Generation of vectorial optical fields with complex spatial distribution in the cross section is of great interest in areas where exotic optical fields are desired, including particle manipulation, optical nanofabrication, beam shaping and optical imaging. In this work, a vectorial optical field generator capable of creating arbitrarily complex beam cross section is designed, built and tested. Based on two reflective phase-only liquid crystal spatial light modulators, this generator is capable of controlling all the parameters of the spatial distributions of an optical field, including the phase, amplitude and polarization (ellipticity and orientation) on a pixel-by-pixel basis. Various optical fields containing phase, amplitude and/or polarization modulations are successfully generated and tested using Stokes parameter measurement to demonstrate the capability and versatility of this optical field generator.
© 2013 Optical Society of America
Optical trapping and manipulation of particles in colloidal and biomedical sciences are made possible by the shaping of light . Shaped optical fields have enabled researchers to better understand the biophysics and colloidal dynamics through the trapping, guiding or patterning of molecules or nano/micro particles. Spatial engineering of focal field intensity has been studied to reach resolution far beyond diffraction limit in microscope system . Vortex beam, also known as “twisted light”, has also drawn a lot of interest owing to its spiral phase wavefront carrying orbital angular momentums . Better integrity of vortex beam through propagation in turbulent atmosphere has been shown  and a lot of research has been done using the orbital angular momentum as information carrier for free space communication due to its orthogonality and multiplexing capability . Besides the manipulation of intensity and phase, as the vector nature of electromagnetic wave, the state of polarization (SOP) also plays an important role in flattop generation [6, 7], focus engineering , optical tweezing and surface plasmon sensing using cylindrical vectorial beams , and spatially resolved ellipsometry . All the above applications require the spatial modulations of certain aspect of optical fields.
Tremendous amount of research has been conducted to develop versatile systems for the generation of optical fields with exotic properties. A diffractive optical element (DOE) based interferometric method was introduced to generate radially and azimuthally polarized vector beams . A robust interferometric method using a spatial light modulator (SLM) was demonstrated to tailor optical vector fields . Arbitrary vector fields with inhomogeneous distribution of linear polarization were realized using a SLM and an interferometric arrangement . An interferometric method using optical Fourier polarization processor was reported in Ref. 14. A recent study showed that in order to fully control the SOP, two spatially addressable retarders need to be used . Researchers constructed such a system consisting of two SLMs with the fast axes 45° from each other and were able to generate optical fields that cover the entire Poincaré sphere. Another non-interferometric method for vector field generation was also proposed in . However, the complete control of both polarization elevation angle and ellipticity can only be realized by modifying the experimental setup. Complete amplitude, phase and polarization control was reported  with the help of a double modulation system that requires two transmissive SLMs whose modulation depth can be controlled to achieve amplitude modulation for certain diffraction order. However, this approach comes with the limitation that each area must contain a large number of periods to achieve the ideal diffraction efficiency, which leads to optical field with limited spatial resolution for practical applications. Very recently, a technique for generating arbitrary intensity and polarization was reported using transmissive SLMs and Mach-Zehnder interferometry setup . Nevertheless, due to the nature of the technique, the absolute phase of each electric field component does not cover the entire 2π range. As a result, a complete phase control cannot be fully realized. The drawbacks of this technique also include relatively low transmittance and poor spatial resolution. All the existing techniques have limitations and cannot be used to generate a spatially-variant arbitrary vectorial field with high spatial resolution on a pixel-by-pixel basis. In this work, we propose and demonstrate a Vectorial Optical Field Generator (VOF-Gen) that is capable of creating arbitrary beam with independent controls of phase, amplitude and polarization on the pixel level utilizing high-resolution reflective phase-only liquid crystal spatial light modulator (LC-SLM).
