Tomographic polarization analyzer by polarization-mode-frequency mapping

Hailong Zhou; Mian Liu; Jianji Dong; Xinliang Zhang

doi:10.1364/OE.25.014023

1. Introduction

The spatial variance of polarization state is an important indicator in diverse areas, such as optical biopsy [1], material characterization [2], extra-solar planet detection and remote mapping of planetary magnetospheres [3,4]. Typically, the issue of polarization measurement has been addressed by dividing the signal up in space or in time [4]. There are many techniques for division-of-space polarimeters that divide the input beam into multiple optical paths and project them simultaneously to different polarization analyzers in a spatial domain or a spectral domain, such as simultaneous detection of Stokes parameters by using multiple detectors [3,5–9], by spatial modulation [10–17], or by spectral modulation [18]. There are also some integrated schemes proposed to make the device compact and conducive to on-chip integration [19–22]. However, these methods need multiple optical polarization elements and an array of photodetectors (PDs). For the division-of-time polarimeters, the polarization of light is modulated by using rotating elements [23], photoelastic modulators [24], or liquid crystal retarders [25] to acquire time sequential data. Then the full-Stokes vector can be calculated by a Fourier-transform spectrometer or by a digital processing system. However, the measured results will be deteriorated if any one of the sampling points has any large deviation. So the time sequential techniques are prone to loose measurement due to very limited sampling points and mechanical instabilities in the sampling. In general, these two kinds of polarization analyzers require multi-step measurements step by step or require multiple PDs, each targeting one of its vector components.

Inspired by tomographic reconstruction techniques widely used in image science [26], we demonstrate a tomographic polarization analyzer by polarization-mode-frequency mapping with time-varying spatial modulation. The unknown light is decomposed into two orthogonal circularly polarized components, and converted to two orbital angular momentum (OAM) modes by a q-plate. Subsequently, we employ time-varying spatial modulation to map the OAM modes to two beating signals with different frequencies. Finally, we complete tomographic reconstruction of the polarization state from the time-varying intensity collected by a PD. The time-varying spatial modulation can be achieved by a programmable device or a spinning object. Both methods are discussed in detail. Compared to the traditional methods, our scheme can directly measure the Jones matrix of light with high precision. Due to the high volume time sampling, our polarization analyzer can resist the systematic and random error in the tomographic reconstruction and offer a high-accuracy measurement.

2. Principle

Figure 1(a) presents the schematic diagram of the proposed polarization analyzer. When an unknown polarized light is input on the device, the input light can be always decomposed into the superposition of the right and left circularly polarized (LCP and RCP) components, and the two orthogonally circular polarizations of light can be converted to two OAM modes by a q-plate [27]. Here the OAM mode (labeled as ${OAM}_{l}$ ) is characterized by $exp (i l θ)$ , where $θ$ is the angular coordinate and $l$ is the topological charge (TC) [28]. On the other hand, the OAM modes can be mapped to different frequencies by using specially designed time-varying spatial modulation. By combining these two aspects, a tomographic polarization analyzer can be designed by polarization-mode-frequency mapping. In the following, we present a simple derivation to explain how the scheme can work. The Jones matrix of input unknown polarization can be written as

E_{i n} = a [\begin{matrix} 1 \\ i \end{matrix}] + b [\begin{matrix} 1 \\ - i \end{matrix}],

where

a

and

b

are the weight coefficients of LCP and RCP components respectively. After passing through a q-plate, the light is expressed by [27]

E_{o 1} = (\begin{matrix} \cos (2 β) & \sin (2 β) \\ \sin (2 β) & - \cos (2 β) \end{matrix}) E_{i n} = a \exp (i 2 β) [\begin{matrix} 1 \\ - i \end{matrix}] + b \exp (- i 2 β) [\begin{matrix} 1 \\ i \end{matrix}] .

Here,

β = q θ + α_{0}

,

θ

is the azimuth angle,

q

is the order of q-plate and

α_{0}

is a constant. One can see that the LCP and RCP light are converted to two OAM modes (labeled as

{OAM}_{2 q}

and

{OAM}_{- 2 q}

) respectively. Then we use a time-varying spatial modulation to measure the OAM distribution. The modulation function of the spatial modulation is designed by

M (θ, t) = \sum_{n \in N} A_{n} \exp (i n θ) [\exp (- i n Ω t) + \exp (i m Ω t)] .

