Setup
The measurement setup is based on a modified Mach–Zehnder interferometer. The modifications (Fig. 2) are necessary to minimize the influence of the elements of the measurement setup onto the polarization distribution to be measured. In the reference arm the orientation angle of the linear polarization is controlled by the rotation of a half-wave plate (HWP). In the object arm the mirror (M1) is moveable to allow for phase shifting. An almost normal incidence (5°) on the mirrors is necessary to suppress the influence of the mirror reflections on the polarization states as indicated by the Fresnel coefficients (Fig. 3). A Wollaston prism is used as beam splitter that allows the adjustment of the visibility of the interferograms by controlling the intensity ratio between object and reference wave. The adjustment is done by rotating a HWP in front of the prism. The polarization states of the partial beams behind the prism have well known orientation and are of high quality. As a beam combiner, a specially designed phase grating is used. The influence of the grating parameters such as the period, duty cycle, and etching depth on TE and TM polarization have been investigated through a rigorous diffraction method regarding diffraction efficiency and phase retardation, and
Fig. 2. Sketch of a Mach–Zehnder interferometer adapted to the measurement task. Wollaston prism as a beam splitter, reference arm with an additional HWP, moveable mirror M1 in object arm, phase grating as a beam combiner.
Fig. 3. Fresnel coefficients as a function of the angle of incidence.
the grating is designed to conserve any polarization patterns present in the incident beams. The polarization state of light can be chosen to be linear or circular (with a help of a quarter wave plate), depending on the polarization element being measured. Currently, the following average values for repeatability and reproducibility are reached: • repeatability: Φ 0.01 rad, Ω 0.004 rad; • reproducibility: Φ 0.02 rad, Ω 0.006 rad. 4. Measurement Results
We present results of three different measurement algorithms. N b5 from the first class and two from the second: N 3;3 and N 5;4 . Two different polarization elements were measured. The simulations of the robustness of individual algorithms [11] indicate that small step errors by the algorithm N b5 of the first class lead to relatively large deviations in the measurement results. The algorithms of the second class should deliver even better results with respect to robustness against phase and polarization errors. With an increasing number of intensity images the measurement results should become more robust. A. Objects to be Measured
The first device under test was a radial polarizer (radPOL) made from a subwavelength aluminum radial grating on a glass substrate for the wavelength λ 633 nm [3]. The circular input polarization is converted into linearly polarized light parallel to the grating vector Fig. 4 left]: This optical element generates an additional vortex phase term in form of a 2π spiral together with the desired polarization distribution [3,6]. This phase vortex can be removed in the measured data by subtracting it via software. A more elegant solution is to repeat the measurement with an illumination of reverse handed circular polarization. By reversing the sense of rotation of the polarization, the sign of the geometric phase changes. By adding these two measured phases, the geometric term cancels and we obtain the pure object phase. The second element is made of the semiconductor material Si3 N4 . The nanostructured Si3 N4 layer on a glass substrate generates a phase shift of π, so the structure can be considered as a HWP with a locally variant orientation [12],radial HWP (radHWP) (Fig. 5). 10 May 2014 / Vol. 53, No. 14 / APPLIED OPTICS
3127
Fig. 4. Schematic representation of the effect of a radPOL. RadPOL with an additional phase vortex is generated from right and left circular input polarizations, respectively.
Therefore, the incident linear polarization is rotated by twice the angle α, where α is the angle between the polarization direction of light and the local grating vector of the element structure. The local orientation of the output polarization can be changed by the rotation of the local subwavelength grating with respect to the input polarization. This element allows the generation of radial or azimuthal polarization or an “intermediate state” by just changing the orientation of polarization of the incident wave. B.
N b5 Algorithm
The N b5 algorithm with only five pictures is the algorithm with the smallest number of measurements in order to measure both phase and polarization. The used steps are δφ
2π 5
δω
4π : 5
and
The measured phase distribution and the orientation of the polarization are demonstrated in Fig. 6 for radPOL and in Fig. 7 for radHWP. This phase vortex can be removed in the measured data by subtracting it via software to get the deviations of the distribution from the ideal spiral form.
Fig. 6. Orientation of polarization (left) and relative phase (right) measured with N b5 algorithm for the element radPOL.
C.
N 5 ;4 Algorithm
By the N 5 ;4 algorithm five phase images with δφ
π 2
are taken each of four polarization positions with π δω : 2 This method averages over the phase range of 2π so the first and the last phase image by each polarization are identical. From a total of 20 images the phase and the orientation of the polarization was calculated, the deviations from the ideal 2π spiral are shown in Fig. 8 for radPOL and for radHWP in Fig. 9. The 2π spirals in polarization distributions behind the radPOL and the radHWP were removed numerically by the software. The phase distribution behind the radPOL is the result of the sum of two measured phase results with left and right circularly polarized incident light. As described in Section 4.A, the additional 2π geometric phase terms of both results have opposite signs and can be eliminated by the sum. D.
N 3;3 Algorithm
The N 3;3 -algorithm with δφ;ω
2π 3
is the minimal measurement algorithm of the second class, for the calculation only nine images are used. The calculated orientation of polarization
Fig. 5. Functional sketch of the generation of radial (top), azimuthal (bottom) polarization and an “intermediate state” (middle) from linearly polarized light. 3128
APPLIED OPTICS / Vol. 53, No. 14 / 10 May 2014
Fig. 7. Orientation of polarization (left) and relative phase (right) measured with N b5 algorithm for the element radHWP.