Vectorial optical field can be represented as a superposition of two orthogonal polarization components. Using Jones vector representation, the desired field can be written as:
2.1. Liquid crystal spatial light modulator
As a key component for the generator, LC-SLM (Holoeye HEO 1080P) is used as a variable and addressable retarder. This LC-SLM is a phase-only reflective liquid crystal device featuring a HDTV resolution of 1920 x 1080 with pixel pitch of 8 μm and fill factor of 87%. The retardation for each SLM pixel in the SLM can be described as a function of the voltage (V) applied: , where d is the thickness of the LC layer, ne and no are the extraordinary and ordinary refractive indices of the LC retarder, respectively. Due to the birefringent nature, the LC-SLM in our system only responds to the horizontal polarization parallel to the LC directors, meaning that the horizontal component (x-component) of the reflected beam will carry the wavefront specified by the SLM while the vertical one (y-component) will be simply reflected unaffected. Since four degrees of freedom in Eq. (1) need to be independently controlled in the system, four reflections are required where each SLM section is loaded with one of the phase patterns for the modulations of phase, amplitude, polarization ratio and retardation.
2.2. Spatially variant polarization rotator
Our proposed system relies on one key component called Polarization Rotator (PR) based on the concept of a pure polarization rotator in order to realize the amplitude modulation and linear polarization rotation. Pure polarization rotator that consists of a quarter-wave plate (QWP), a variable optical retarder with fast axis at 45° and another QWP with its fast axis perpendicular to that of the first QWP has been proposed to achieve fast, non-mechanical polarization rotation [19, 20]. In our design, the variable retarder is replaced with the LC-SLM to realize spatially variant polarization rotation function on a pixel-by-pixel basis (Fig. 1(a)). The fast axis of the QWP is along 45° with respect to the horizontal axis. The incident light passes through the QWP with fast axis oriented at 45° (upper part in Fig. 1(b)) in the laboratory coordinate (x, y), and gets reflected off the SLM surface. The reflected light goes through the same QWP for the second time (lower part in Fig. 1(b)). However, due to the opposite propagation direction, the new laboratory coordinate (x’, y’) is a mirror image of coordinate (x, y) about the y axis. Therefore, the fast axis of the QWP has been effectively rotated to 135° in the coordinate (x’, y’). Thus, the Jones matrix representation of the PR can be calculated as:
Equation (2) describes the Jones matrix of the PR in terms of a spatially variant rotation matrix R with an extra phase term arising from the geometric effect. It is important to keep this additional geometric phase term in mind, as it needs to be compensated in many applications. The rotation matrix R indicates an effective polarization rotation of at each pixel. The PR setup is also used to calibrate the look up table (gamma curves) for both SLM panels. By precisely measuring the amount of rotation based on the nulling effect with a linear analyzer for each gray level, we are able to calibrate the gamma curves so that the gray level and the actual phase imposed by the SLM are precisely correlated.
2.3. System flow chart
The VOF-Gen consists of the light source, 4 SLM sections as subsystems to realize the modulation of all degrees of freedom for arbitrary optical field generation, 4-f imaging systems  to relay the optical field from one SLM section to the next or to the output plane and a CCD camera for detection. The SLM sections (subsystems) are integrated to achieve the complete modulation functionality as described in Fig. 2. For a desired complex optical field that we want to generate, the phase patterns for the SLMs are calculated based on a collimated Gaussian input beam. Then the phase patterns are loaded onto the generator and the modulations of phase, amplitude, polarization ratio and retardation are realized through SLM Sections 1 to 4 as shown in the flow chart, respectively. At the output plane, the beam is expected to have the desired complex optical field distribution and can be analyzed thereafter. The 4-f imaging systems are introduced to relay between SLM sections to minimize the diffraction effects. Details of the modulations and phase pattern calculations are given in the following.
2.4. Modulation of light
2.4.1. Phase modulation (SLM Section 1)
Phase modulation can be readily realized as the phase information loaded on the SLM will be directly imposed onto the horizontal component of the reflected beam. This is done in SLM Section 1 with a horizontally polarized, well-collimated Gaussian input beam. The resulting field can be represented in terms of Jones Vector as:Eq. (1), but also a pre-compensation phase that are due to the geometric phase effect for the polarization rotator described in Section 2.2 above.
