A scheme is proposed to design a polarization grating that splits an incident linearly polarized beam to an array of linearly polarized beams of identical intensity distribution and various azimuth angles of linear polarization. The grating is equivalent to a wave plate with space-variant azimuth angle and space-variant phase retardation. The linear polarization states of all split beams make the grating suitable for coherent beam combining architectures based on Dammann gratings. © 2015 Optical Society of America OCIS codes: (050.1950) Diffraction gratings; (140.3298) Laser beam combining. http://dx.doi.org/10.1364/AO.54.002495

1. Introduction

Space-variant polarization optical elements have attracted an increasing amount of attention and have found applications in polarimetry [1], image encryption [2], beam shaping [3,4], bioimaging [5], spot array generation [6,7], and magneto-optic data storage [8]. Polarization diffraction gratings periodically modulate the polarization state of an incident beam and have been designed for measuring Stokes parameters [1], tailoring optical vector beams [9–11], and splitting laser beams as duplicators or triplicators [12]. Common approaches to physical realization of polarization gratings include reconfigurable liquid crystal spatial light modulators and subwavelengthlength-period stripe gratings with lift-off or dry etching techniques [7,8]. Liquid crystal devices offer the flexibility of real-time programmable polarization control, whereas subwavelength structures enable continuous polarization variation. Polarization grating beam splitters provide higher efficiencies and better uniformity than their phase grating counterparts because they utilize the optimization of two independent polarizations and offer both phase and amplitude control by transferring a portion of energy from one component to the other without losing any energy [13,14]. Polarization grating beam splitters can be categorized to polarizers with space-variant transmission

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axis, wave plates with space-variant azimuth angle, and wave plates with space-variant phase retardation. Polarizers with space-variant transmission axis have inherent loss and will not be discussed in this paper. Wave plates with space-variant azimuth angle have a periodically varying angle between the x axis and the local slow axis, whereas the phase retardation between the two polarization components is fixed [7,13,15]. Wave plates with space-variant phase retardation have a periodically varying retardation between the two polarization components, whereas the angle between the x axis and the local slow axis is unchanged [12,16–18]. We propose a scheme to design a polarization grating that is equivalent to a wave plate with spacevariant azimuth angle and space-variant phase retardation. An incident linearly polarized beam is split into an array of linearly polarized beams of identical intensity distribution and different azimuth angles of linear polarization. The grating is not limited to a duplicator or triplicator and the number of split beams is arbitrary. The polarization states of split beams are all linearly polarized, which makes the grating suitable for utilization in coherent beam combining architectures based on Dammann gratings. 2. Theory

Polarization gratings apply space-variant control of polarization state to an incident beam. The transmission Jones matrix of a polarization grating is given by [19] 20 March 2015 / Vol. 54, No. 9 / APPLIED OPTICS

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Ax 0 Tx expjαR−φx Rφx 0 Bx cosφx − sinφx Ax 0 cosφx sinφx expjα sinφx cosφx 0 Bx − sinφx cosφx Axcos2 φx Bxsin2 φx Ax − Bx sinφx cosφx expjα : Ax − Bx sinφx cosφx Axsin2 φx Bxcos2 φx Ax and Bx are the complex amplitude modulation for the two orthogonal components vibrating along the slow and fast axes, respectively. R is the rotation matrix. φx is the angle between the x axis and the local slow axis. α is the phase bias. Ax and Bx satisfy the constraints jAxj ≤ 1 and jBxj ≤ 1. For polarizers with space-variant transmission axis, Ax 1 and Bx 0. For wave plates with spacevariant azimuth angle, Ax exp−jΓ∕2 and Bx expjΓ∕2 where Γ is a fixed term. φx is space variant. For wave plates with space-variant phase retardation, Ax exp−jΓx∕2, Bx expjΓx∕2, and φ is a fixed term. The design of a polarization grating beam splitter realizes a high diffraction efficiency and great uniformity by opti-

optimization include the azimuth angles of linear polarization of each split beam and the normalized x and y components of the optical field in the front focal plane, given by

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V x ρx expjθx ;

(2)

V y ρy expjθy :

(3)

