1. Introduction

The techniques of imaging polarimeter are capable of recording light polarization of a scene, which contains unique information related to surface features, shapes, shadows and roughness. Therefore it improves the ability of target recognition in a complex background, and also provides valuable information in environmental monitoring, remote sensing, biomedical imaging and military target detection [1–6]. After extensive research, several techniques of imaging polarimeter including division-of-time, division-of-amplitude, division-of-aperture and division-of-focal- plane (DoFP) have emerged in the past decades [7–11]. Among them, DoFP polarimeters adopt four independent micro-polarizers on the top of neighboring pixels of the image sensor to assess the four components of a Stokes vector. This approach has significant advantages in terms of manufacturing cost, volume, weight, power dissipation and system integration. As early as in 1994, reactive-ion-etching (RIE) was used to pattern dichroic polymer film and form a micropolarimeter array capable of extracting (S0, S1, S2) for partial-linear polarization imaging [12–14]. Subsequent approaches include using evaporated aluminum or gold film to form the birefringent micropolarizer array for the polarization image [15,16]. But the birefringence of metallic film originating from absorbing the light polarization is parallel to the grating stripe, which cannot distinguish the phase information of light. The first full-Stokes DoFP polarimeter emerged in 2006 [17] and adopted sub-wavelength gratings in fused silica as the micro polarization elements. However, it was difficult to control the morphology and depth of the grating in fused silica by the technology of electron beam direct writing. Substantial errors existed in the experimental results. In addition, by patterning a liquid crystal layer on top of a visible-regime metal-wire-grid polarizer, X. Zhao et al. demonstrated the full-Stokes measurement in principle in 2010 [18]. However, until 2012, first practical full-Stokes DoFP imaging was presented in experiments by using patterned liquid crystal polymer (LCP) polarizers and retarders [19]. But this approach was limited in a handful of research fields and applications owing to the complicated fabrication of the LCP polarizers and retarders [20,21].

On the other hand, with the rapid development of ultra-fast laser technology, the femtosecond laser processing of inorganic non-metallic materials has grown up to be a hot research field [22,23]. Depending on the amount of deposited energy, three distinct types of modifications can be induced in the bulk of transparent materials. In particular, moderate fluence results in the spontaneous formation of gratings with sub-wavelength refractive index distribution [24–26], dubbed nanogratings, which have the characteristics of high transmittance, rewritability and thermal stability. More importantly, the gratings exhibit an obvious optical birefringence effect, which can be exactly obtained by empirically controlling the optical fluence of the femtosecond laser. On the processing area, the optical slow and fast axes align parallel and perpendicular to the nanograting corrugation, respectively. Therefore, both the optical axis and phase delay induced by the self-organized nanograting are designable [27,28]. Although the mechanism of nanograting formatting is not fully understood, many applications based on the self-organized nanogratings have been developed in recent years [29–31].

In this paper, we proposed a novel DoFP polarization imaging technique based on the self-organized nanograting array in fused silica. With the periodical arrangement of four independent self-organized nanogratings on the focal plane of an imaging camera, polarization information can be retrieved from the light intensity pattern after a subsequent polarizer. To investigate the optical properties of the DoFP polarimeter, we firstly modeled the phase delay of the nanograting based on the Rigorous Coupled-Wave Analysis (RCWA). Secondly, the incident angle, spectral and thermal dependencies of the nanograting were discussed. At last, the calculation of the determinant of the Müller matrix puts forward a promising approach for a broadband polarimeter in the visible range.

2. Theory of the DoFP polarimeter

In a DoFP polarimeter, the combination of the wave-plates and polarizers are integrated on the focal plane of the imaging objective. Because of the sub-wavelength structure, only zero-order diffraction exists in the transmitted light of the self-organized nanograting. The optical properties of the nanograting are similar to a wave-plate with specific phase delay and optical axis direction. At least four independent measurements are required for full-Stokes’ vector detection in theory. The scheme based on the self-organized nanograting array is illustrated in Fig. 1. Four nanogratings with the same phase delay and different optical axis orientations form a combination, which is arranged periodically on the two-dimensional focal plane of the imaging objective. Moreover, the subsequent linear polarizer transfers the transmitted light to the intensity pattern recorded by an adjacent camera. The size of the nanogratings is as same as the single pixel of the camera to provide maximum imaging resolution.

 

Fig. 1 The schematic diagram of a DoFP polarimeter based on the self-organized nanograting array.