2.4.2. Amplitude modulation (SLM Section 2)
Amplitude modulation is achieved by putting a linear polarizer with polarization axis oriented along horizontal direction after a PR setup utilizing the second SLM section. For horizontally polarized input field defined in Eq. (3), the resulting output field can be represented in Jones Vector format as:Equation (4) shows that amplitude modulation can be achieved with the sine function while the output is still horizontally polarized. For ϕ2 = 0, 0 amplitude can be obtained and unit amplitude is expected for ϕ2 = π. Recall the general expression for an arbitrarily desired field as shown in Eq. (1), compared to the definition for the desired field, ϕ2(x, y) can be calculated via the following expression:
2.4.3. Polarization ratio modulation (SLM Section 3)
As described in Eq. (4), the output field of SLM Section 2 is horizontally polarized. Using another PR setup consisting of the third SLM section, the SOP at each pixel can be linearly rotated to any direction prescribed by the local phase pattern to realize the desired polarization ratio distribution between the x- and y-polarization components. Assuming the phase pattern for SLM Section 3 is ϕ3(x, y), the output field of SLM Section 3 is given by:Eq. (1) as:
2.4.4. Phase retardation modulation (SLM Section 4)
Phase retardation can be introduced by directly shining the linearly polarized output field of SLM Section 3 as shown in Eq. (6) to the last SLM section due to the birefringence nature of the LC molecules. Assuming the phase pattern is ϕ4(x, y) for SLM Section 4, the final output field of the generator can be written as:Eq. (1).
As we previously discussed, the first SLM section is responsible for the phase modulation. As we can see in Eq. (8), the phase of the final output will have the following expression:
For any desired output field with arbitrary spatial distributions of phase, amplitude and polarization, Eqs. (5), (7), (9) and (11) can be used to calculate the required phase patterns. By loading the phase patterns onto each of the SLM sections of the VOF-Gen, arbitrarily complex desired output field can thus be generated.
3. Experimental setup
From the discussions above, in general four SLMs would be needed in order to fully control all of the degree of freedoms to create an arbitrarily complex optical field. However, taking the advantage of the HDTV format of the Holoeye HEO 1080P SLM, in our VOF-Gen setup two SLM panels are used with each of the SLM panel divided into two halves. Each half of the SLM panels is used to realize the control of one degree of freedom. This architecture utilizes the high resolution of the SLM panel while keeps the complexity of the experimental setup is kept manageable. The schematic diagram of the VOF-Gen is shown in Fig. 3. He-Ne laser of 632.8 nm wavelength is used as the input. Polarizer P1 and half wave plate λ/2 are used in combination to adjust input Gaussian beam intensity. Non-polarizing beam splitters (NPBSs) are used to properly direct the beam, thanks to its insensitivity of the polarization direction of the input beam. The SLM panels are divided into 4 sections, as shown in Fig. 3. The input beam first incidents on SLM Section 1 where phase modulation can be directly obtained. Lens L1 and Mirror M1 are used as a 4-f system. The optical field at the SLM surface in SLM Section 1 can be relayed to SLM Section 2 by carefully controlling the distances from the SLM to L1 and from L1 to M1 to be both equal to the focal length of L1, which is 300 mm. The QWP λ/4 combined with SLM Section 2 works as a PR discussed in Section. 2.2. Amplitude modulation is achieved in this SLM section by using the PR setup and a polarizer P2 while the output beam is still polarized horizontally. The second 4-f system comprised of lenses L2 and L3 is used to image the optical field from SLM Section 2 to SLM Section 3, where polarization rotation is obtained with a second PR setup. Lens L4 and Mirror M2 work as another 4-f system to relay the field at SLM Section 3 to SLM Section 4. Retardation is added to the optical field after being reflected from SLM Section 4. Finally, lenses L5 and L6 are used to relay the field from SLM Section 4 to the Detector (LBA-FW-SCOR by Spiricon) as the last 4-f imaging system in the entire VOF-Gen.
As previously mentioned, the VOF-Gen consists of two SLM panels. Therefore a simultaneous and independent control of both panels is required. This is realized through a color channel coding scheme such that the phase pattern for SLM 1 is coded into the green color (green channel) while the pattern for SLM 2 is coded in the red color (red channel). Then the two colors are combined to generate a color image as the overall phase pattern. Note that we have divided each SLM panel into two halves. Therefore the control signal (overall phase pattern) is also multiplexed spatially into right and left halves to control the left and right sections of both SLMs, respectively. Thus the entire VOF-Gen can be operated with one computer that is capable of outputting 1920 x 1080 resolution color graphics. In order to generate arbitrary beams, diffraction effects have to be taken into consideration. The diffraction needs to be minimized so that sharp edges or high frequency information in phase, amplitude and polarization can survive. This is achieved by the four 4-f imaging systems used in our setup. Spatial filters SF1 and SF2 located in the Fourier planes of the 4-f systems are used to suppress the interference caused by bulk cube beam splitters. Opaque cardboards (shown as black bars between NPBSs in Fig. 3) are placed to block the direct illumination.