Fig. 8. Orientation of polarization after addition of reverse spiral term (left) and phase distribution after addition of a measured phase with an illumination of reverse circular polarization (right) measured with N 5 ;4 algorithm of radPOL.
Fig. 11. Orientation of polarization (left) without the spiral term and relative phase (right) measured with N 3;3 algorithm.
In this paper three different algorithms from both classes were presented. All presented measurement results (see Figs. 6–11) calculated from different numbers of polarization orientations and phase steps correspond to the expectations and are consistent with the theory. The polarization has the form of a 2π spiral, the small deviations from the ideal radial orientation can be explained by small structural defects. The visible stripes in both distributions are the artifacts caused by lithographic fabrication of the grid used for beam combination.
The residual polarizations in Figs. 9 and 11 show a systematic structure. It seems plausible to attribute this residual systematic to a small ellipticity still present in the polarization field. The effect of such small deviations from the assumption of local linear polarization is currently under investigation. The object phase of radPOL is superimposed with the geometrical term (see Fig. 6). This additional phase vortex can be eliminated in the two ways described in Section 4. If we compare the pure object phases after applying these two methods, then it becomes clear that the results are very similar. By forming the difference of two measurement results of different algorithms, it can be seen whether the data coincide or not. Figure 12 shows the difference of results calculated with N 3;3 and N 6;6 algorithms. The rms values of both data are in the same order of magnitude as the reproducibility and do not have any areas with significant mismatch. Figure 12 shows the difference of phase and polarization results calculated with N 3;3 and N 6;6 algorithms. The robustness against stepping errors was analyzed by extra measurements with misaligned phase and polarization steps. The robustness increased with an increasing number of measurement steps, as predicted in the simulation [11]. Especially for the N b5 algorithm, the steps must be adjusted very precisely in order to get good results. On the other hand, it turned out that a relatively large number of measuring steps in both the phase as well as the polarization do not necessarily lead to an improvement of the measurement accuracy. The longer the measurement time, the greater the influence of external disturbances on the measurement result such as air turbulence and vibrations. The values
Fig. 10. Orientation of polarization (left) and relative phase (right) measured with N 3;3 algorithm of radPOL after addition of reverse spiral term.
Fig. 12. Difference of results measured with N 3;3 and N 6;6 algorithms showing polarization (left) and phase (right), rms values are 0.005 and 0.01, respectively.
Fig. 9. Orientation of polarization after addition of reverse spiral term (left) and phase (right) measured with N 5 ;4 algorithm of radHWP.
and relative phase measured with N 3;3 algorithm of radPOL and radHWP, after addition of reverse spiral term, are shown in Figs. 10 and 11, respectively. The spiral phase term behind the radPOL and of the polarization spirals behind both elements were subtracted by the software. 5. Discussion
10 May 2014 / Vol. 53, No. 14 / APPLIED OPTICS
3129
of repeatability and reproducibility do not become better by a larger number of measuring steps and stay approximately constant.
5.
6. Conclusion
The first measurements show very good results. It was experimentally shown that just five or nine shots, respectively, are sufficient for the simultaneous measurement of the phase and the local orientation of the linear polarization. The next step is the extension of the measurement algorithm to arbitrary states of local elliptical polarization.
6. 7. 8. 9.
References 1. 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). 2. J. A. Davis, D. E. McNamara, D. M. Cottrell, and T. Sonehara, “Two-dimensional polarization encoding with a phase-only liquid-crystal spatial light modulator,” Appl. Opt. 39, 1549– 1554 (2000). 3. Z. Ghadyani, I. Vartiainen, I. Harder, W. Iff, A. Berger, N. Lindlein, and M. Kuittinen, “Concentric ring metal grating for generating radially polarized light,” Appl. Opt. 50, 2451–2457 (2011). 4. Z. Ghadyani, S. Dmitriev, N. Lindlein, G. Leuchs, O. Rusina, and I. Harder, “Discontinuous space variant sub-wavelength
3130
APPLIED OPTICS / Vol. 53, No. 14 / 10 May 2014
10.
11.
12.
structures for generating radially polarized light in visible region,” J. Eur. Opt. Soc. 6, 11041 (2011). Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Spacevariant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. 27, 1141–1143 (2002). G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Formation of helical beams by use of Pancharatnam-Berry phase optical elements,” Opt. Lett. 27, 1875–1877 (2002). D. Goldstein and D. H. Goldstein, Polarized Light (Dekker, 2011). H. Schreiber and J. H. Bruning, Phase Shifting Interferometry (Wiley, 2006). D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, and M. Duparré, “Measuring the spatial polarization distribution of multimode beams emerging from passive step-index large-mode-area fibers,” Opt. Lett. 35, 3429–3431 (2010). C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013). A. Berger, V. Nercissian, K. Mantel, and I. Harder, “Evaluation algorithms for multistep measurement of spatially varying linear polarization and phase,” Opt. Lett. 37, 4140–4142 (2012). S. Dmitriev, I. Harder, and N. Lindlein, “Artificial wave plates made from sub-wavelength structures,” in DGaO Proceedings (2012), http://www.dgao‑proceedings.de/download/113/ 113_p16.pd.
