Experimental implementation of tightly focused beams with unpolarized transversal component at any plane

Rosario Martínez-Herrero; David Maluenda; Ignasi Juvells; Artur Carnicer

doi:10.1364/OE.22.032419

1. Introduction

Three dimensional electromagnetic field distributions generated at the focal region of a high numerical aperture (NA) focused system has been extensively investigated in the last years [1–21]. Non-paraxial fields have demonstrated very useful in many fields for instance in high-resolution microscopy, particle trapping, high-density recording, tomography, electron acceleration, nonlinear optics, and optical tweezers. In most of the cases the study is concentrated on fully polarized fields. However, partially polarized tightly focused fields are attracting some more attention [22–28]. Tailoring three dimensional vectorial electromagnetic distributions with specified characteristics (shape, polarization, coherence, angular moment, etcetera) requires a careful design of the incident beam. The objective of this paper is to develop a method for designing focused fields with a non-polarized transversal component at any plane in the focal area and a non-zero longitudinal component on axis. This kind of fields may be useful in tomography, plasmonics spectroscopy, or invisibility cloaking [29–31].

Very recently we developed a framework for analyzing focused fields with specific polarization features when the optical system is illuminated with partially polarized light. This method relates the circular content of the incident beam with the circular and longitudinal components of the focused field [32]. Using this formalism, in this paper we derive analytical expressions that link the coherence-polarization properties of a quasi-monochromatic statistically stationary incident beam and the transversal part of a focused field. Taking these equations into account, we provide sufficient conditions for the incident beam so as to obtain fields in the focal area whose transversal component remains unpolarized for any z. These class of beams are implemented experimentally by means of spatial light modulators (SLM) displaying computer generated holograms. Numerical calculations and experimental results are compared and analysed.

Accordingly, the paper is organized as follows: in section 2 we review the description of highly focused partially polarized beams in terms of the angular spectrum. In section 3, this framework is used for designing focused fields whose transversal part remains unpolarized for any z. Section 4 includes an explanation of the optical setup used for generating these kind of beams. Experimental results are presented and discussed. Finally, the main conclusions of this paper are summarized in section 5.

2. Review of basic concepts

The electric field distribution at any point in the focal region of a high numerical aperture focusing system is given by the well know Richards-Wolf integral [33]

E (r, φ, z) = A \int_{0}^{θ_{0}} \int_{0}^{2 π} P (θ) E_{0} (θ, ϕ) e^{i k sin θ r cos (φ - ϕ)} e^{- i k cos θ z} sin θ d θ d ϕ,

where A is a constant, related to the focal length and the wavelength, k is the wave number, r and φ denote here the polar coordinates at the focal plane, and angles ϕ, θ and θ₀ are represented in Fig. 1. P(θ) denotes the so called apodization function obtained from energy conservation and geometric considerations. According to the characteristics of the problem, it can be advisable to write the angular spectrum E₀ in terms of the circular content of the beam instead of using the conventional radial and azimuthal description [34]:

E_{0} (θ, ϕ) = g_{1} (θ, ϕ) v_{1} (θ, ϕ) + g_{2} (θ, ϕ) v_{2} (θ, ϕ),

where g₁ and g₂ are, respectively, the right-hand and left-hand circular components of the incident field and v₁ and v₂ are mutually orthogonal unitary vectors that can be described in terms of the radial and azimuthal vectors e₁ and e₂ by means of

v_{1} (θ, ϕ) = \frac{e^{i ϕ}}{\sqrt{2}} (e_{2} + i e_{1})

v_{2} (θ, ϕ) = \frac{e^{- i ϕ}}{\sqrt{2}} (e_{2} - i e_{1}),

where

e_{1} (ϕ) = (- sin ϕ, cos ϕ, 0)

e_{2} (θ, ϕ) = (cos θ cos ϕ, cos θ sin ϕ, sin θ) .

v_m(θ, ϕ)exp(ikr · s) (m = 1, 2) are circularly polarized planar waves propagating along the direction defined by s = (sinθ cosϕ,sinθ sinϕ, −cosθ). Angular spectrum E₀ can be understood as the superposition of right and left circularly polarized plane waves. Notice that in the paraxial limit (θ → 0),

v_{1} = \frac{1}{\sqrt{2}} (1, i, 0)

and

v_{2} = \frac{1}{\sqrt{2}} (1, - i, 0)

are the usual transversal circular plane waves.