4. Experimental results and discussions
4.1. Spatially variant phase modulation: vortex generation
We first demonstrate the phase modulation capability with the generation of optical vortex beams. Optical vortices with spiral wavefront carry orbital angular momentum (OAM). If the phase of the beam has an azimuthal dependence of , where φ is the azimuthal angle, then it’s said to have an OAM of lħ or a topological charge of l. Vortex beam with topological charge l in the far field will have Laguerre Gaussian distribution of LG0l. Here only the first SLM section is controlled with a spiral phase and the phase patterns for the rest of VOF-Gen remain flat. The generated field is focused by a lens and the intensity is recorded at the focal plane. Vortex beams with topological charges 1, 10 and 15 are generated and the far field intensities are shown in Fig. 4. As we can see, LG0,1, LG0,10 and LG0,15 are observed in the focal plane of the lens. As the topological charge increases, the size of the dark center also increases. A weak central spot is also observed, which may be caused by the direct reflection due to the filling factor, finite pixel size and level of quantization of the SLM. The relative amplitude of the central spot is more pronounced for higher topological charges due to the enlarged ring area.
4.2. Spatially variant amplitude modulation
In order to demonstrate the functionality of amplitude modulation, we designed an “EO” logo binary amplitude pattern. SLM Section 2 is loaded with “EO” shaped π-phase pattern for 100% transmission and zero for the rest of the window. SLM Section 1 is loaded with a pre-compensation phase and SLM Sections 3 and 4 both have flat phase. The output is directly captured by the CCD camera shown in Fig. 5. Note that the entire window would have been illuminated by the input Gaussian beam without the amplitude modulation. The result shows that fine features such as sharp edges in amplitude modulation are well preserved in the output beam.
4.3. Spatially variant polarization rotation: radially polarized beam
Cylindrical Vector (CV) beams are a group of beams whose spatially variant SOP possesses cylindrical symmetry . Due to the unique properties when focused by a high NA objective , there has been a great increase in the research of the CV beams recently, which has led to applications in high-resolution imaging , plasmonic focusing  and particle manipulation . Numerous approaches have been proposed to generate such beams, including both active [26, 27] and passive methods [10, 28].
We illustrate the capability of using our generator to create these CV beams through the generation of a radially polarized beam, one important subset of CV beams. Essentially the CV beams such as the radial polarization can be generated with the polarization rotation function of the SLM Section 3. A horizontally polarized input beam can be locally rotated pixel-by-pixel to the desired polarization direction. However, as we pointed out in Eq. (2), the spatially variant polarization rotation introduces an additional geometric phase. It is important to pre-compensate this additional phase by the phase pattern for SLM Section 1, as shown in Eq. (11).
When radially polarized beam is focused by low numerical aperture (NA) lens, a doughnut distribution will be resulted in the focal plane owing to the polarization singularity at the center. In order to generate radially polarized beam, the phase pattern for SLM Section 3 ϕ3 will have an azimuthal dependence of 2φ according to Eq. (7), where φ is the azimuth angle. Based on Eq. (8), we know that the extra phase introduced by SLM Section 3 will carry a phase with azimuthal dependence of φ, in other words, a spiral phase with topological charge l = 1. In order to generate true radial polarization with flat phase, a spiral phase with topological charge l = −1 needs to be incorporated in the pre-compensation phase. To verify the phase pre-compensation, we generate radially polarized beams without and with the pre-compensation phase. The far field intensities are captured at the focal plane of a lens, as shown in Fig. 6.
From Fig. 6(a), we can see that without pre-compensation, a bright spot is obtained when such beam is focused. This can be understood as the additional spiral phase cancelled the polarization singularity at the center of the focused radially polarized beam. Once the pre-compensation phase is introduced, a doughnut distribution is obtained as expected (shown in Fig. 6(b)). This confirms that the geometrical phase generated due to the operation of SLM Section 3 is successfully compensated. In general, the phase pre-compensation scheme can be used to compensate any additional phases that are introduced in the following modulation steps from SLM Sections 2, 3 and 4, as shown in Eq. (11). Moreover, this confirmation of the phase pre-compensation also serves as another evidence of the phase modulation capability discussed in Section 4.1.