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In theory, the detected polarization S(in) is transferred to S(out) by a combination of the nanogratings and polarizer. The measurements can be expressed as Eq. (1).

Here M is the Müller matrix for the detection light path. Superscript i = 1,2,3,4, represents different independent measurement. Δ,θ denote the phase delay and the optical axis of the nanograting, and φ is the axis angle of the polarizer. Müller matrix has 4 × 4 components as follows:

Only the light intensity (i.e., S₀) can be detected by the image sensor, so we rewrite the first line of Eq. (2) of four measurements as Eq. (3):

In the experiments, light intensities of four measurements were recorded and used to calculate the input polarization with Eq. (3). M_det contains the contributions of the nanograting and polarizer:

Retarder-similar optical properties of the nanogratings were reported at various visible wavelengths [28–32]. A single nanograting can be modeled by a retarder with phase delay Δ and optic axis angle of θ as Eq. (5). Usually, microscopic inhomogeneities and induced defect absorption cause scattering loss in nanograting, which can be described with a polarization-independent transmissivity T in Eq. (5). However, scattering loss can be easily calibrated in experiments, and has limited influence on optical properties of polarization imaging, T = 1 was adopted in our simulations.

The Müller matrix for the polarizer can be given by:

The necessary condition for the existence of solution for Eq. (3) is that the determinant of M_det is nonzero. This condition can also be used to estimate the independence of four measurements. If |M_det| $\ne$0, the solution of Eq. (3) is:

Here $M_{\det }^{*}$ the adjoint matrix of M_det.

3. Model of the self-organized nanograting

Typically, three regimes of structural changes can be observed with the tightly focusing ultrafast lasers into fused silica. A homogeneous refractive index increase, birefringent index change and permanent voids occur with the increase of the laser energy. The self-organized nanograting accompanied with birefringent index emerges at intermediate energies, which usually has two sub-wavelength refractive index periods in the directions along light propagation and polarization, respectively. Because the depth of nanograting is much larger than the refractive index period in the direction along light propagation, we used an effective depth instead in the simulations. Therefore, the structure of nanograting can be simplified as shown in Fig. 2. The refractive index of the gray area is modified from n₁, the value of bulk fused silica, to n₂ by the ultrafast laser. In experiments, by cutting or grinding the dielectric to the modified zone as well as polishing and etching the surface, the nanostructure can be observed with an atomic force microscope or a scanning electron microscope [24,27]. The results showed that the modified planes were ≤ 10 nm in thickness. The period can be roughly described as T = λ/2n, and it decreases with the increase of ultrafast laser fluence [24,27,28]. For a Ti:sapphire laser with 800 nm wavelength, the nanograting period is about 267 nm, a sub-wavelength grating for the visible and infrared light.

 

Fig. 2 The nanostructure of the self-organized nanograting in the simulations.

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The phase delay induced by the self-organized nanograting is related to its micro refractive index structure. In this article, RCWA was introduced to simulate the sub-wavelength grating instead of the traditional optical diffraction theory. This method was founded by M. Moharma et al. in 1982 [33,34], and its stability, convergence and computing efficiency were improved greatly in 1990s [35–37]. It was extensively used to improve the polarization characteristics of sub-wavelength gratings [38–40]. In general, the RCWA can be implemented in several steps [41]. Firstly, the permittivity and the electromagnetic field are expressed by the Fourier series of the grating period:

Here the grating period is supposed to be along the x-direction, and $m$is an integer. Then the Fourier series are used in Maxwell’s equations to get equations of the discrete coefficients of the reflected and transmitted field. Furthermore, the boundary condition is applied to solve the equations. Finally, the reflectivity and transmittance of the field are calculated by the Poynting vector $\overrightarrow{S}=\overrightarrow{E}\times \overrightarrow{H}$. The diffraction efficiencies (DE) of the TE, TM wave are:

Here i is the order of the grating diffraction. T_TE,i and T_TM,i denote the transmission coefficient of TE and TM waves. k₀ is the wave vector of the incident field, and k_II,zi is the component of the transmitted wave vector in the normal direction. n_I and n_II are the refraction indexes of the materials in the input and output region. θ is the incident angle. Accuracy of RCWA calculation depends on the truncation error of Fourier series. According to the analysis in [35], the truncation error is less than 1% in a RCWA using 25 Fourier orders. In this study, 40 Fourier orders are adopted, the truncation error can be estimated to be less than 0.5%.