Fig. 1 Geometry, bases and variables involved.

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The use of this circular basis is particularly useful in the analysis of the angular momentum and when dealing with vortex beams [34–36]. Moreover, this basis is very suitable for designing the polarization characteristics of the transverse component of a highly focused beam [18, 32]. We rewrite the field at the vicinity of the focus plane E as follows:

E = E_{c} (\begin{matrix} \frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\ \frac{1}{\sqrt{2}} & \frac{- i}{\sqrt{2}} & 0 \\ 0 & 0 & 1 \end{matrix}) .

where E_c = (E₊, E₋, E_z); here E₊ and E₋ represent the right and left circular content of the transverse field at the vicinity of the focus plane, and E_z is the magnitude of the longitudinal component. Taking into account Eqs. (2) and (5), Eq. (1) is rewritten as

E_{c} (r, ϕ, z) = A \int_{0}^{θ_{0}} \int_{0}^{2 π} g (θ, ϕ) \hat{A} (θ, ϕ) P (θ) e^{i k sin θ r cos (ϕ - φ)} e^{- i k cos θ z} sin θ d θ d ϕ,

where g = (g₁, g₂) and Â(θ, ϕ) is a 2×3 matrix given by

\hat{A} (θ, ϕ) = (\begin{matrix} {cos}^{2} (θ / 2) & - {sin}^{2} (θ / 2) e^{2 i ϕ} & \frac{1}{\sqrt{2}} sin θ e^{i ϕ} \\ - {sin}^{2} (θ / 2) e^{- 2 i ϕ} & {cos}^{2} (θ / 2) & \frac{1}{\sqrt{2}} sin θ e^{- i ϕ} \end{matrix}) .

Thus, Eq. (6) relates the circular components of incident field with the circular and longitudinal content of focused field.

Let us next consider a quasi-monocromatic statistically stationary incident beam. On the one hand, the coherence-polarization features of the input paraxial beam are characterized by means of the 2×2 matrix Ĝ defined as

\hat{G} (θ_{1}, ϕ_{1}, θ_{2}, ϕ_{2}) = 〈 g^{†} (θ_{1}, ϕ_{1}) g (θ_{2}, ϕ_{2}) 〉,

where the dagger † stands for transpose complex conjugate and the angular brackets indicate statistical average. On the other hand, the coherence-polarization properties of the focused field are fully contained in the 3×3 cross-spectral density matrix

\hat{W_{c}}

given by

\hat{W_{c}} (r_{1}, φ_{1}, r_{2}, φ_{2}, z) = 〈 {E_{c}}^{†} (r_{1}, φ_{1}, z) E_{c} (r_{2}, φ_{2}, z) 〉 .

By using Eqs. (6), (8) and (9) the relationship between

\hat{W_{c}}

and Ĝ is obtained:

\begin{array}{l} \hat{W_{c}} (r_{1}, φ_{1}, r_{2}, φ_{2}, z) = {| A |}^{2} \int_{0}^{θ_{0}} \int_{0}^{θ_{0}} \int_{0}^{2 π} \int_{0}^{2 π} \hat{A} {(θ_{1}, ϕ_{1})}^{†} \hat{G} (θ_{1}, ϕ_{1}, θ_{2}, ϕ_{2}) \hat{A} (θ_{2}, ϕ_{2}) \\ P (θ_{1}) P (θ_{2}) exp (- i k r_{1} \cdot s_{1}) exp (i k_{1} r_{2} \cdot s_{2}) sin θ_{1} sin θ_{2} d θ_{1} d θ_{2} d ϕ_{1} d ϕ_{2} . \end{array}

where r_j = (r_j cosϕ_j, r_j sinϕ_j,z) and s_j = (sinθ_j cosφ_j, sinθ_j sinφ_j, −cosθ_j) with j = 1, 2. Eq. (10) relates coherence-polarization properties of the incident and the focused field in terms of the circular polarization content.