The generated radially polarized beam is shown in Fig. 7(a). The arrows in Fig. 7(b) indicate the directions of the linear analyzer in front of the camera and the intensity of each linear polarization component is shown respectively. The polarization map is given in Fig. 7(c), which is calculated based on partial Stokes parameter measurement of S0, S1 and S2. The orientation of the lines indicates the local polarization direction while the length of lines indicates the local intensity. As shown in the figure, radial polarization in the output field is observed and the spatially variant polarization rotation capability is demonstrated.
4.4. Spatially variant phase retardation
Right-hand circular polarization (RCP) and left-hand circular polarization (LCP) have phase retardations of + π/2 and -π/2, respectively. To demonstrate the capability of optical field with spatially variant phase retardation, a Taiji pattern is generated using VOF-Gen with one half polarized in right-hand circular polarization (RCP) and the other half in left-hand circular polarization (LCP). The total field and the RCP and LCP components are shown in Fig. 8. Circular analyzers consisting of a QWP and a linear polarizer are used, where the angle between the fast axis of the QWP and the transmission axis of the polarizer is set at + π/4 and –π/4 to examine the RCP and LCP, respectively.
4.5. Complex vectorial optical field generation with multiple parameters
At this point, the modulation of each individual degree of freedom in determining an arbitrarily complex optical field has been successfully demonstrated. In the next two examples, we would like to generate optical fields that require controls of multiple parameters. First we demonstrate a ring-shaped optical field with the local SOP continuously varying from azimuthal to radial direction as the radius increases. The experimental result is shown in Fig. 9. The total field, filed distribution after polarizer at 0°, 45°, 90° and 135° and the local SOP map are shown in Figs. 9(a)-9(c), respectively. It clearly shows the gradual change from azimuthal to radial for local SOP.
In the next example, we generate a double ring pattern where the inner ring is azimuthally polarized with 0.5 relative amplitude while the outer ring is radially polarized with unit relative amplitude shown in Fig. 10. The total field is shown in Fig. 10(a), the linear polarization components are shown in Fig. 10(b) where the polarization axes are indicated by the arrows and the polarization map is given in Fig. 10(c). As we can see from this example, the outer ring has a higher intensity than the inner one as opposed to the fundamental Gaussian distribution and the polarization distribution follows the design.
As a final demonstration, complex vectorial optical field with local polarization elevation angle along radial direction and constant ellipticity of π/10 is designed. In other words, the SOP at each location is elliptical with constant ellipticity and the major axis of the ellipse is always along the radial direction. The ideal field distribution with polarization map, the experimental result and the histogram of the ellipticity (in unit of π radian) experimentally generated are shown in Figs. 11(a)-11(c), respectively. In this case, the full Stokes parameter measurement of S0, S1, S2 and S3 is performed to reveal the spatial distribution of the SOP of the generated beam. It can be shown that the experimental results generally agree with the design. The histogram of the ellipticity peaks around 0.1π, which shows the generation of the designed ellipticity. At some points the local SOP is slightly different from the expected. This may be due to the fact that the interference patterns caused by the LC-SLM surface can significantly change the SOP and the vibration also affects the accuracy of the full Stokes parameters measurement.
In summary, we reported a novel and versatile vectorial optical field generator (VOF-Gen) that is capable of generating arbitrary optical fields by spatially modulating all aspects of optical field (including phase, amplitude, polarization ratio and retardation) on a pixel-by-pixel basis. Various complex vector fields are generated and tested to demonstrate the functionality and flexibility of the proposed VOF-Gen. To the best of our knowledge, this is the first successful experimental demonstration of an arbitrary vectorial optical field generation system that is capable of tailoring all the spatial aspects of optical fields with high spatial resolution. This arbitrary complex optical field generator may find extensive applications in areas where exotic input fields are required, such as optical imaging, sensing, particle manipulation and beam shaping, etc.
W. Han has been supported in part by the University of Dayton Office for Graduate Academic Affairs through the Graduate Student Summer Fellowship Program and Dissertation Year Fellowship Program.
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