## Abstract

This paper describes a Stokes vector measurement method based on a snapshot polarization-sensitive spectral interferometry. We measure perpendicular linearly polarized complex wave information of an anisotropic object in the spectral domain from which an accurate Stokes vector can be extracted. The proposed Stokes vector measurement method is robust to the object plane 3-D pose variation and external noise, and it provides a reliable snapshot solution in numerous spectral polarization-related applications.

© 2014 Optical Society of America

## 1. Introduction

Polarization-sensitive spectral measurement technology is among the most precise and promising solutions in various applications. Specular spectroscopic ellipsometry, which has been used mainly for scatterometry in the semiconductor industry, can measure topographic data relevant to subwavelength periodic structures and thin films [1,2]. Scatterometry is based on polarization-sensitive spectroscopy and is regarded as an extremely sensitive solution that can be used in nanometrology complementary to scanning electron microscopy (SEM) and atomic force microscopy (AFM). There have been numerous attempts to combine polarization measurement techniques with interferometry during the last four decades. Optical coherence tomography (OCT) is an interferometric technique that produces depth-resolved images of backscattering from biological tissues. Although OCT is concerned primarily with measuring the envelope of the interference fringes rather than the phase of the backscattered light, polarization information is needed to obtain additional, important information on the polarization properties of the tissue. Spectral-domain polarization-sensitive optical coherence tomography (SD PS-OCT) has been studied intensively for biomedical tissue imaging [3–8]. Such polarization-sensitive measurement techniques have also been applied in other fields such as real-time, high-sensitivity surface plasmon resonance (SPR) biosensing [9,10] and circular dichroism (CD) measurements. Some of these techniques are based on spectral interferometry [10,11], and their fundamental schemes originate from interferometric ellipsometry [12–15].

Typical spectroscopic polarization measurement schemes employ mechanical rotating mechanisms or electrical modulation devices to extract a Stokes vector [16]. Recently, new attempts have been proposed to replace such scanning approaches with spectral interferometry based on a channeled spectrum generated by multiple thick wave plates [17], birefringence crystals [18], and dual-spectrum sensing modules combined with a Michelson interferometer [19,20]. The Δ(*k*) extraction method by using a dual-spectrum sensing-based spectral interferometric scheme provides a reliable phase-resolved polarization-sensitive measurement capability [20]. Among those schemes, only Oka’s [17] provides a snapshot Stokes vector measurement capability; however, this scheme’s inherent limited spectral resolution due to the channeled spectrum complexity is a disadvantage.

In this paper, we describe a novel Stokes vector measurement method based on snapshot polarization-sensitive spectral interferometry. Our proposed scheme provides Stokes vector measurement capability for reflective anisotropic objects while maintaining the inherent high spectral resolution of the spectrometers. Notably, this study describes an entirely calibrated Stokes vector measurement method that is robust to object 3-D-pose variation. In addition, our proposed approach has inherent robustness to external vibration, which enables us to obtain the Stokes vector precisely and accurately without using a vibration-free optical table. This paper demonstrates experimentally how the reliable Stokes vector is extracted accurately based on the spectral interferometry. We expect that the proposed Stokes vector measurement method can be extended to numerous fields related to interferometric spectro-polarimetry and ellipsometry that require high-speed, compact snapshot capability.

## 2. Snapshot Stokes vector measurement: pre-preparation steps

The polarization-sensitive spectral interferometric scheme for extracting a Stokes vector, depicted in Fig. 1, has no moving parts and no complicated specially manufactured optical components. The system consists of three parts, namely collimating optics with a broadband light source, a Michelson interferometer, and a perpendicular linearly polarized dual-spectrum sensing module.

It requires only dual-spectral data that can be captured simultaneously by using the sensing module. A 100W Tungsten-Halogen lamp is used as the broadband light source, which is connected to the optical fiber inlet through a multimode fiber. After passing through the collimating optics, the illuminating beam enters the Michelson interferometer and the interfered wave travels to the dual-spectrum sensing module. The dual-spectrum sensing module is comprised of a polka-dot beam splitter, two perpendicular linearly polarized Glan-Thompson polarizers, two parabolic mirrors with a focal length of 50 mm, two multimode optical fibers with a diameter of 1000 μm, and two 1,251-pixel array sensor spectrometers with a spectral measurement range from 524 to 785 nm. Equation (1) describes the interfered spectrum for the TM polarized wave *I _{TM}*(

*k*) measured by the snapshot polarization-sensitive spectral interferometer.