3. Focused fields with unpolarized transversal component

In this section we concentrate on the transverse field components of Eq. (10):

\begin{array}{l} \hat{W_{c T}} (r_{1}, φ_{1}, r_{2}, φ_{2}, z) = {| A |}^{2} \int_{0}^{θ_{0}} \int_{0}^{θ_{0}} \int_{0}^{2 π} \int_{0}^{2 π} \hat{A_{T}} {(θ_{1}, ϕ_{1})}^{†} \hat{G} (θ_{1}, ϕ_{1}, θ_{2}, ϕ_{2}) \hat{A_{T}} (θ_{2}, ϕ_{2}) \\ P (θ_{1}) P (θ_{2}) exp (- i k r_{1} \cdot s_{1}) exp (i k r_{2} \cdot s_{2}) sin θ_{1} sin θ_{2} d θ_{1} d θ_{2} d ϕ_{1} d ϕ_{2} . \end{array}

where

\hat{W_{c T}}

is a 2×2 matrix containing the transverse part of

\hat{W_{c}}

, and

\hat{A_{T}}

is the 2×2 matrix defined as

\hat{A_{T}} (θ, ϕ) = (\begin{matrix} {cos}^{2} (θ / 2) & - {sin}^{2} (θ / 2) e^{2 i ϕ} \\ - {sin}^{2} (θ / 2) e^{- 2 i ϕ} & {cos}^{2} (θ / 2) \end{matrix}) .

Let us now introduce

\hat{G_{T}}

defined as

\hat{G_{T}} (θ_{1}, ϕ_{1}, θ_{2}, ϕ_{2}) = \hat{A_{T}} {(θ_{1}, ϕ_{1})}^{†} \hat{G} (θ_{1}, ϕ_{1}, θ_{2}, ϕ_{2}) \hat{A_{T}} (θ_{2}, ϕ_{2})

thus Eq. (11) can now be written in a more compact way

\begin{array}{l} \hat{W_{c T}} (r_{1}, φ_{1}, r_{2}, φ_{2}, z) = {| A |}^{2} \int_{0}^{θ_{0}} \int_{0}^{θ_{0}} \int_{0}^{2 π} \int_{0}^{2 π} \hat{G_{T}} (θ_{1}, ϕ_{1}, θ_{2}, ϕ_{2}) P (θ_{1}) P (θ_{2}) \\ exp (- i k r_{1} \cdot s_{1}) exp (i k r_{2} \cdot s_{2}) sin θ_{1} sin θ_{2} d θ_{1} d θ_{2} d ϕ_{1} d ϕ_{2} . \end{array}

To analyse the polarization of the transversal component we write the matrix $\hat{W_{c T}} (r, φ, r, φ, z)$ , namely $\hat{W_{c T}} (r, φ, z)$ , in terms of the circular Stokes parameters $C_{n} (r, z) = tr (\hat{W_{c T}} (r, φ, z) σ_{n})$ with n = 0, 1, 2, 3; σ_n are the Pauli matrices [37]. According to Eq. (14),

\begin{array}{l} C_{n} (r, z) = {| A |}^{2} \int_{0}^{θ_{0}} \int_{0}^{θ_{0}} \int_{0}^{2 π} \int_{0}^{2 π} C_{n T} (θ_{1}, ϕ_{1}, θ_{2}, ϕ_{2}) P (θ_{1}) P (θ_{2}) \\ exp (- i k r_{1} \cdot s_{1}) exp (i k r_{2} \cdot s_{2}) sin θ_{1} sin θ_{2} d θ_{1} d θ_{2} d ϕ_{1} d ϕ_{2} \end{array}

being C_nT

C_{n T} (θ_{1}, ϕ_{1}, θ_{2}, ϕ_{2}) = tr (\hat{G_{T}} (θ_{1}, ϕ_{1}, θ_{2}, ϕ_{2}) σ_{n}) .