The incident beam is supposed to be linearly polarized in x direction and its Jones vector is 10. By inserting the expressions of Ax and Bx to Eq. (1), the transmission Jones matrix of the grating can be written as

cos2 φ exp−jΓ∕2 sin2 φ expjΓ∕2 Tx expjα −j sin2φ sinΓ∕2 mizing the parameters in the transmission matrix, such as Γ (x) or φ (x). We propose a scheme that does not rely on optimizing parameters in the transmission matrix. The grating is placed in the front focal plane of a lens. The split beams in the back focal plane are assumed to have identical intensity distributions and various azimuth angles of linear polarization. By tuning the azimuth angles of linear polarization of each split beam in the back focal plane in order to match their combined intensity distribution in the front focal plane to that of the incident beam, the design parameters of the polarization grating are obtained accordingly. Figure 1 explains the optimization process to design a polarization grating beam splitter. The procedure is equivalent to the vectorial Fourier analysis introduced in [11,20,21]. The grating design is valid for fully polarized and fully coherent light in paraxial systems. The x and y components in the back focal plane are Fourier transforms related to the x and y components in the front focal plane, respectively. By tuning the azimuth angles of linear polarization of each beam in the back focal plane, their combined intensity distribution in the front focal plane fits a single Gaussian intensity distribution very closely. Although the optical field in the front focal plane has a Gaussian intensity distribution, its polarization states are space variant, which results from the conversion from a normal incident Gaussian beam by the polarization grating. The results of the

(1)

−j sin2φ sinΓ∕2 : sin2 φ exp−jΓ∕2 cos2 φ expjΓ∕2

(4)

The Jones vector of the optical field in the front focal plane equals VV xy , where V x cos2 φ exp−jΓ∕2 sin2 φ expjΓ∕2 expjα ρx expjθx ;

(5)

V y −j sin2φ sinΓ∕2 expjα ρy expjθy : (6) From Eqs. (5) and (6), a set of grating design parameters can be calculated as follows: for phase bias, π (7) α θy ; 2 for phase retardation, π ; Γ 2 arccos ρx cos θx − θy − 2

(8)

and for azimuth angle, φ

ρy 1 : arcsin sinΓ∕2 2

(9)

3. Simulation

The simplest case is a duplicator that splits an incident linearly polarized beam to two orthogonal linearly polarized beams. The azimuth angles of

Fig. 3. Space-variant phase retardation of a duplicator (one period).

Fig. 1. Scheme to design a polarization grating. F.T., Fourier transform.

linear polarization of split beams are 0 and π2. No optimization is required and the grating parameters can be derived according to Eqs. (7)–(9) after calculating the optical field distribution in the front focal plane. Figures 2–4 show one period of the three parameters of a polarization grating duplicator: the phase bias α, the phase retardation Γ, and the azimuth angle φ. The parameters used in the simulation include the wavelength (632.8 nm), the FWHM of individual beam in the back focal plane (20 μm), the beam spacing in the back focal plane (80 μm), the focal length (20 cm), and the FWHM of the incident Gaussian beam (4.81 mm). The grating length is much longer than the width of the incident beam. The ratio of the period of the phase bias, the phase retardation, and the azimuth angle is 4∶2∶1. In [12], another solution of a polarization grating duplicator is presented. A comparison of the two solutions is listed in Table 1. The optimization simulation of a triplicator utilizes the simulated annealing algorithm. The simulation parameters remain unchanged except that the number of split beams is three. The optimization process

Fig. 4. Space-variant azimuth angle of a duplicator (one period).

tunes the azimuth angles of linear polarization of the three split Gaussian beams and attempts to fit their combined intensity distribution in the front focal plane to a Gaussian shape as closely as possible. After the optimization, the azimuth angles of linear polarization of the three split beams are obtained as 6.26, 3.906, and 4.69 in radians. Figures 5–7 show one period of the three parameters of a polarization grating triplicator: the phase bias α, the phase retardation Γ, and the azimuth angle φ. The three space-variant parameters are all periodic and their periods are identical (1.5836 mm). Figure 8 shows the intensity distributions of the split beams in the back focal plane. The peak intensities and the azimuth angles of linear polarization are indicated in the figure. The uniformity of split beams is 1.00, defined as the ratio between the minimal and the maximum peak intensity within the array [7]. Table 1.

Comparison of Duplicator Solutions in This Paper and in [12]

Incident Beam This paper [12] Fig. 2. Space-variant phase bias of a duplicator (one period).

