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Abstract
Vector optical field has recently gained interest in a variety of application fields due to its novel characteristics. Conventional approaches of generating vector optical fields have difficulties in forming highly continuous polarization and suffer from the issue of high energy utilization rates. In order to address these issues, in this study a single optical path was proposed to generate vector optical fields where the birefringent phase plate modulated a linear polarized light into a vector optical field, which was then demodulated to a non-uniform linear polarization distribution of the vector optical field by the polarization demodulation module. Both a theoretical model and numerical simulations of the vector optical field generator were developed, illustrating the relationship between the polarization distribution of the target vector optical field and the depth distribution of the birefringent phase plate. Furthermore, the birefringent phase plate with predefined surface distributions was fabricated by grayscale exposure and ion etching. The generated vector optical field was experimentally characterized, capable of producing continuous polarization with high light energy utilization ratio, consistent with simulations. This new approach may have the potential of being widely used in future studies of generating well-controlled vector optical fields.
© 2017 Optical Society of America
1. Introduction
Vector optical field plays key roles in light wave diffraction and space-time evolution, leading to various novel properties in comparison to the scalar optical field. In 2000, the Brown Group of the Rochester University in the United States reported that a vector optical field with a radial polarization had a super-diffraction-limited focusing effect and a strong longitudinal field under tight focusing conditions [1]. The discovery of this phenomenon has led to the applications of vector optical fields to light trapping [2–4], laser processing [5,6], super-resolution microscopy [7–9], optical communication [10,11], particle acceleration [12], and fluorescence imaging [13].
Initially, the Mach-Zehnder interferometer [14] was used in combination of optical elements with special optical effects (e.g., diffractive elements [15], dove prisms [16], cylindrical grating prisms [17]) to generate high-quality vector optical fields. However, these optical units made the optical path bulky and prone to environmental disturbances. With the rapid development of electronic industry and semiconductor devices, the efficiency, regulation and stability of liquid crystal type spatial light modulator (SLM) are emerging and becoming more perfect, which makes the generation of vector optical field more flexible and greatly. In 2007, Maurer et al. utilized the spatial light modulator to achieve the generation of randomly distributed vector optical field [18]. Wang Haitian’s group proposed a method based on 4f system and the common-path interference to produce the vector optical field [19–21]. The biggest advantage of this method is to overcome the optical path of the disturbance caused by the deviation. However, the damage threshold of liquid crystals is low, which limits its practical applications.
Different from traditional interference superpositions, there were many methods used single light path to generate vector optical field such as method by using the biaxial crystals [22]. This method is based on the conversion of the Bessel function, which can generate high efficiency vector optical fields. With the development of micro-nano optics, the metasurfce was also used to produce vector optical fields, which can greatly simplify the system [23–27]. More specifically, metasurface was formed by fabricating micro-nanostructures on the substrate to abtain equivalent birefringence, and thus to modulate polarization. Since the polarization continuity of vector optical fields is closely related to the size of the Metasurface (roughly 100 nanometers), which cannot be produced by the traditional processing technology, metasurface based approaches is still in the stage of scientific research. In addition, the previous researchers also fabricated q-plates by using birefringent liquid crystal materials [28–31] to modulate the polarization of the incident light by changing the optical axis of a liquid crystal unit. Although this method can generate vector optical fields with high light energy utilization, it is still limited by the light energy threshold of the liquid crystal material, and thus it is difficult to generate a high power vector optical field.
In this paper, we proposed an approach of generating vector optical fields relying on a single optical path, which contains only a birefringent phase plate, and a demodulation module. The birefringent phase palte is continuous and function in the manner of pure phase modulation, which can greatly reduce the loss of energy and improve the continuity of polarization. Also, the light energy threshold of the birefringent material is high, which can generate a high power vector optical field. Based on the theories of optical transmissions and electromagnetics, a theoretical model was developed. The surface distribution of the birefringent phase plate was designed and fabricated by grayscale exposure and ion etching. The generated vector optical field was characterized, indicating a continuous polarization state and a high utilization ratio of light energy, which effectively validated the feasibility of this method.