C_nT are the so-called circular generalized Stokes parameters as they are the analogue of the generalized Stokes parameters [37]. Eq. (15) is one of the main results of this paper, since provides the relationship between the polarization properties of the incident beam and the transversal part of focused field, in terms of the circular Stokes parameters.

An interesting application of the formalism developed above is the design of a a field in the focal area whose transversal part is non-polarized. Taken Eq. (15) into account, the transversal field is unpolarized when C_n(r, z) = 0 for n = 1, 2, 3. To this end we consider a beam whose circular generalized Stokes parameters are given by

C_{0 T} (θ_{1}, ϕ_{1}, θ_{2}, ϕ_{2}) = 4 h (θ_{1}, θ_{2}) cos (m (ϕ_{1} - ϕ_{2}))

C_{1 T} (θ_{1}, ϕ_{1}, θ_{2}, ϕ_{2}) = 4 i h (θ_{1}, θ_{2}) sin (m (ϕ_{1} - ϕ_{2}))

C_{2 T} (θ_{1}, ϕ_{1}, θ_{2}, ϕ_{2}) = C_{3 T} (θ_{1}, ϕ_{1}, θ_{2}, ϕ_{2}) = 0

here h(θ₁, θ₂) is a nonnegative definite function fulfilling h(θ₁, θ₂) = h(θ₂, θ₁)^*. These Stokes parameters correspond to a matrix

\hat{G_{T}}

given by

\hat{G_{T}} (θ_{1}, ϕ_{1}, θ_{2}, ϕ_{2}) = h (θ_{1}, θ_{2}) {\hat{U}}^{†} (ϕ_{1}) \hat{U} (ϕ_{2})

where

\hat{U} (ϕ) = (\begin{matrix} - i e^{- i m ϕ} & i e^{i m ϕ} \\ e^{- i m ϕ} & e^{i m ϕ} \end{matrix}) .

Substituting Eq. (17) into (15) we get

\begin{array}{l} C_{0} (r, z) = 16 π^{2} {| A |}^{2} \int_{0}^{θ_{0}} \int_{0}^{θ_{0}} P (θ_{1}) P (θ_{2}) h (θ_{1}, θ_{2}) J_{m} (k r sin θ_{1}) J_{m} (k r sin θ_{2}) \\ exp (- i k r z cos θ_{2}) exp (i k z cos θ_{1}) sin θ_{1} sin θ_{2} d θ_{1} d θ_{2} \end{array}

C_{1} (r, z) = C_{2} (r, z) = C_{3} (r, z) = 0

Hence, a field whose transversal part is unpolarized for any z and vanishes on axis is obtained, regardless of the form of function h. According to Eqs. (13) and (18), the matrix of incident beam Ĝ now reads

\hat{G} (θ_{1}, ϕ_{1}, θ_{2}, ϕ_{2}) = h (θ_{1}, θ_{2}) {(\hat{A_{T}} {(θ_{2}, ϕ_{2})}^{†})}^{- 1} {\hat{U}}^{†} (ϕ_{1}) \hat{U} (ϕ_{2}) \hat{A_{T}} {(θ_{2}, ϕ_{2})}^{- 1} .