For a subwavelength grating, the threshold period under which only the transmitted and the reflected zero orders are non-evanescent [42] is ${\Delta }_{th}={\lambda }_{0}k/\left(n\pm \sin \theta \right)$, where ${\lambda }_0$is the wavelength of the incident light. $k$is the order of diffraction and only the zero order diffraction exists when $k=1$. For example, the threshold period is 364 nm for the fused silica at 532 nm incident light. In order to calculate the light field distributions in the nanograting, assuming a grating with the period of 300 nm and a duty cycle of 10/300. The grating thickness of 3000 nm is chosen for the clear display of the simulated light field. The refractive index modification in Fig. 3(a) is set as n₂-n₁ = 0.19 corresponding to an 800nm wavelength processing laser with 0.65 μJ and 150-fs pulses according to the literature [28]. The absorption in the modified areas usually induces a slight decrease of the grating transmittance, which means that a complex refractive index exists in these areas. Because we only concern the phase delay of the self-organized grating, an imaginary part of the refractive index $\kappa =0$is adopted in our simulations. The amplitudes of Ey field of TE wave (with polarization perpendicular to the grating walls) and the Hy field of TM was (with polarization paralleled to the grating walls) were plotted in Fig. 3(b) and Fig. 3(c), respectively. Both fields are partly confined in the modified areas with higher refractive index. And the phase delay between two waves due to the propagation speed difference can be seen in Fig. 3.

 

Fig. 3 The refractive index map of the nanograting(a), and the simulated field distribution of TM wave (b) and TE wave (c) in the nanograting.

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The transmitted amplitudes and phases of TE and TM waves can be calculated by Eqs. (9) and (10). Diffraction waves of $\pm$20 orders were presented in Fig. 4, where more than 97% transmitted amplitude converges on the zero-order diffraction. The horizontal ordinate is the order of diffraction. The phase delay at the high order diffraction is meaningless due to the near-zero transmittance, but the value between zero-order TE and TM waves is about 2.44 degrees for a grating of 3000 nm depth in our simulation. It originates the propagation speed difference between two waves as mentioned above and is linearly proportional to the grating depth. So a specific phase delay can be obtained by adjusting the grating depth in our simulation.

 

Fig. 4 Calculation of the amplitude and phase of the transmitted TE and TM wave of a subwavelength dielectric grating.

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Well-known results on the optimization of the retardance for a complete Stokes polarimeter were presented by D. S. Sabatke in 2000 [43]. A retardance of 132 degrees and retarder orientation angles of $\pm$51.7 degrees and $\pm$15.1 degrees were found to be optimal when four measurements were used. This combination corresponds to maximum |M_det| in Eq. (7) and a minimum signal-to-noise ratio. Therefore our simulations were implemented around the phase delay of 132 degrees induced by the nanograting. The phase delay induced by the nanograting can be written as:

Here n_Te and n_Tm are the effective refractive index for TE and TM waves, respectively. λ is the wavelength of incident light. L is the depth of the nanograting. Based on the linear relation of the phase delay and grating depth, a depth of 162 μm is chosen to access a phase delay of 132 degrees in the 300 nm period and 10/300 duty cycle grating, at 532 nm incident light in further simulations.

Experiments showed that the period was roughly equal to λ/2n. And it was inversely proportional to the number of processing laser pulses per second, i.e. the writing velocity. However, the thickness of the grating wall was kept unchanged during the ultrafast processing [24]. So the thickness of the grating wall was fixed at 10 nm and the refractive index modification was 0.19 in our simulations. The period and wavelength dependence of the grating phase delay were analyzed and presented in Fig. 5. The phase delay decreases rapidly with the wavelength of incident light. Considering the material dispersion in fused silica [44]:

 

Fig. 5 The wavelength and period dependence of the self-organized grating.

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The wavelength λ is in the unit of μm. The simulated results agree with the definition of the phase delay $2\pi \cdot n/\lambda$. The phase delay of a normal wave-plate made of fused silica was calculated as a comparison. It can be seen from Fig. 5(a) that only slight difference exists at the longer wavelengths. As shown in Fig. 5(b), even the period varying from 100 to 400 nm, less than 9% phase delay changes at 400 nm and less than 2% at 800 nm. Because the thickness of the grating wall is much less than the period, the phase delay is not sensitive to the period. The modification increases in the shorter wavelength due to the larger overall phase delay.