Here,

Ω

is a constant indicating the modulation period,

m

is a constant integer to generate a reference frequency and

t

denotes the time. For simplicity, we assume that

β = 0.5 θ

and

A_{- 1} = A_{1} = 1

. Figure 1(b) shows the phase masks of the modulation function dependent on time. So when the light after q-plate is modulated by the time-varying spatial modulation and then goes through a single mode filtering and a horizontal polarizer, the output light is an x-polarized fundamental mode, expressed by

E_{o 2} = a \exp (i Ω t) + b \exp (- i Ω t) + B \exp (i m Ω t) .

Here,

B = a + b

. The total intensity is given by

\begin{array}{l} I = {| a |}^{2} + {| b |}^{2} + {| B |}^{2} + 2 | a b | \cos [2 Ω t + Λ (a b^{*})] + \\ 2 | a B | \cos [(m - 1) Ω t + Λ (a^{*} B)] + 2 | b B | \cos [(m + 1) Ω t + Λ (b^{*} B)], \end{array}

where

Λ (.)

returns the phase angle. From Eq. (5), we can see that the weight coefficients of LCP and RCP components of input light are proportional to coefficients of two beating signals with frequencies of

(m \mp 1) Ω / 2 π

respectively. The amplitude ratio (define as

γ = | b / a |

) and phase difference (define as

Δ ϕ = angle (b / a)

) can be obtained via tomographic reconstruction of the polarization state from the time-varying intensity signal collected by a PD with spectral analysis. Namely, the Jones matrix of input polarization can be retrieved according to the amplitude ratio and phase difference.

Fig. 1 (a) Schematic diagram of the polarization analyzer. (b) Phase masks of the modulation function dependent on time.

Download Full Size | PDF

3. Experimental results

An experiment is carried out to demonstrate the scheme. The experimental setup is presented in Fig. 2. The light emitted from a laser (wavelength of 1550 nm) is coupled into the free space by a single mode fiber (SMF1) and a fiber coupler (C1). The polarization of light is adjusted by a half wave plate (HWP1) and a quarter wave plate (QWP). At this point, the tested polarization is prepared. Subsequently the tested light is split into two paths by a BS (BS1). One light is modulated as the reference mode (OAM₄) by a computer generated hologram (CGH) [29]. In order to ensure that the reference mode has a high-weight horizontal component, another half wave plate (HWP2) is used to adjust the polarization state. The other path of the tested light is converted into the superposition of two OAM modes by a q-plate. Then, the light illuminates a spatial light modulator (SLM) and passes a horizontal polarizer (Pol.1). Here, the SLM is used to carry the time-varying spatial modulation, which is acquired based on Taylor–Fourier expansion [30]. The method can encode the amplitude and the phase of an optical field into a phase-only hologram, which allows the exact control of spatial transverse modes. It can add an arbitrary intensity and phase modulation to the input light. Another SMF (SMF2) and fiber coupler (C2) are used to select the fundamental mode and then a PD is used to receive the power of fundamental mode. The input polarization can be retrieved by Fourier analysis of the signal collected by the PD. To capture the light patterns, two beam splitters (BS2 and BS3) are used to combine the light after the q-plate and the reference mode to generate interference patterns. Another linear polarizer (Pol.2) and a charge-coupled device (CCD) are placed in one of the output paths of BS3 to capture the light patterns. It's worth noting that the polarization analyzer only contains the part of “Polarization measurement”. The parts of “Polarization preparation” and “Pattern capture” are only used to support the experiment. In the experiment, we set the reference frequency at $4 Ω / 2 π$ , namely, $m = 4$ .

Fig. 2 Experimental setup. Polarization preparation: preparing the test light with various polarization states. Polarization measurement: measuring the input polarization state. Pattern capture: capturing the light patterns.