Using Eqs. (10) and (21), and after some calculations, the following expression for the longitudinal component of the focused field is obtained:

\begin{matrix} \hat{W_{c}} {(r_{1}, ϕ_{1}, r_{2}, ϕ_{2}, z)}_{z z} = 8 π^{2} {| A |}^{2} cos ((m - 1) (ϕ_{1} - ϕ_{2})) \int_{0}^{θ_{0}} \int_{0}^{θ_{0}} P (θ_{1}) P (θ_{2}) h (θ_{1}, θ_{2}) tan θ_{1} tan θ_{2} \\ exp (- i k r z cos θ_{2}) exp (i k z cos θ_{1}) J_{m - 1} (k r_{1} sin θ_{1}) J_{m - 1} (k r_{2} sin θ_{2}) sin θ_{1} sin θ_{2} d θ_{1} d θ_{2} \end{matrix}

From Eq. (22) we conclude that, on axis, the irradiance of the longitudinal component vanishes when m ≠ 1. However, for m = 1 the field is purely longitudinal on axis, and its irradiance becomes

\begin{array}{l} I (0, z) = tr (\hat{W_{c}} (0, φ, 0, φ, z)) = 8 π^{2} {| A |}^{2} \int_{0}^{θ_{0}} \int_{0}^{θ_{0}} P (θ_{1}) P (θ_{2}) h (θ_{1}, θ_{2}) tan θ_{1} tan θ_{2} \\ exp (- i k r z cos θ_{2}) exp (i k z cos θ_{1}) sin θ_{1} sin θ_{2} d θ_{1} d θ_{2} \end{array}

Since fields with longitudinal component are relevant in several applications, from this point onwards, we analyse the case m = 1 thus Eq. (21) becomes

\hat{G} (θ_{1}, ϕ_{1}, θ_{2}, ϕ_{2}) = h (θ_{1}, θ_{2}) (\begin{array}{l} a e^{i (ϕ_{1} - ϕ_{2})} & b e^{i (ϕ_{1} + ϕ_{2})} \\ b e^{- i (ϕ_{1} + ϕ_{2})} & a e^{- i (ϕ_{1} - ϕ_{2})} \end{array})

with

\begin{array}{l} a = 1 + cos θ_{1} cos θ_{2} \\ b = 1 - cos θ_{1} cos θ_{2} \end{array}

A calculation of the beam proposed in Eq. (24) has been carried out using Eq. (10) and assuming a constant value for function h. The results are presented in Fig. 2. The first row shows the intensity of the focused field at z = 0 and z = 2λ, i.e.

tr (\hat{W_{c}} (r, φ, r, φ, z))

. In the second row, the profiles of the irradiance and the transversal

\hat{W_{c}} {(r, φ, r, φ, z)}_{11} + \hat{W_{c}} {(r, φ, r, φ, z)}_{22}

and longitudinal components

\hat{W_{c}} {(r, φ, r, φ, z)}_{z z}

are presented. Note that for the beams we are handling, the magnitudes presented here only depend on r and z. As expected, the intensity of the transverse component is always zero at r = 0. In the next section a practical implementation of such kind of beams is explained.

Fig. 2 Numerical simulation of Eq. (10) for the beam proposed in Eq. (24), NA=0.85. First row: false color representation of the intensity at z = 0 and z = 2λ; second row: profiles of the irradiance and the transversal and longitudinal components.

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4. Experimental implementation

Figure 3 depicts an experimental setup based on a Mach-Zehnder interferometer able to generate arbitrary spatially-variant polarized focused beams. An extended explanation on how this system is used to generate beams with arbitrary polarization and shape can be found in [17]. An unpolarized quasi monochromatic input beam is split into two beams by means of polarizing beam splitter PBS₁. Reflected by mirrors M₁ or M₂, the split beam passes through wave plates HWP and QWP which rotate the oscillating plane and set the modulator to the required desired modulation curve. Then, light passes through a translucent SLM (Holoeye HEO 0017) displaying cell-based double-pixel holograms to encode complex transmittances Es_x and Es_y [38]. Using this holographic procedure, the displays can access nearly all possible complex values within a circle of transmittance T = 0.3, as explained in [18]. The beams are subsequently recombined by means of polarizing beam splitter PBS₂ and fed into a 4f system. A spatial filter removes higher-order terms whereas allowing pass the synthesized field. The irradiance of this beam can be observed by means of camera CCD₁. Afterwards, the beam is focused by means of a high numerical aperture microscope lens NA=0.85. This objective obeys the sine condition thus $P (θ) = \sqrt{cos θ}$ . The field in the focal area is reflected on a glass surface and imaged on camera CCD₂. Polarization analysis is carried out by means of polarizer LP and a quarter-wave plate.