2. Principle
The proposed vector optical field generator mainly includes two modules: a birefringent phase plate and a polarization demodulation module, wherein the birefringent phase plate is the core element (see Fig. 1). The birefringent materials have two different refractive index of ne and no for ordinary- and extraordinary-polarized lights, respectively, which makes two vertical polarization components of the incident light have different phase distributions. As shown in Fig. 1, after the linearly polarized light passes through the birefringent phase plate, a light field containing key information about the polarization distribution of the target vectors, including circular polarization (CP), elliptical polarization (EP) and linearly polarized light (LP) is generated. In this study, in order to produce the non-uniform linearly polarization distribution, a polarized wave demodulation module composed of a quarter wavelength plate and a half wavelength plate was added to the optical path, producing a well-regulated linearly polarized vector optical field.
Fig. 1 Schematic of the vector optical field generator based on a single optical path where a linear polarized light was modulated to a light field containing polarization information of multiple target vectors by a birefringent phase plate and then demodulated to a non-uniform linearly polarization distribution of the vector optical field by a polarization demodulation module.
In order to illustrate the working mechanism, a theoretical model was developed. Under the condition that the polarization angle of the incident light is 45° to the x-axis, the Jones matrix Ei of the incident light is expressed as:
Since the fast axis of the birefringent phase plate is in the x-axis direction, the Jones matrix G0 of the birefringent phase plate is expressed as:
Due to the birefringent material used in the phase plate, the phase delay between the ordinary- and the extraordinary-polarized lights can be generated. In Eq. (2), δ1(x, y) is the phase modulation of the ordinary-polarized light, and the δ2(x, y) is the phase delay between the ordinary light and the extraordinary-polarized light. Thus, the incident linearly polarized light is modulated to a vector containing a variety of polarization information after passing through the birefringent phase plate. In order to obtain the non-uniform linearly polarized distribution in the target vector optical field, the polarized wave demodulation module composed of a quarter wavelength plate and a half wavelength plate is added to the optical path. The fast axis of the quarter wavelength plate is 22.5° to the x-axis, and the fast axis of the half wavelength plate is 45° to the x-axis. The Jones matrixes G1 and G2 of wavelength plates can be expressed as:
Then, the Jones matrix Eo of the output light passing through the birefringent phase plate and the polarization demodulation structure is expressed as:
As can be seen from Eq. (4), the polarization distribution of the output is related to the amount of phase delay δ2(x, y), which is then modulated by the depth distribution of the birefringent phase plate. Then the depth distribution of the birefringent phase plate is obtained according to the target vector. If the polarization distribution of the target vector optical field is E(x, y), it can be expressed as:
Where θ(x, y) is the polarization angle of the linearly polarized light at each position in the vector optical field. By comparing Eqs. (4) and (5), it is observed that Eq. (4) contains a phase factor that does not affect the polarization state and polarization distribution information. In order to obtain the polarization distribution of the target vector optical field, the polarization distribution information of Eqs. (4) and (5) is compared, and the following relationship is obtained:
The amount of phase delay δ2(x, y) has the following relationship with the depth distribution h(x,y) of the birefringent phase plate:
Substituting Eq. (6) into Eq. (7), the relationship between the polarization distribution of the target vector optical field and the depth distribution of the birefringent phase plate is obtained as follows:
In Eq. (8), the one-to-one correspondence between the vector optical field distribution and the depth distribution of the birefringent phase plate is established, which can prepare arbitrary vector optical fields, providing directory guidance for the fabrication of the corresponding birefringent phase plates. The continuous polarization of the vector optical field means that the depth distribution of the birefringent phase plate must be continuous. So the sampling interval of the mask design needs to be small enough to ensure the continuity of birefringent phase plate pattern, thus ensuring that the resulting vector optical field is continuous.
3. Simulation
In this paper, a radial vector optical field was used as the simulation target. The distribution of the radial vector optical field is shown in Fig. 2(a) and the polarization distribution was expressed as:
Fig. 2 Schematic of the numerical simulations include (a) polarization distribution of the radial vector optical light and (b) depth distribution of the birefringent phase plate.
The working wavelength was 532 nm and the birefringent material was YVO4 with the refractive index of no and ne quantified as 1.9929 and 2.2154, respectively. The birefringence difference of yttrium vanadate was significantly, which dramatically reduced the fabrication depth of the phase plates, effectively decreasing the production difficulty.