*k*is a wavenumber defined by 2π/

*λ*.

*E*(

_{R_TM}*k*) and

*E*(

_{O_TM}*k*) represent the complex reference wave traveling to the reference mirror and the complex object wave that goes to the object for the TM polarized state, respectively. The coherence function γ

*(*

_{TM}*k*,

*h*,

*α*,

*β*) and the total phase Φ

*(*

_{TM}*k*,

*h*,

*α*,

*β*) of the TM polarized wave are all functions of the wavenumber

*k*, the spectral carrier frequency

*h*, and the tilted object plane angles

*α*and

*β*as illustrated in Fig. 1.

*(*

_{TM}*k*) represented in Eq. (2) contains four terms. The first term 2

*kh*corresponds to the spectral carrier-related, high-frequency term generated by

*h*. The second term

*ϕ*(

_{TM_Obj anisotropy}*k*) represents the phase shift due to the object’s anisotropic characteristics. The third term

*ϕ*(

_{TM_R}*k*) signifies the reference mirror phase shift. Finally, the fourth term

*ϕ*(

_{TM_Remaining optics}*k*,

*α*,

*β*) denotes the phase shift of the TM polarization wave due to the remaining optical parts including the beam splitters, polarizer, optical fiber, and spectrometer. To attain the TE spectral data, we replace the subscript TM with TE in Eq. (1) and Eq. (2).

Figure 2 shows the raw spectral interference signals of the TM polarization wave acquired by the TM channel in the dual-spectrum sensing module by varying the object plane 3-D pose that is represented by *h*, *α*, and *β*; for this, we used a plane mirror. Such high-frequency spectral interference signals can be acquired by generating the spectral carrier frequency *h* defined as the distance between the virtual reference plane (i.e., the *x*-*y* plane) and the object plane as described in Fig. 1. Here, we observe that the spectral intensity distribution varies corresponding to *h* since the spectral (temporal) coherence characteristics of the proposed system varies. As illustrated in Fig. 2(a), the amount of overall coherence tends to decrease as *h* increases when the object plane aligns parallel to the reference plane. Likewise, the spectral interference signal distribution varies according to the tilting angle of the object plane because the spatially generated interferogram enters the spectrum sensing channel and the tilted object’s spatially summed interference signal produces the reduced spectral interference contrast as depicted in Fig. 2(b). The spectral interference signals for the TM and TE polarization waves strongly depend on the object plane 3-D pose. Preventing 3-D pose dependency is one of the key issues with obtaining an accurate, reliable Stokes vector measurement. To obtain the Stokes vector of the reflected wave from an anisotropic object, theoretically, we need to extract the two perpendicular linearly polarized complex object waves *E _{O_TM}*(

*k*) and

*E*(

_{O_TM}*k*) described in Eq. (3) [21].

*E*(

_{O_TM}*k*) and

*E*(

_{O_TE}*k*) denote |

*E*(

_{O_TM}*k*)|e

*and |*

^{iϕTM_Obj anisotropy(k)}*E*(

_{O_TE}*k*)|e

*, respectively. To measure the Stokes vector based on the proposed snapshot scheme, the four terms |*

^{iϕTE_Obj anisotropy(k)}*E*(

_{O_TM}*k*)|, |

*E*(

_{O_TE}*k*)|,

*ϕTM_Obj anisotropy(k)*, and

*ϕTE_Obj anisotropy(k)*need to be extracted simultaneously from the two interfered spectra captured by the dual-spectrum sensing module.

As a pre-preparation step, conducted without a vibration-free optical table, to extract the Stokes vector based on the proposed snapshot scheme, we need to measure two systematic spectral functions *C*(*k*) defined as the ratio between |*E _{O_TM}* (

*k*)| and |

*E*(

_{O_TE}*k*)|, and

*Γ*(

*k*,

*h*,

*α*,

*β*) denoting the coherence function ratio between γ

*(*

_{TM}*k*,

*h*,

*α*,

*β*) and γ

*(*

_{TE}*k*,

*h*,

*α*,

*β*). To obtain the two systematic functions

*C*(

*k*) and

*Γ*(

*k*,

*h,α*,

*β*), we capture six spectra

*I*(

_{TM}*k*),

*I*(

_{TE}*k*), |

*E*(

_{R_TM}*k*)|, |

*E*(

_{R_TE}*k*)|, |

*E*(

_{O_TM}*k*)|, and |

*E*(

_{O_TE}*k*)| for a non-patterned object such as a reflective plane mirror. First, the object amplitude ratio