Download Full Size | PDF

Firstly, we set the tested light as RCP or LCP light by using the quarter wave plate. When the tested light passes through the q-plate, it will be converted into an OAM mode, as shown in Fig. 3(a) or Fig. 3(c). One can see that the intensity patterns are annular for both tested light. The annular patterns are caused by the center phase singularities of OAM modes [27, 31]. Similar to the interference with a plane wave [32], we know that when the OAM mode (OAM_l) interfere with a reference mode (OAM₄), there will be $| l - 4 |$ bright fringes along the azimuth angle (0-2π) and the sign of $l - 4$ is depended on the bending direction of fringes. In our experiment, we make the OAM mode interfere with a reference mode (OAM₄) by using the setup in Fig. 2. One can see that there are five and three bent fringes along the azimuth angle for the RCP and LCP tested light respectively. Therefore, the interference patterns in Fig. 3(b, d) manifest that the OAM modes are OAM_-1 and OAM₁ for RCP and LCP tested light respectively. It proves that the q-plate can indeed convert the two circularly polarized light into two OAM modes. From Eq. (3), we know that the time-varying spatial modulation is a periodical function. Therefore, we can scan the temporal parameter of $Ω t$ in one period from 0 to $2 π$ by refreshing the SLM patterns, and set totally 120 sampling points within one period. The simultaneous intensity and phase modulation expressed by Eq. (3) can be achieved with a single phase-only SLM based on Taylor–Fourier expansion [30]. Figure 1(b) shows the phase masks dependent on time with $Ω t$ = 0, π/2, π, 3π/2, 2π. The harmonic distribution can be calculated by orthogonal integral or Fourier analysis. Figures 4(a) and (b) present the received signal by the PD and the corresponding harmonic distribution when an RCP light is incident. We can see that the received signal is a fifth harmonic signal with a strong direct current component. And the amplitudes of other high harmonic components are quite low compared to that of the fifth harmonic component, which can be seen from the harmonic distribution in Fig. 4(b). So we can deduce that the tested light is an RCP light. For the LCP light, as shown in Figs. 4(c) and (d), only the third harmonic component and the direct current component are dominant, so the tested light is deduced as an LCP light.

Fig. 3 (a, c) The intensity patterns of the converted OAM modes when an RCP or LCP light is incident. (b, d) The corresponding interference patterns with the reference mode (OAM₄).

Download Full Size | PDF

Fig. 4 (a) The received signal by the PD and (b) the corresponding harmonic distribution when a RCP light is incident. (c, d) The results for LCP light. (e, f) The results for a 45° linearly polarized light.

Download Full Size | PDF

When an unknown polarized light is incident on the device, it can be always decomposed into the superposition of an LCP light and an RCP light according to Eq. (1). So the received signal by the PD will always contains a fifth harmonic component and a third harmonic component. More importantly, the amplitude ratio of the two harmonic components is exact the one of the two circularly polarized components, and so is the phase difference. In the experiment, we prepare different linearly and elliptically polarized light to verify the scheme. The theoretical polarization states are preset and determined by tuning the half wave plate and the quarter wave plate. Figure 4 (e) shows the received temporal intensity signal by the PD when a 45° linearly polarized light is preset and incident on our measurement system. From the received signal, we can see that many unequal peaks appear. And from the harmonic distribution shown in Fig. 4(f), we can see there are three high harmonic components dominant, namely the second, the third and the fifth harmonic components. The second harmonic component comes from the beating of the two converted signals with frequencies at $\pm 1 Ω / 2 π$ . The other two are the beating between the signal with the reference frequency at $m Ω / 2 π$ (m = 4) and the two converted signals with frequencies at $\pm 1 Ω / 2 π$ respectively. The amplitude ratio of the two harmonic components is about 0.9314 and the phase difference is about 96°. According to Eq. (1), we can deduce that the measured Jones matrix of input light is ${[\begin{matrix} 1 & 1.1 * \exp (0 .07 i) \end{matrix}]}^{T}$ , which matches well with the theoretical one ( ${[\begin{matrix} 1 & 1 \end{matrix}]}^{T}$ ).

We also further measure other different input polarization states including linear polarizations and elliptical polarizations, as shown in Fig. 5. The “theory” donates the theoretical polarization states which are obtained by tuning the half wave plate and the quarter wave plate. The “experiment” donates the measured results by our method. We can see that the experimental results agree well with the theoretical ones, which proves that our scheme has the ability to effectively measure the unknown polarization state of light.