As a polarization component mounted in the focal plane of the imaging objective, the acceptance angle is an interesting parameter for most applications. We calculated the phase delay modification while the incident angles change from zero to 40 degrees. The results were plotted in Fig. 6(a). The phase delay varies −7.7%, −6.7%, −4.4%, 1.9% and 90% with the incident angle at the wavelengths of 800, 700, 600, 500 and 400 nm, respectively. The most stable phase delay appears near 520 nm, which has the best response for a detector in the visible range. In the blue end, the phase delay increased fastly with the incident angle, so an acceptance angle smaller than 20 degrees is expected in this bandwidth. The temperature dependence of the phase delay also can be estimated in our model. The refractive index of fused silica varies with the temperature as [45]:

 

Fig. 6 The incident angle and temperature dependence of the self-organized grating.

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Here T is the temperature in the unit of Kelvin. As an approximation, the refractive index modification was kept as 0.19 for various temperature. The thermal expansion of the grating structure was neglected because of the insensitivity of the phase delay to the grating period. Simulated results in Fig. 6(b) show that smaller than −0.012 degree/°C is found from −50 to 800°C in the whole visible range. The phase delay declines linearly with the temperature, but only a small influence induced by the temperature. It indicates that the self-organized grating has a remarkable thermal stability for most applications.

4. Broadband DoFP polarimeter based on self-organized nanogratings

Above analyses indicate that the self-organized grating has an excellent thermal stability and large acceptance angle in the visible range. The properties are in favor of a practical polarimeter based on the self-organized grating array. In a DoFP polarimeter, a cell for the polarization detection is composed of four identical nanogratings oriented in different directions and a linear polarizer. Because the linear polarizer usually has the high extinction ratio spanning the whole visible range, the spectral property of the polarimeter is only limited by the gratings. In theory, the systematic Müller matrix M_det in Eq. (4) can be calibrated in a polarimeter using several light sources with standard polarization, and the measured Stokes vector can be retrieved with Eq. (7). However, the error of inversion calculation depends on the determinant of M_det. Fixing the directions of gratings on the optimized $\pm$51.7 degrees and $\pm$15.1 degrees, the variation of |M_det| with the phase delay was calculated by Eqs. (4-6) and shown in Fig. 7. The maximum determinant of 0.2 appears at the phase delay of 132 degrees, which agrees with Sabatke’s theory. Half width of the determinant contains the phase delay ranging from 92 to 164 degrees. According to Eq. (7), the error of Stoke vector retrieval is smaller than twice of the optimized one at 132 degrees in this region. In the view of spectral bandwidth, we found that the small error was kept even with the incident wavelength ranging from 438 nm to 741 nm in Fig. 5(a), i.e., the whole visible range.

 

Fig. 7 Variation of the Müller matrix determinant with the phase delay of the self-organized gratings.

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Based on Sabatke’s theory, the four rows of the measurement matrix specify the four vertices of a tetrahedron inscribed in the sphere. Furthermore, the volume of this tetrahedron is proportional to the determinant of the measurement matrix [43]. Figure 8 shows a regular tetrahedron with the optimized phase delay of 132 degrees in (a), which has the maximum volume on the Poincaré sphere, and two smaller tetrahedrons at phase delay of 92 and 164 degrees in (b) and (c), respectively. The red curve shows the locus of coordinates on the sphere for a given phase delay. Because the tetrahedrons in Fig. 8(b) and (c) correspond to the polarization measurements at wavelength of 741 nm and 438 nm, respectively, half volumes of the tetrahedrons in Fig. 8(b) and (c) validate the small retrieve error and broad spectrum property of the proposed DoFP polarimeter.

 

Fig. 8 Müller matrix shows as a tetrahedron (blue lines) on a Poincaré sphere: (a) for the phase delay of 132 degrees, (b) for the phase delay of 92 degrees, and (c) for the phase delay of 164 degrees. The red curve shows the locus of coordinates on the sphere for a given phase delay.

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5. Conclusions

In this article, a numerical study of a DoFP polarimeter based on the self-organized nanograting array was presented. The nanograting array was produced by the ultrafast laser and constructs a two-dimensional phase modulator in the focal plane of the imaging polarimeter. At first, the systemic Müller matrix for the polarimeter composed of the nanograting array and linear polarizer was presented. Secondly, based on the RCWA theory, comprehensive analyses on the incident light spectrum, grating microstructure, incident light angle and temperature dependence of the naogratings were simulated. In the end, the spectral property of the proposed DoFP polarimeter was investigated by the calculation of Müller matrix determinant in the visible range. The thermal stability, large acceptance angle and broadband properties of the DoFP polarimeter based on the self-organized nanograting array were revealed by our simulations.