*C*(

*k*) can be calculated directly by using the two measured spectra |

*E*(

_{O_TM}*k*)| and |

*E*(

_{O_TE}*k*)| for the non-patterned object. Ideally,

*C*(

*k*) should be a unit function when the dual-spectrum sensing channels provide exactly identical spectral responses. However,

*C*(

*k*) becomes a function of the wavenumber since the spectral response of all the optical components in each spectrometer such as a grating and a CCD sensor are not identical. Second, we extract the coherence functions γ

*(*

_{TM}*k*,

*h*,

*α*,

*β*) and γ

*(*

_{TE}*k*,

*h*,

*α*,

*β*). Since we can extract the amplitude and phase of I

*(*

_{TM_mod}*k*) and I

*(*

_{TE_mod}*k*) described in Eq. (4) by using the Fourier transform method [22–24], subsequently we can extract both coherence functions γ

*(*

_{TM}*k*,

*h*,

*α*,

*β*) and γ

*(*

_{TE}*k*,

*h*,

*α*,

*β*) separately from the two modified spectra.

*k*,

*h*,

*α*,

*β*) between the two coherence functions γ

*(*

_{TM}*k*,

*h*,

*α*,

*β*) and γ

*(*

_{TE}*k*,

*h*,

*α*,

*β*), which we measure by varying the vertical position of the object

*h*from 25 to 55 μm at 15-μm intervals with an object plane tilt of ~0.01°. For this measurement, we use an aluminum-coated non-patterned plane mirror as the object. As shown in Fig. 3(a), Γ(

*k*,

*h*,

*α*,

*β*) is a function of

*h*since the temporal coherence length of the spectral interferometric system is dependent on the spectrometers’ spectral resolution; the dual spectrometers used in the system inherently have different spectral resolutions. The measured object’s tilt also causes interference contrast variation as described in Fig. 2(b) by the tilt dependency of the coherence functions γ

*(*

_{TM}*k*,

*h*,

*α*,

*β*) and γ

*(*

_{TE}*k*,

*h*,

*α*,

*β*). However, we note that Γ(

*k*,

*h*,

*α*,

*β*) is a systematic function of

*h*that is independent of the object tilting

*α*and

*β*variation. As illustrated in Fig. 3(b), the contrast change of the spectral interference in TM polarization matches the contrast change in TE polarization. Ideally, the spatial interferogram variation caused by the object plane tilt affects both TM and TE spectrum sensing channels equally since the spatially summed spectral energy to both channels is equal. Therefore, we conclude that Γ(

*k*,

*h*,

*α*,

*β*) depends on the

*h*variation solely, and it can be stored in advance by using the non-patterned plane mirror so that it can be used to calculate the Stokes vector of other anisotropic objects. We assume that

*E*(

_{R_TM}*k*) and

*E*(

_{R_TE}*k*) are known spectra since the two reference waves can be measured in advance and used as fixed signals.

As the next pre-preparation step, we extract the total phase functions Φ* _{TM}*(

*k*,

*h*,

*α*,

*β*) and Φ

*(*

_{TE}*k*,

*h*,

*α*,

*β*) from the non-patterned plane object’s two modified spectra

*I*(

_{TM_mod}*k*) and

*I*(

_{TE_mod}*k*) described in Eq. (4). We newly define Φ

*(*

_{TM_non_pattern}*k*,

*h*,

*α*,

*β*) and Φ

*(*

_{TE_non_pattern}*k*,

*h*,

*α*,

*β*) denoting 2

*kh*-

*ϕ*(

_{TM_R}*k*) +

*ϕ*(

_{TM_Remaining optics}*k*) and 2

*kh*-

*ϕ*(

_{TE_R}*k*) +

*ϕ*(

_{TE_Remaining optics}*k*), respectively, which we extracted from the non-patterned object to measure the Stokes vector of an anisotropic object. Here, the total phase functions Φ

*(*

_{TM_non_pattern}*k*,

*h*,

*α*,

*β*) and Φ

*(*

_{TE_non_pattern}*k*,

*h*,

*α*,

*β*) become independent of the object plane tilt by employing a polka-dot beam splitter in the dual-spectrum sensing module [20].