Fig. 5 (a) The measured amplitude ratio and (b) the phase difference for different input polarization states.

Download Full Size | PDF

4. Polarization analyzer with a spinning object

In the aforementioned scheme, the setup is quite simple but a programmable SLM is needed to carry the time-varying spatial modulation, which makes this scheme high-cost. In fact, the SLM can be replaced by a spinning object if the reference frequency term in Eq. (3) is removed from the time-varying spatial modulation. In the case, the modulation function of the time-varying spatial modulation is expressed by

M (θ, t) = \sum_{n \in N} A_{n} \exp (i n θ) \exp (- i n Ω t) = \sum_{n \in N} A_{n} \exp [i n (θ - Ω t)]

Here,

Ω

is the rotation speed,

t

denotes the time and

Ω t

denotes the rotation angle. So this kind of time-varying spatial modulation can be replaced by a spinning object, as shown in Fig. 6(a), which is similar to our previous works [32, 33]. In our previous works, we proposed an OAM complex spectrum analyzer that enables simultaneous measurements of the power and phase distributions of OAM modes by mapping the modes to frequencies with the rotational Doppler Effect [32, 33], in which a spinning object with an optically rough surface may induce a Doppler shift in light reflected parallel to the rotation axis, provided that the light carries OAM [34–38]. According to this work, we can introduce a reference light to regenerate the reference frequency. The reference light, expressed by

E_{r} = B \exp (i m θ) {[1, 0]}^{T}

, will illuminate the spinning object simultaneously with

E_{o 1}

, where

B

is the complex amplitude and

m

is the mode index. For simplicity, we assume that

β = 0.5 θ

and

A_{- 1} = A_{1} = A_{- m} = 1

. So after single mode filtering and horizontal polarization filtering, the output x-polarized light will have the similar expression by Eq. (4). Namely, by introducing a reference light, the time-varying spatial modulation can be implemented by a spinning object equivalently.

Fig. 6 (a) Schematic diagram and (b) Experimental setup of the polarization analyzer by using a spinning object.

Download Full Size | PDF

The experimental setup for spinning object is shown in Fig. 6(b). The parts of Polarization preparation and Pattern capture are similar to the aforementioned scheme. In the Polarization measurement, the tested light is split into two paths by a BS (BS1). One light is converted into the superposition of two OAM modes by a q-plate and the other one is modulated as the reference mode (OAM₄) by a computer generated hologram (CGH). In order to adjust the reference mode with a high-weight horizontal component, another half wave plate (HWP2) is used to adjust the polarization state. Then the two light beams are combined into one by another BS (BS2). Subsequently, the combined light passes a horizontal polarizer (Pol.1) and illuminates a spatial light modulator (SLM). Here, the SLM is used to emulate the spinning object by rotating the carried pattern, which is expressed in Eq. (6). Another SMF (SMF2) is used to select the fundamental mode and a PD is used to receive the power of fundamental mode. The input polarization can be retrieved by Fourier analysis of the signal collected by the PD.

The measured results are shown in Fig. 7, when an RCP, an LCP or a −45° linearly polarized light is incident. We can see only the fifth harmonic component and the direct current component are dominant for RCP light illumination, so the tested light is deduced as an RCP light. Only the third harmonic component and the direct current component are dominant for LCP light illumination, so the tested light is an LCP light. When a −45° linearly polarized light is incident, there are three high harmonic components dominant, namely the second, the third and the fifth harmonic components. The second harmonic component comes from the beating of the two converted signals with frequencies at $\pm 1 Ω / 2 π$ . The other two are the beating between the signal with the reference frequency at $m Ω / 2 π$ and the two converted signals with frequencies at $\pm 1 Ω / 2 π$ respectively. The amplitude ratio of the two harmonic components is about 1.017 and the phase difference is about −93°. According to Eq. (1), we can deduce that the measured Jones matrix of polarization light is ${[\begin{matrix} 1 & 1 .054 * \exp (-3 .125 i) \end{matrix}]}^{T}$ , which matches well with the theoretical one ( ${[\begin{matrix} 1 & - 1 \end{matrix}]}^{T}$ ). Similarly, different input polarization states are measured and the final results are presented in Fig. 8. We can see that the experimental results agree well with the theoretical ones, proving the new scheme with a spinning object can also implement the polarization analysis.