Fig. 3 Experimental setup

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In order to synthesize the beam of Eq. (24) the following complex valued distributions are coded on each SLM

E s_{x} = i sin ϕ cos θ + cos ϕ

E s_{y} = - i cos ϕ cos θ + sin ϕ

The observation plane (a cover slip) is mounted on a stage that enables to modify the observation distance z. Figure 4 shows the profile of C₀ measured at z = 2λ. The distance of the observation plane with respect to the focal plane is estimated by comparing the experimental light distribution with the numerical evaluation of the angular average of the transversal irradiance. As shown in Fig. 4, the model developed reproduces with precision the profile of the transversal irradiance despite the fact that the value of this distribution at r = 0 is not zero. The intensity detected at the center is compatible with certain amounts of background noise and spherical aberration of the imaging system. The focal plane cannot be retrieved with enough accuracy because of the insufficient resolution and limited bandwidth of the camera, and the lack of precision along the z–axis due to the equipment used.

Fig. 4 Experimental C₀ distribution at the focal area z = 2λ.

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Figure 5 summarizes the measures of the transversal Stokes parameters. The first row of Fig. 5 shows the parameters of the laser source measured in a plane between the optical fiber and lens L₁. The second and third rows display the Stokes parameters measured in two transverse planes in the focal area, in particular at z = 2λ and z = 4λ. In the three cases considered, parameters S₁, S₂ and S₃ are close to zero at any point. Note that parameters C₀, C₁, C₂ and C₃ used in this paper are related to the conventional Stokes parameters by means of S₀=C₀, S₁=C₂, S₂=C₃ and S₃=C₁. Table 1 indicates the averaged values of the Stokes parameters. For completeness, the averaged value of the degree of transversal polarization P₂_D is also presented. Accordingly, the source beam is mostly unpolarized and the behavior of the Stokes parameters in the focal area is compatible with our thesis: the transverse component of the focused field remains unpolarized on axis.

Fig. 5 Measure of the transversal Stokes parameters. First row: laser source; second and third rows: focal area of the microscope lens.

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Table 1. Averaged normalized Stokes parameters.

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5. Concluding remarks

We developed a suitable framework for designing some transversal features of the field distribution in the focal region of a high numerical aperture lens when the incoming beam is partially polarized. In particular, we propose a field with a non-polarized transversal part that preserves this characteristic for any z and non-zero longitudinal component on axis. The combined use of an interferometric setup for generating beams with arbitrary polarization and digital holography techniques enables the generation of such field in practice. Numerical and experimental results are provided, showing a good agreement between theoretical predictions and the experimental behavior of the beam.

Acknowledgments

This work has been supported by the Ministerio de Ciencia e Innovación of Spain, project FIS2010-17543, and Ministerio de Economía y Competitividad projects FIS2013-46475.

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	S₁	S₂	S₃	P_2D
He-Ne laser source	−0.06	0.04	0.04	0.10
z = 2λ	0.08	−0.02	0.01	0.08
z = 4λ	−0.04	0.03	−0.01	0.10

Experimental implementation of tightly focused beams with unpolarized transversal component at any plane

Abstract

1. Introduction

2. Review of basic concepts

3. Focused fields with unpolarized transversal component

4. Experimental implementation

5. Concluding remarks

Acknowledgments

References

Cited By

Figures (5)

Tables (1)

Equations (32)

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