Based on the aforementioned analysis, the depth distribution h (x, y) of the birefringent phase plate was:
From Eq. (10), the corresponding depth distribution of the required birefringent phase plate was shown in Fig. 2(b). The depth profile was in a spiral format with a maximum depth of 2.39 μm. The simulation of the light propagation in free space was carried out by coding in MATLAB. The distance between the receiving surface and the exit surface of the system was 0.25 m, and the simulation results were shown in Fig. 3.
Fig. 3 Numerical simulations of light fields generated by passing through the polarizers with a group of angles including (a) no analyzer, (b) 0°, (c) 90°, (d) 45° and (e) 135°.
Figure 3 shows the light fields generated by passing through the polarizers with different analyzer directions after 0.25 m propagation in free space. In the absence of an analyzer, the resulting light field does not appear the extinction position. By adding the analyzer into the optical path and changing the polarization directions, the extinction position was changed accordingly, indicating that the polarization state of the optical field was not uniformly distributed. According to the relationship between the direction of the analyzer and the inspection position, it was observed that the polarization distribution of the light field was in a radial distribution, which met the design requirements.
Since the final vector optical field distribution was determined by the depth distribution of the phase plate, the optical field distribution caused by the depth error of the phase plate was simulated with simulation results as follows
Figure 4 shows the distribution of the light field after the output light passed through the analyzer in different directions under the different depth (h) of the birefringent phase plate. The direction of the analyzer was 0°, 45°, 90°, and 135° to the x-axis, respectively and h0 was the maximum depth. In case of h = h0, the polarization of the light field was distributed in a strict radial direction, and the simulation result was shown in Fig. 4(a). In case of h = 1.2h0 (see Fig. 4(b)), the polarization state was overlapped at the position of phase mutation, and the polarization angle θ at continuous phase plate depth became larger. When the direction of the analyzer was 0°, the area with decreased light intensities was rotated in the direction of a shallow depth. When the inspection direction was 90°, the region with decreased light intensities became larger. In case of h = 0.8h0 (see Fig. 4(c)), the polarization state was missing at the position of phase mutation, and the polarization angle θ at continuous phase plate depth becomes smaller. When the direction of the analyzer was 0°, the area with decreased light intensities was rotated in the direction of higher depth. When the inspection direction was 90°, the region with decreased light intensities became smaller.
Fig. 4 The simulated distributions of the light field after the output light passed through the analyzer in a function of analyzers’ angels (θ) and birefringent phase plate depth (h) including (a1) h = h0, θ = 0°; (a2) h = h0, θ = 90°; (a3) h = h0, θ = 45°; (a4) h = h0, θ = 135°; (b1) h = 1.2h0, θ = 0°; (b2) h = 1.2h0, θ = 90°; (b3) h = 1.2h0, θ = 45°; (b4) h = 1.2h0, θ = 135°; (c1) h = 0.8h0, θ = 0°; (c2) h = 0.8h0, θ = 90°; (c3) h = 0.8h0, θ = 45°; and (c4) h = 0.8h0, θ = 135°.
By analyzing the depth differences, it was observed that if the polarization angle error was less than 5% with a corresponding depth error of 120 nm for birefringent phase plates, the design requirements were effectively met.
4. Fabrication and experiment
4.1 Fabrication process
In order to validate the feasibility of the aforementioned design, the birefringent phase plate shown in Fig. 5(b) was fabricated. The material of the birefringent phase plate was YVO4, and the corresponding area was 2 cm × 2 cm × 0.5 mm. The preparation was carried out by photolithography and ion etching. The mask used in lithography was shown in Fig. 5(a) where the corresponding structure in the mask was in a gray-scale distribution, enabling the modulation of the exposure dose. More specifically, as shown in Fig. 5(a), the dark parts indicate low exposure doses and the light parts indicate high exposure doses. The amounts of exposure doses corresponded to the sizes of the relief depth after development, and then the 3D structures were transferred from the photoresist to the YVO4 material by the subsequent ion beam etching.
In order to validate the simulation results, the corresponding experiments were conducted where YVO4 was chosen as the substrate and AZ9260 was used as the photoresist, which was spin-coated at a speed of 2000 rpm for 30 sec. Key parameters of prebake temperature, prebake duration, and obtained photoresist thickness were 100°C, 30 min, and 4.9 μm, respectively. A gray-scale mask was fabricated to modulate the dose of exposure according to the surface profile of the birefringent phase plate (see Fig. 5(a)). After exposure and development, photoresist patterns were obtained. Then the etching of YVO4 was conducted, which transferred the patterns of photoresist to the YVO4 substrate (see Fig. 5(b)).