Fig. 7 (a) The received signal by the PD and (b) the corresponding harmonic distribution when an RCP light is incident for the polarization analyzer by using a spinning object. (c, d) The results for LCP light. (e, f) The results for a −45° linearly polarized light.

Download Full Size | PDF

Fig. 8 (a) The measured amplitude ratio and (a) the phase difference for different input polarizations for the polarization analyzer by using a spinning object.

Download Full Size | PDF

In fact, the principles of our two schemes are similar, where both are based on the polarization-mode-frequency mapping. The differences are in the measurement of OAM complex spectrum. The comparisons of the two schemes are listed in Table 1. The original scheme can generate the reference frequency by the time-varying spatial modulation, so no reference light and interferometer are required, which make the setup quite simple and robust, but a programmable device is needed to generate the time-varying spatial modulation. In general, a programmable device is high-cost. In order to achieve the same function in a low-cost way, the new scheme uses a spinning object to replace the programmable device and regenerates the reference frequency by introducing a reference light. In contrast, it makes the optical path quite complex and an interferometer is required, resulting in that the device is susceptible to outside interference. In addition, the use of a real spinning object would increase the requirements regarding the mechanical performance of the system or introduce mechanical vibrations, resulting in an unstable working state. But a transmissive spinning object such as a phase plate will reduce the influence because the mechanical vibrations will almost not affect the output direction of transmissive light.

Table 1. Comparisons of the two schemes.

View Table

In the experiment, we set the parameter $m = 4$ and $β = 0.5 θ$ . In fact, other parameter settings are still feasible, only if the three beating frequencies in Eq. (5) are all different. In addition, if the condition: $A_{- 1} = A_{1}$ or $A_{- 1} = A_{1} = A_{- m} = 1$ cannot be satisfied, the scheme can still work with a system calibration [33].

Our method can also be extended to measure the non-homogeneous polarization state in the near future, such as the non-homogeneous polarization state: ${[\begin{matrix} \cos l θ & \sin l θ \end{matrix}]}^{T}$ and ${[\begin{matrix} \sin l θ & - \cos l θ \end{matrix}]}^{T}$ (i.e., the vector beams) [39, 40]. We can measure the OAM mode distribution of x- and y-polarized components by using the same method respectively. And then we convert the OAM mode distribution into the trigonometric function distribution to get the distribution of $\cos l θ$ and $\sin l θ$ , namely the distribution of two non-homogeneous polarization states.

5. Conclusions

In summary, we put forward a tomographic polarization analyzer by polarization-mode-frequency mapping. The two orthogonal circularly polarized components of light are converted to two OAM modes by a q-plate, and then the OAM modes are mapping to two beating signals with different frequencies based on time-varying spatial modulation carried by a programmable SLM. The polarization of light can be retrieved by tomographic reconstruction of the signal collected by a PD. Further, a cheap spinning object can be used to replace the programmable device but an additional reference light is needed to regenerate the reference frequency, which makes the setup more complex. Comparisons of the two methods are discussed in detail. Both methods can work well. Compared to the traditional methods, the scheme can directly measure the Jones matrix of light and is precise enough due to the high volume time sampling and resistance to the systematic and random error.

Funding

National Natural Science Foundation of China (NSFC) (61622502 and 61475052).

References and links

1. M.-R. Antonelli, A. Pierangelo, T. Novikova, P. Validire, A. Benali, B. Gayet, and A. De Martino, “Mueller matrix imaging of human colon tissue for cancer diagnostics: how Monte Carlo modeling can help in the interpretation of experimental data,” Opt. Express 18(10), 10200–10208 (2010). [CrossRef] [PubMed]

2. R. M. Azzam and N. M. Bashara, Ellipsometry and polarized light (North-Holland. sole distributors for the USA and Canada, Elsevier Science Publishing Co., Inc., 1987).