Linearly polarized, azimuth angle of linear polarization 0 Linearly polarized, azimuth angle of linear polarization π∕4

Diffraction Orders

Space-Variant Variables

−1, 1

Phase bias, phase retardation, azimuth angle Phase retardation

0, 1

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Fig. 5. Space-variant phase bias of a triplicator (one period).

Fig. 8. Beam splitting by a continuous polarization grating.

divided by 32. Therefore, the phase bias quantization π step size is 16 . The phase retardation quantization step size is 0.165 rad. The azimuth angle quantization step size is 1.3°. Figure 9 shows the beam splitting in the back focal plane by a quantized polarization grating. The peak intensities and the azimuth angles of linear polarization are indicated in the figure. The uniformity is decreased to 0.96. 4. Experiment

The polarization grating can be realized by subwavelength stripe structures and fabricated by photolithography and dry etching techniques [15,22]. The orientation of stripes corresponds to the azimuth angle parameter. The phase retardation and the bias are determined by the duty cycle and etching depth [7]. To analyze the practicality of the polarization grating triplicator, the calculation of quantization errors is added to the simulation. Similar to [3], 32 discrete levels replace the continuous variation of all parameters of the grating. The lateral quantization step size is 49.5 μm. This is determined by the period divided by 32. The quantization step sizes of other parameters are determined by the difference between their maximum and minimum values

To verify the validity of the polarization grating design, we set up an experiment to prove the theoretical calculation (Fig. 10). A linearly polarized laser beam from a He–Ne laser is converted to a circularly polarized beam by a quarter-wave plate and then spatially filtered to obtain a collimated plane wave. The plane wave hits a polarization plate that has three equally spaced 2 mm × 2 mm apertures with linear polarizers superimposed. Adjacent apertures are separated by 2 mm. After the polarization plate, a coherent array of three linearly polarized beams of equal power and various azimuth angles is obtained and then focused by a 20 cm lens. The intensity distribution in the back focal plane of lens 2 is magnified by a microscope objective and observed by a CCD camera. The results are analyzed by a Thorlabs beam profiler. In theory, a linearly polarized Gaussian beam can be split to an array of Gaussian beams with various azimuth angles of linear polarization; a linear polarized beam of sinc amplitude can be split to an array of

Fig. 7. Space-variant azimuth angle of a triplicator (one period).

Fig. 9. Beam splitting by a quantized polarization grating.

Fig. 6. Space-variant phase retardation of a triplicator (one period).

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retardation. The number of split beams is arbitrary, although a larger array takes longer to be optimized. All split beams are linearly polarized, which makes the grating suitable for applications such as coherent beam combining. The next step of the project is to fabricate a subwavelength stripe grating and accomplish related measurements.

Fig. 10. Experiment setup.

The author would like to acknowledge support from “The Fundamental Research Funds for the Central Universities, HUST: 0118182022” and “Specialized Research Fund for the Doctoral Program of Higher Education: 0214182024.” References

Fig. 11. Intensity distribution in the back focal plane of lens 2.

truncated plane waves with different azimuth angles of linear polarization. In the case of a triplicator, these azimuth angles of linear polarization are 90.0°, 135.0°, and 0.0°. If the polarizers attached to the polarization plate are adjusted correctly in the experiment, it is supposed to observe a sinc2 intensity distribution in the back focal plane of lens 2. In Section 3, the size of the incident beam is 4.81 mm, and the size of each split beam is 20 μm. The experiment is performed in the reverse direction of beam splitting. The size of each split beam is enlarged to 2 mm, which facilitates making a polarization plate. According to the similarity theorem of Fourier transform, the adjustment in size does not prevent validating the theory. Figure 11 shows the intensity distribution in the back focal plane of lens 2. The blue dashed curve corresponds to the theory and the red solid curve represents the experimental measurements. The polarization states of several marked locations are indicated at the top. The case with two polarization apertures is analyzed in [20] and proves the validity of designing a polarization grating duplicator with this method. 5. Conclusion

We propose a scheme to design a polarization grating that splits a linearly polarized incident beam to an array of linearly polarized beams with various azimuth angles. The grating is equivalent to a wave plate with space-variant azimuth angle and space-variant phase

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