4.2 Surface profile test
A step profilometer (ALPHA-Step IQ) was used to measure the surface profiles, validating the spiral structure of birefringent phase plate (see Fig. 6(a)). From the results of the staircase instrument, the finely and sleekly profile of birefringent phase plate, and the depth was 2.33 μm, which was within the allowable range of the errors. The roughness of the birefringent phase plate was tested by step profile (see Fig. 6(b)). From the figure we can see that in the test distance of 3mm the surface roughness is about ten nanometers which can satisfy the real application.
4.3 Experiment
In order to test the optical effect of the obtained phase plate, the optical path was constructed as shown in Fig. 7.
In this optical path, a laser was first expanded by a beam expander system (BE), and then modulated by a polarizer into a uniform linearly polarized light with the polarization angle of 45°. The beam was transmitted through the fabricated birefringent phase plate and the polarization demodulation structure consisting of a quarter wave plate (QWP) and a half wave plate (HWP). In order to verify the polarization pattern of the output beam, different analyzers were placed before the screen: a linear polarizer oriented at 0°, 45°, 90° and 135°. The distances from the exit surface of the system to screen were 0.25 m with the results shown in Fig. 8.
Fig. 8 The experiment results: (a)Intensity distribution of the obtained vector optical field ;(b)comparision of experimentally obtained intendity distribution with the theory results; and the obtained optical fields with a group of analyzer angles including (d) 0°, (e) 90°, (f) 45° and (g) 135°.
As can be seen from Fig. 8, the experimental results were consistent with numerical simulations with insignificant differences due to manufacturing errors. These results validated the aforementioned methodologies and the correctness of the theoretical derivations.
In this system, the birefringent phase plate and demodulation modules realize the polarization modulation. Since the birefringent phase plate was continuous and functioned in the manner of pure phase modulation, the theoretical light energy utilization rate was close to 100% and there was no occurrence of sub-class diffractions. There is no loss of light energy when the reflections inherent to the surface of the element were removed. In experiments, all phase plates were not coated with anti-reflective films, and thus the light energy utilization rate was lower than what was initially expected.After the measurement, the energy of incident light and outgoing light is 1.5mw and 1.39mw respectively, so the light energy utilization rate is 92.67%.
The depth of birefingent phase plate is 2.33 μm, and the corresponding phase is from 0 to 9.7π at the position of mutation. There is a phase singularity area corresponding to the dark line in the experiment, and the width of the dark line is 35 μm. From the theoretical analysis, the generated vector optical light field also contains a spiral phase factor, and the topological charge of the spiral phase is 9.84. Because of the existence of the spiral phase, there is a dark spot in the experimental light field. The diameter of the dark spot is 1704 μm, and this result is consistent with the theoretical results (see Fig. 8 (b)). The presence of these phases causes the intensity of the light field to appear discontinuously, but it does not affect the polarization distribution of the light field.Since the distribution of the polarization state of the vector optical field is related to the depth distribution of the birefringent phase plate. According to the results of the face slice detection, the surface type has good continuity, and the generated vector optical light field also has good polarization continuity. About the stability of the system, this paper mainly considers whether it will be interfered by the conditions of environment. The system is a single optical path and fewer factors will affect it than the traditional interference superposition, so this system is stable.
5. Summary
In this paper, a methodology based on a single light path of the birefringent phase plate was demonstrated to generate vector optical fields. Both numerical simulations and experimental results confirmed the methodology in generating vector optical fields featured with high continuities and low energy utilization rates. The simplification of the system can greatly promote the application of vector optical fields in various fields.
Funding
National Natural Science Foundation of China (NSFC) (61505214, 61605211); Applied Basic Research Programs of Department of Science and Technology of Sichuan Province (Nos. 2016JY0175, 2016RZ0067, 2017JY0058).
Acknowledgments
Thanks for the support of the Youth Innovation Promotion Association Chinese Academy of Sciences (CAS) and the CAS “Light of West China” Program. The authors also thank their colleagues for their discussions and suggestions to this research.
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