3. J. D. Perreault, “Triple Wollaston-prism complete-Stokes imaging polarimeter,” Opt. Lett. 38(19), 3874–3877 (2013). [CrossRef] [PubMed]

4. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45(22), 5453–5469 (2006). [CrossRef] [PubMed]

5. A. Peinado, A. Turpin, A. Lizana, E. Fernández, J. Mompart, and J. Campos, “Conical refraction as a tool for polarization metrology,” Opt. Lett. 38(20), 4100–4103 (2013). [CrossRef] [PubMed]

6. R. M. A. Azzam, “Arrangement of four photodetectors for measuring the state of polarization of light,” Opt. Lett. 10(7), 309–311 (1985). [CrossRef] [PubMed]

7. R. M. A. Azzam, “Division-of-amplitude Photopolarimeter (DOAP) for the Simultaneous Measurement of All Four Stokes Parameters of Light,” Optica Acta: International Journal of Optics 29(5), 685–689 (1982). [CrossRef]

8. E. Compain and B. Drevillon, “Broadband Division-of-Amplitude Polarimeter Based on Uncoated Prisms,” Appl. Opt. 37(25), 5938–5944 (1998). [CrossRef] [PubMed]

9. O. Morel, R. Seulin, and D. Fofi, “Handy method to calibrate division-of-amplitude polarimeters for the first three Stokes parameters,” Opt. Express 24(12), 13634–13646 (2016). [CrossRef] [PubMed]

10. V. Gruev, A. Ortu, N. Lazarus, J. Van der Spiegel, and N. Engheta, “Fabrication of a dual-tier thin film micropolarization array,” Opt. Express 15(8), 4994–5007 (2007). [CrossRef] [PubMed]

11. J. S. Tyo, C. F. LaCasse, and B. M. Ratliff, “Total elimination of sampling errors in polarization imagery obtained with integrated microgrid polarimeters,” Opt. Lett. 34(20), 3187–3189 (2009). [CrossRef] [PubMed]

12. T. Wakayama, Y. Otani, and T. Yoshizawa, “Axisymmetrical Mueller matrix polarimeter,” in 2009), 74610M–74610M–74618. [CrossRef]

13. I. Nishiyama, N. Yoshida, Y. Otani, and N. Umeda, “Single-shot birefringence measurement using radial polarizer fabricated by direct atomic force microscope stroking method,” Meas. Sci. Technol. 18(6), 1673–1677 (2007). [CrossRef]

14. T. Mu, C. Zhang, Q. Li, and R. Liang, “Error analysis of single-snapshot full-Stokes division-of-aperture imaging polarimeters,” Opt. Express 23(8), 10822–10835 (2015). [CrossRef] [PubMed]

15. X. Meng, J. Li, D. Liu, and R. Zhu, “Fourier transform imaging spectropolarimeter using simultaneous polarization modulation,” Opt. Lett. 38(5), 778–780 (2013). [CrossRef] [PubMed]

16. R. Perkins and V. Gruev, “Signal-to-noise analysis of Stokes parameters in division of focal plane polarimeters,” Opt. Express 18(25), 25815–25824 (2010). [CrossRef] [PubMed]

17. W.-L. Hsu, G. Myhre, K. Balakrishnan, N. Brock, M. Ibn-Elhaj, and S. Pau, “Full-Stokes imaging polarimeter using an array of elliptical polarizer,” Opt. Express 22(3), 3063–3074 (2014). [CrossRef] [PubMed]

18. F. Snik, T. Karalidi, and C. U. Keller, “Spectral modulation for full linear polarimetry,” Appl. Opt. 48(7), 1337–1346 (2009). [CrossRef] [PubMed]

19. J. P. Balthasar Mueller, K. Leosson, and F. Capasso, “Ultracompact metasurface in-line polarimeter,” Optica 3(1), 42–47 (2016). [CrossRef]

20. D. A. LeMaster and K. Hirakawa, “Improved microgrid arrangement for integrated imaging polarimeters,” Opt. Lett. 39(7), 1811–1814 (2014). [CrossRef] [PubMed]

21. S. Wei, Z. Yang, and M. Zhao, “Design of ultracompact polarimeters based on dielectric metasurfaces,” Opt. Lett. 42(8), 1580–1583 (2017). [CrossRef] [PubMed]

