Synthesis of focused beam with controllable arbitrary homogeneous polarization using engineered vectorial optical fields

Guanghao Rui; Jian Chen; Xiaoyan Wang; Bing Gu; Yiping Cui; Qiwen Zhan

doi:10.1364/OE.24.023667

1. Introduction

Over the past few decades, optical microscopy has become an important and widely used tool in scientific research due to its nondestructive nature, which is useful when dealing with precious and irreplaceable samples. It is often desirable to control the focal field so as to impact the focal spot size, relative strength and phase of transverse and longitudinal field components, and the interaction with the sample, leading to the development of microscopy with novel functionalities and performances, such as coherent anti-Stokes Raman spectroscopy [1], third harmonic generation microscopy [2–4], and stimulation emission depletion microscopy [5,6] etc. Moreover, plenty of laser beam shaping systems have been proposed to transform the intensity distribution of an incident laser beam into a specific intensity distribution at a particular target plane, which is in demand for applications such as laser thermal annealing, laser fusion, material processing, holography, optical gauging, and optical recording. In addition to the optimization of the focus shape, size and intensity distribution, polarization of the focal field is another important parameter that deserves attention. In principle, full control of the state of polarization (SOP) in the focal plane could provide much richer information in optical microscopy and significantly expand its functionality. Beam shaping in the focal region of a high numerical aperture (NA) objective lens requires careful design of the input field. Tremendous amount of research has been conducted to develop versatile systems for the generation of optical fields with exotic properties. For example, a diffractive optical element based interferometric method was introduced to generate cylindrical vector beam [7]. Several methods have been demonstrated to develop systems for the generation of arbitrary optical vector fields with spatial light modulators (SLMs) [8–12]. Besides, the inverse design of the complete shaping of the optical focal field with the prescribed features has also been proposed [12–15]. In this work, we develop a method for inverse calculating the input field from a predetermined polarization in the focal volume. By carefully engineering the spatial distribution of the input field in terms of SOP and amplitude, arbitrary homogeneous polarization can be obtained at any transverse plane in the focal region. This versatile method may find many important applications in the fields of microscopy, optical tweezers, and nano-optics etc.

2. Inverse calculation of the required input field

Highly focused polarized beams can be numerically analyzed with the Richard-Wolf vectorial diffraction method [16]. The geometry of the problem is shown in Fig. 1. The electric field of a generalized beam illumination can be expressed as:

E_{i} = e^{i m φ} (f_{r} {\vec{e}}_{r} + f_{φ} {\vec{e}}_{φ}),

where m indicates the topological charge, f_r and f_φ are the radial and azimuthal components of the incident field, respectively. An aplanatic objective lens produces the input beam with a planar wavefront over the pupil into a spherical wave converging to the focal point. The field near the focus can be expressed as a linear combination of the focal fields arising from the radial and azimuthal polarization components:

E (r, ϕ, z) = - i A \int_{0}^{θ_{\max}} \int_{0}^{2 π} P (θ) [f_{r} (θ, φ) {\vec{e}}_{r} + f_{φ} (θ, φ) {\vec{e}}_{φ}] \times e^{i k r \sin θ \cos (ϕ - φ)} e^{i k z \cos θ} e^{i m φ} \sin θ d θ d φ,

where θ_max is the maximal angle corresponding to the NA of the objective lens, k is the wave number, f_r and f_φ are the radial and azimuthal components of the incident field, respectively. A is a constant given by the objective lens focal length f and wavelength λ as:

A = \frac{π f l_{0}}{λ},

where l₀ being the peak field amplitude and the pupil plane. P(θ) is the pupil apodization function that strongly depends on the objective lens design. In this case, sine condition lens is used therefore P(θ) = cos^1/2θ.

Fig. 1 Focusing of a vectorial optical beam.

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To analyze the transvers polarization structure of the focal field, a pair of orthogonal base vectors ( ${\vec{e}}_{a}$ , ${\vec{e}}_{b}$ ) on the Poincaré sphere is applied:

{\begin{cases} {\vec{e}}_{a} = (\cos α \cos ε - i \sin α \sin ε) {\vec{e}}_{x} + (\sin α \cos ε + i \cos α \sin ε) {\vec{e}}_{y} \\ {\vec{e}}_{b} = (- \sin α \cos ε + i \cos α \sin ε) {\vec{e}}_{x} + (\cos α \cos ε + i \sin α \sin ε) {\vec{e}}_{y} \end{cases} .

where α and ε are the elevation angle and the ellipticity of the polarization, respectively. Vector