## 3. Stokes vector measurement of reflective anisotropic objects

Once we obtain the total phase functions Φ* _{TM_non_pattern}*(

*k*,

*h*,

*α*,

*β*) and Φ

*(*

_{TE_non_pattern}*k*,

*h*,

*α*,

*β*) as well as the two systematic calibration functions

*C*(

*k*) and

*Γ*(

*k*,

*h*) through the pre-preparation steps by using a non-patterned object, we can measure the Stokes vector of reflective anisotropic objects, such as a reflective grating, by using our proposed snapshot approach. To extract the Stokes vector of the anisotropic object, we again acquire the two interfered spectra

*I*(

_{TM}*k*) and

*I*(

_{TE}*k*), which are captured simultaneously for the anisotropic object. We need to subtract the dual spectrometers’ dark noise signals from the two interfered spectra for accurate measurements. By using the Fourier transform method, we can extract the total amplitude functions

*A*(

_{TM}*k*) and

*A*(

_{TE}*k*), which equal 2γ

*(*

_{TM}*k*)|

*E*(

_{R_TM}*k*)||

*E*(

_{O_TM}*k*)| and 2γ

*(*

_{TE}*k*)|

*E*(

_{R_TE}*k*)||

*E*(

_{O_TE}*k*)|, respectively [22–24]. Since we already know the two systematic calibration functions

*C*(

*k*) and

*Γ*(

*k*,

*h*) as well as the amplitudes of the perpendicular linearly polarized reference waves |

*E*(

_{R_TM}*k*)| and |

*E*(

_{R_TE}*k*)|, we can obtain the measured |

*E*(

^{m}_{O_TM}*k*)| and |

*E*(

^{m}_{O_TE}*k*)| for the anisotropic object defined as follows.

*E*(

^{C}_{O_TE}*k*)| is obtained by |

*E*(

_{O_TE}*k*)| ×

*C*(

*k*). The total phase functions Φ

*(*

_{TM}*k,h*) and Φ

*(*

_{TE}*k,h*) for the patterned anisotropic object can also be extracted from the same Fourier transform procedure used to obtain

*A*(

_{TM}*k*) and

*A*(

_{TE}*k*). Then, we calculate the phase difference between Φ

*(*

_{TM_pattern}*k*,

*h*) and Φ

*(*

_{TM_non_pattern}*k*,

*h*) in the TM polarization wave, which is defined as ΔΦ

*(*

_{TM}*k,h*). Likewise, ΔΦ

*(*

_{TE}*k,h*) can be obtained as follows.

*h*and

_{1}*h*denote the applied spectral carrier frequency when the non-patterned object is measured and when the spectra acquisition is made for the patterned object, respectively. The object 3-D-pose difference caused by the spectra captured time difference between the non-patterned object and the patterned object measurements inevitably creates the unwanted phase shift represented as

_{2}*ak*+

*b*. Here,

*a*and

*b*are arbitrary real values denoting 2(

*h*-

_{2}*h*) and {

_{1}*ϕ*(

_{TM_Remaining optics}*k*)}

*- {*

_{2}*ϕ*(

_{TM_Remaining optics}*k*)}

*, respectively. By using our proposed snapshot scheme, we can measure*

_{1}*E*(

^{m}_{O_TM}*k*) and

*E*(

^{m}_{O_TE}*k*) described in Eq. (7) separately instead of using the object complex wave defined in Eq. (3).

We can accurately extract the normalized Stokes vector described in Eq. (8) by using the measured complex object wave represented in Eq. (7) since *E ^{m}_{O_TM}*(

*k*)

*E*(

^{m}_{O_TM}*k*)* equals {

*E*(

_{O_TM}*k*)

*E*(

_{O_TM}*k*)*} × {γ

*(*

_{TM}*k,h*)}

^{2}and

*E*(

^{m}_{O_TM}*k*)

*E*(

^{m}_{O_TE}*k*)* equals {

*E*(

_{O_TM}*k*)

*E*(

^{C}_{O_TE}*k*)*} × {γ

*(*

_{TM}*k,h*)}

^{2}. Likewise,

*E*(

^{m}_{O_TE}*k*)

*E*(

^{m}_{O_TM}*k*)* equals {

*E*(

^{C}_{O_TE}*k*)

*E*(

_{O_TM}*k*)*} × {γ

*(*

_{TM}*k,h*)}

^{2}. Figure 4 illustrates how the discrepancy in the phase between Eq. (3) and Eq. (7) can be compensated in the complex plane analysis. We can remove the discrepancy in the amplitude since the measured Stokes vector is normalized by

*S*(

^{m}_{0}*k*) denoting

*I*(

_{0}*k*) × {γ

*(*

_{TM}*k,h*)}

^{2}. Here, the total energy

*I*(

_{0}*k*) signifies

*E*(

^{m}_{O_TM}*k*)

*E*(

^{m}_{O_TM}*k*)* +

*E*(

^{m}_{O_TE}*k*)

*E*(

^{m}_{O_TE}*k*)*.