22. O. Arteaga, B. M. Maoz, S. Nichols, G. Markovich, and B. Kahr, “Complete polarimetry on the asymmetric transmission through subwavelength hole arrays,” Opt. Express 22(11), 13719–13732 (2014). [CrossRef] [PubMed]

23. X. Meng, J. Li, H. Song, and R. Zhu, “Full-Stokes Fourier-transform imaging spectropolarimeter using a time-division polarization modulator,” Appl. Opt. 53(24), 5275–5282 (2014). [CrossRef] [PubMed]

24. S. Alali, T. Yang, and I. A. Vitkin, “Rapid time-gated polarimetric Stokes imaging using photoelastic modulators,” Opt. Lett. 38(16), 2997–3000 (2013). [CrossRef] [PubMed]

25. A. De Martino, Y.-K. Kim, E. Garcia-Caurel, B. Laude, and B. Drévillon, “Optimized Mueller polarimeter with liquid crystals,” Opt. Lett. 28(8), 616–618 (2003). [CrossRef] [PubMed]

26. G. T. Herman, Fundamentals of Computerized Tomography: Image Reconstruction from Projections (Springer, 2009), pp. 64–68.

27. L. Marrucci, C. Manzo, and D. Paparo, “Optical Spin-to-Orbital Angular Momentum Conversion in Inhomogeneous Anisotropic Media,” Phys. Rev. Lett. 96(16), 163905 (2006). [CrossRef] [PubMed]

28. L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]

29. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992). [CrossRef] [PubMed]

30. E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, “Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram,” Opt. Lett. 38(18), 3546–3549 (2013). [CrossRef] [PubMed]

31. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161–204 (2011). [CrossRef]

32. H. Zhou, D. Fu, J. Dong, P. Zhang, and X. Zhang, “Theoretical analysis and experimental verification on optical rotational Doppler effect,” Opt. Express 24(9), 10050–10056 (2016). [CrossRef] [PubMed]

33. H. Zhou, D. Fu, J. Dong, P. Zhang, X. Chen, X. Cai, F. Li, and X. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6(4), e16251 (2017). [CrossRef]

34. C. Rosales-Guzmán, N. Hermosa, A. Belmonte, and J. P. Torres, “Direction-sensitive transverse velocity measurement by phase-modulated structured light beams,” Opt. Lett. 39(18), 5415–5418 (2014). [CrossRef] [PubMed]

35. M. P. J. Lavery, S. M. Barnett, F. C. Speirits, and M. J. Padgett, “Observation of the rotational Doppler shift of a white-light, orbital-angular-momentum-carrying beam backscattered from a rotating body,” Optica 1(1), 1–4 (2014). [CrossRef]

36. M. P. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padgett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341(6145), 537–540 (2013). [CrossRef] [PubMed]

37. A. Belmonte and J. P. Torres, “Optical Doppler shift with structured light,” Opt. Lett. 36(22), 4437–4439 (2011). [CrossRef] [PubMed]

38. C. Rosales-Guzmán, N. Hermosa, A. Belmonte, and J. P. Torres, “Experimental detection of transverse particle movement with structured light,” Sci. Rep. 3(1), 2815 (2013). [CrossRef] [PubMed]

39. G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett. 32(11), 1468–1470 (2007). [CrossRef] [PubMed]

40. S. Fu, C. Gao, Y. Shi, K. Dai, L. Zhong, and S. Zhang, “Generating polarization vortices by using helical beams and a Twyman Green interferometer,” Opt. Lett. 40(8), 1775–1778 (2015). [CrossRef] [PubMed]

	Time-varying spatial modulation	Spinning object
Principle	Polarization-mode-frequency mapping	Polarization-mode-frequency mapping
Reference light	Not required	Required
Programmable device	Required	Not required
Setup	Simple	Complex
Cost	High	Low
Stability	Robust	Susceptible to interference

Tomographic polarization analyzer by polarization-mode-frequency mapping

Abstract

1. Introduction

2. Principle

3. Experimental results

4. Polarization analyzer with a spinning object

5. Conclusions

Funding

References and links

Cited By

Figures (8)

Tables (1)

Equations (6)

Optics Express