{\vec{e}}_{r}

and

{\vec{e}}_{φ}

can be rewritten as:

\begin{array}{l} {\vec{e}}_{r} (θ, φ) = - \sin φ {\vec{e}}_{x} + \cos φ {\vec{e}}_{y} \\ = {\begin{cases} [- \cos ε \sin (φ - α) - i \sin ε \cos (φ - α)] {\vec{e}}_{a} \\ [\cos ε \cos (φ - α) + i \sin ε \sin (φ - α)] {\vec{e}}_{b} \end{cases} \\ {\vec{e}}_{φ} (θ, φ) = \cos θ \cos φ {\vec{e}}_{x} + \cos θ \sin φ {\vec{e}}_{y} + \sin θ {\vec{e}}_{z} \\ = {\begin{cases} \cos θ [\cos ε \cos (φ - α) - i \sin ε \sin (φ - α)] {\vec{e}}_{a} \\ \cos θ [\cos ε \sin (φ - α) - i \sin ε \cos (φ - α)] {\vec{e}}_{b} \\ \sin θ {\vec{e}}_{z} \end{cases} \end{array}

The electric field after refraction of the objective lens can be expressed as:

\begin{array}{l} E_{t} = e^{i m φ} \times \\ {\begin{cases} f_{r} \cos θ [\cos ε \cos (φ - α) - i \sin ε \sin (φ - α)] + f_{φ} [- \cos ε \sin (φ - α) - i \sin ε \cos (φ - α)] {\vec{e}}_{a} \\ f_{r} \cos θ [\cos ε \sin (φ - α) - i \sin ε \cos (φ - α)] + f_{φ} [\cos ε \cos (φ - α) + i \sin ε \sin (φ - α)] {\vec{e}}_{b} \\ f_{r} \sin θ {\vec{e}}_{z} \end{cases} \end{array}

Assuming the desired SOP of in the focal volume is purely

{\vec{e}}_{a}

, then the following relations need to be satisfied:

{\begin{cases} f_{r} = h (θ, φ) \\ f_{φ} = - \frac{\cos ε \sin (φ - α) - i \sin ε \cos (φ - α)}{\cos ε \cos (φ - α) + i \sin ε \sin (φ - α)} \cos θ \cdot h (θ, φ) \end{cases}

where h(θ, φ) is an arbitrary function. Substituting Eq. (7) into Eq. (1) and (2), the focal field can be expressed as:

E = - \frac{i A}{π} \int_{0}^{θ_{\max}} \int_{0}^{2 π} e^{i k [z \cos θ + r \sin θ \cos (φ - ϕ)]} e^{i m φ} h (θ, φ) \times {\begin{cases} \cos θ / [\cos ε \cos (φ - α) + i \sin ε \sin (φ - α)] {\vec{e}}_{a} \\ 0 {\vec{e}}_{b} \\ \sin θ {\vec{e}}_{z} \end{cases}

These analytical derivations illustrate that arbitrary homogeneous polarization can be generated at any transverse plane in the focal volume, as long as adjusting the SOP distribution of the input field according to the selected polarization basis.

3. Numerical modeling

Figure 2(a) shows the illumination pattern of the input beam that is required to generate elliptically polarized focused beam with m = 1, α = π/4 and ε = π/8. The SOPs at the beam cross-section are indicated by the polarization ellipses. The statistical data of Fig. 2(a) in terms of elevation angle and ellipticity are summarized in the histograms of Figs. 2(b) and 2(c). The histogram of the ellipticity and elevation angle peak around [0.115π, 0.125π] and [35°, 45°], respectively. Assuming the input beam is tightly focused by a high NA objective lens (NA = 0.8), the propagation characteristics of the focused beam is illustrated in Figs. 2(d)-2(f). Clearly the focused beam expands its size but maintains the designed ellipticalpolarization while propagating at difference transverse planes (z = 1.5λ, z = 2.5λ, and z = 5λ) in the focal volume. In addition, focused beam with other homogeneous polarizations are also considered. The ideal input field distribution with polarization map, the histogram of ellipticity and elevation angle designed for synthesizing specific linearly (m = 1, α = π/4 and ε = 0) and circularly (m = 1, α = 0 and ε = π/4) polarized focal field are shown in Figs. 3 and 4, respectively. The properties of circularly polarized vortex beam in cylindrical polarization bases can be found in [17]. Note that the ellipticity and elevation angle peak for linear polarization are around [-0.005π, 0.005π] and [35°,45°], respectively. For circular polarization, the ellipticity is peaked around [0.235π, 0.245π] while the elevation angles are equally distributed.

Fig. 2 Synthesis of elliptically polarized focused beam with m = 1, α = π/4 and ε = π/8. (a) Intensity distribution with polarization map, histograms of (b) ellipticity and (c) elevation angle of the ideal input beam. (d)-(f) Intensity distributions with polarization map of the focused beam at different transverse planes in the focal region for z = 1.5λ, z = 2.5λ, and z = 5λ.