*E*(

^{m}_{O_TM}*k*) and

*E*(

^{m}_{O_TE}*k*) from the reflective holographic grating object; Fig. 5(a) illustrates |

*E*(

_{O_TM}*k*)| and |

*E*(

_{O_TE}*k*)|, and Fig. 5(b) represents Δ

*Φ*(

_{TM_pattern}*k*,

*h*) and Δ

*Φ*(

_{TE_pattern}*k*,

*h*) for the vertically aligned grating. Likewise, Figs. 5(c) and 5(d) correspond to the measured object amplitude and phase of the horizontally aligned grating object.

As illustrated in Figs. 5(b) and 5(d), the measured phase functions are asymmetrical to the 0° line since the unknown term *ak* + *b* is added in the phase measurement results. However, we note that the phase shift term *ak* + *b* in *E ^{m}_{O_TM}*(

*k*) is same as that in

*E*(

^{m}_{O_TE}*k*) as analyzed in Fig. 4(b). This distinct, separate complex object wave measurement capability enables us to extract the normalized Stokes vector, which means it can measure the range of the Stokes vector.

In order to verify that the proposed scheme can provide a reliable Stokes vector measurement capability, we compare the normalized Stokes vector extracted by using Eq. (8) for a reflective grating with an azimuthal angle of zero to the vector taken with a 90° rotation. Ideally, *S _{1}^{nor}*(

*k*), which denotes the energy difference between the TM and TE polarization states, and

*S*(

_{3}^{nor}*k*), which represents the energy difference between a right-circularly polarized and a left-circularly polarized reflective wave from the grating surface, should be symmetric along the zero line. In contrast,

*S*(

_{2}^{nor}*k*), which denotes the difference between the two perpendicularly polarized waves rotated by 45° with respect to the grating grooves direction (i.e. x-axis) described in Fig. 5(a), should be identical. Figure 6 depicts the measurement results between the vertically and horizontally aligned grating object. The

*S*(

_{1}^{nor}*k*) and

*S*(

_{3}^{nor}*k*) for the 0° azimuthal angle case is almost exactly symmetric along the zero line to that obtained for the 90° azimuthal angle, and the measured

*S*(

_{2}^{nor}*k*) for the 0° azimuthal angle case is nearly equal to that obtained for the 90° azimuthal angle as illustrated in Fig. 6(a). We also conducted consecutive experiments by varying the object tilt angle α to verify the object-tilt tolerant measurement capability. We observe that the Stokes vector can be obtained consistently regardless of the object tilt variation as depicted in Fig. 6(b).

## 4. Conclusions

In this paper, we demonstrate that our proposed system’s ability to directly extract the perpendicular linearly polarized complex wave information produces the snapshot Stokes vector measurement. The dual-spectra simultaneous acquisition scheme that makes the phase shift *ak* + *b* in *E ^{m}_{O_TM}*(

*k*) the same as that in

*E*(

^{m}_{O_TE}*k*) enables us to extract the Stokes vector accurately in measuring anisotropic objects without depolarization effect. We claim that our proposed dual interfered spectra measurement scheme can provide the object plane 3-D pose tolerant snapshot Stokes vector measurement capability within the permissible range of

*h*(~20 to 60 μm) which is mainly decided by spectral resolution of the dual spectrometers, with an object plane tilt of ~0.01°. Remarkably, our proposed system does not require a vibration-free environment through the entire measurement process including the pre-calibration steps. Among the random noises such as external vibration, light source fluctuation, and the CCD sensor electrical noise, the main noise comes from the light source fluctuation. Spectral signals used in this study have been averaged 100 times to reduce random noises. However, accurate Stokes vector can be obtained in a few msec without any serious degradation.

## Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MEST) (2012R1A1B3003346). Additionally, this work was supported under the framework of international cooperation program managed by National Research Foundation of Korea (2013053224) and by the Texas Instruments Distinguished University Chair in Nanoelectronics endowment.

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