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Fig. 3 Synthesis of linearly polarized focused beam with m = 1, α = π/4 and ε = 0. (a) Intensity distribution with polarization map, histograms of (b) ellipticity and (c) elevation angle of the ideal input beam. (d)-(f) Intensity distributions with polarization map of the focused beam at different transverse planes in the focal region for z = 1.5λ, z = 2.5λ, and z = 5λ.

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Fig. 4 Synthesis of circularly polarized focused beam with m = 1, α = 0 and ε = π/4. (a) Intensity distribution with polarization map, histograms of (b) ellipticity and (c) elevation angle of the ideal input beam. (d)-(f) Intensity distributions with polarization map of the focused beam at different transverse planes in the focal region for z = 1.5λ, z = 2.5λ, and z = 5λ.

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4. Experiment results

In the experiment, a vectorial optical field generator (VOF-Gen) is adopted to create arbitrary optical field with independent controls of phase, amplitude and polarization distribution on the pixel level. More details about the VOF-Gen can be found in [11]. The desired input beam coming from the VOF-Gen is introduced to a focusing system and the SOP of the focused beam is analyzed with Stokes parameters measurement. The diagram of the experiment setupis illustrated in Fig. 5. A collimated and expanded linearly polarized He-Ne laser (632.8 nm) is used as the light source. Taking advantage of the HDTV format of the Holoeye HEO 1080P reflective phase-only liquid crystal SLM, each SLM panel is divided into two halves and each half of the SLM panels is used to realize the control of one degree of freedom, enabling the fully control of all the degree of freedoms to create an arbitrary complex optical field. The engineered optical field can be directly observed on CCD camera 1 (Spiricon SP620U). The experimental results of the input beam designed for generating elliptically polarized (m = 1, α = π/4 and ε = π/8) focused beam are shown in Figs. 6(a)-6(c). The full Stokes parameter measurement is performed to reveal the spatial distribution of the SOP of the generated beam. The histogram of the ellipticity and elevation angle peak around [0.115π, 0.125π] and [35°, 45°], respectively, demonstrating the generation of the designed SOP distribution presented in Figs. 2(a)-2(c). A high NA objective lens (Nikon Plan Fluor 50X NA = 0.8) is used to focus the input light onto a silicon wafer with high reflectivity. The reflected beam back-propagates through the objective lens and imaged on the CCD camera 2 (Spiricon SP620U) by lens L7. In order to observe the irradiance pattern at different transverse planes, the silicon wafer is fixed on a three-dimensional stage that is connected to ultra-resolution electrostrictive actuators (Newport AD-30) and controlled with Newport Drive Controller (Model ESA-C), the high precision of which enables one to precisely change the position of the observation plane in the focal volume. Figures 6(d)-6(f) show the captured intensity distributions of the synthesized elliptically polarized vortex beam in the focal region at z = 1.5λ, 2.5λ, and 5λ. The focal plane at z = 0 is estimated by comparing the sizes of the objective aperture and the reflected beam captured before lens L7. However, due to the insufficient resolution of the CCD camera, the pattern exactly at the focal plane is not presented. To reveal the SOP distribution of the focused beam, Stokes parameters are measured by inserting a combination of quarter-wave plate and linear polarizer between lens L7 and CCD camera 2. The full Stokes parameter measurement of S₀, S₁, S₂ and S ₃ is performed by the following relations:

{\begin{cases} S_{0} = I (0^{\circ}, 0^{\circ}) + I (90^{\circ}, 90^{\circ}) \\ S_{1} = I (0^{\circ}, 0^{\circ}) - I (90^{\circ}, 90^{\circ}) \\ S_{2} = I (45^{\circ}, 45^{\circ}) - I (135^{\circ}, 135^{\circ}) \\ S_{3} = I (45^{\circ}, 0^{\circ}) - I (135^{\circ}, 0^{\circ}) \end{cases}

where I(α, β) denotes the intensity measured by the optical power meter (Newport, Model 1918-R) when the axes of the linear polarizer and quarter-wave plate are set at angle of located at α and β with respect to the x axis of the experiment setup. The Stokes images of Fig. 6(a) for the focused field at z = 2.5λ are shown in Fig. 7(a). It can be seen that the values of S₂ and S₃ images are nearly the same and much higher than that of the S₁ image, indicating the elliptical polarization with ellipticity and elevation angle of about π/8 and 45°. In order to quantitatively evaluate the synthesized focused beam in terms of the overall quality of the SOP, the cumulative normalized Stokes parameters are introduced [13]:

P_{i} = \sqrt{\frac{\sum S_{i}^{2} (x_{0}, y_{0})}{\sum S_{0}^{2} (x_{0}, y_{0})}} i = 1, 2, 3

where (x₀, y₀) are the indexes of the pixels of the Stokes image. Figure 7(b) shows the theoretical and experimental values of P₁, P₂ and P₃ of the focused field at different observation planes. Very good agreement between the theoretical predications and the experimental behavior demonstrate that the specific elliptical polarization (m = 1, α = π/4 and ε = π/8) is well maintained by the focused beam at any transverse plane in the focal volume. The average error is calculated to be only about 2%. The slight discrepancy between the theoretical predictions and the experimental results may arise from the imperfection of the input light and measurement error. To validate the versatility of the proposed method, the optical field presented in Figs. 3(a) and 4(a) are also experimentally generated. As shown in Figs. 8(a)-8(c) and Figs. 9(a)-9(c), the ellipticity and elevation angle peak around [-0.005π, 0.005π], [35°,45°], [0.235π, 0.245π] and [5°,15°], respectively. The experimental results of the irradiance patterns of their focal fields at different observation planes are shown in Figs. 8(d)-(f) and Figs. 9(d)-9(f), respectively. Besides, the SOP analyses of the focused beams presented in Fig. 8(e) and Fig. 9(e) are shown in Fig. 10 and Fig. 11, respectively. Clearly, the characters of linear (m = 1, α = π/4 and ε = 0) and circular (m = 1, α = 0 and ε = π/4) polarizations can be recognized since P₂>>P₁ + P₃ and P₃>>P₁ + P₂ at any transverse plane, respectively. The average errors are calculated to be near 2% for both.

Fig. 5 Schematic diagram of the experiment setup. HWP, half wave plate; P, polarizer; L, lens; M, mirror; QWP, quarter wave plate; SF, spatial filter; BS, beam splitter; MO, microscope objective; OP, observation plane.

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Fig. 6 Experimental results of the synthesized elliptically polarized focused beam with m = 1, α = π/4 and ε = π/8. (a) Intensity distribution with polarization map, histograms of (b) ellipticity and (c) elevation angle of the experimentally generated input beam. (d)-(f) Intensity distributions of the focused beam at different transverse planes in the focal region for z = 1.5λ, z = 2.5λ, and z = 5λ.

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Fig. 7 (a) Stokes images of the elliptically polarized focused field at z = 2.5λ. (b) Theoretical and experimental P_i values for the different transverse planes.

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Fig. 8 Experimental results of the synthesized linearly polarized focused beam with m = 1, α = π/4 and ε = 0. (a) Intensity distribution with polarization map, histograms of (b) ellipticity and (c) elevation angle of the experimentally generated input beam. (d)-(f) Intensity distributions of the focused beam at different transverse planes in the focal region for z = 1.5λ, z = 2.5λ, and z = 5λ.

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Fig. 9 Experimental results of the synthesized circularly polarized focused beam with m = 1, α = 0 and ε = π/4. (a) Intensity distribution with polarization map, histograms of (b) ellipticity and (c) elevation angle of the experimentally generated input beam. (d)-(f) Intensity distributions of the focused beam at different transverse planes in the focal region for z = 1.5λ, z = 2.5λ, and z = 5λ.

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Fig. 10 (a) Stokes images of the linearly polarized focused field at z = 2.5λ. (b) Theoretical and experimental P_i values for the different transverse planes.

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Fig. 11 (a) Stokes images of the circularly polarized focused field at z = 2.5λ. (b) Theoretical and experimental P_i values for the different transverse planes.

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5. Conclusions

In this work, we demonstrate that a wide variety of special focal field with specifically engineered polarization distributions can be created with proper vectorial optical fields as illumination. Systematic approaches to find the required vectorial optical fields at the pupil of objective lens for the creation of focal field with various desired SOPs are developed with Richard-Wolf vectorial diffraction method and experimentally demonstrated. The comprehensive focal field engineering techniques presented in this work may find important applications in single molecular imaging, tip enhanced Raman spectroscopy, high resolution optical microscopy, and particle trapping and manipulation.

Funding

National Natural Science Foundation of China (11504049, 11474052); Natural Science Foundation of Jiangsu Province (BK20150593); National Key Basic Research Program of China (2015CB352002); Fundamental Research Funds for the Central Universities (2242015KD001).

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Synthesis of focused beam with controllable arbitrary homogeneous polarization using engineered vectorial optical fields

Abstract

1. Introduction

2. Inverse calculation of the required input field

3. Numerical modeling

4. Experiment results

5. Conclusions

Funding

References and links

Cited By

Figures (11)

Equations (10)

Optics Express