Improvement of lens axicon’s performance for longitudinally polarized beam generation by adding a dedicated phase transmittance

K. B. Rajesh; Z. Jaroszewicz; P. M. Anbarasan

doi:10.1364/OE.18.026799

1. Introduction

In the case of high aperture focusing, polarization effects become particularly important, thus leading to a number of interesting phenomena. The focused radially polarized cylindrical vector (CV) beam has been found to have a fully symmetric focus spot with added resolution enhancement [1] and a strong longitudinal field component at focus in the direction of propagation of the beam. It has been shown that this field component can be used to excite a second harmonic generation [2], Raman spectroscopy [3] and particle acceleration [4]. Most of these near field applications demands sub-wavelength beam of longitudinal polarization with a large depth of focus (DOF). The super-resolution was extensively investigated using amplitude apertures [5,6], phase apertures [6], or their combination [7,8]. The axicon, invented in 1954, can focus a light beam with a longer depth of focus. However, the axial intensity of the beam generated by the axicon increases with the propagation distance and it has difficulty in realizing sub wavelength focusing [9,10]. In recent years, various diffractive lenses with long focal depth and high transverse resolution have been studied extensively [11–14]. The most promising optical elements for imaging with extended depth of focus in real time seem to be optical elements focusing an incident plane wave into a focal line segment. These elements can be regarded as modified lenses with controlled aberrations and was first suggested by Steel in 1960 [15]. It has been thoroughly investigated both analytically and numerically [16–18]. The experimental aspect of designing the lens axicon is investigated in [19]. However, light within the focal range of such systems is substantially transversally Polarized and not in the sub wavelength scale .Recently we have introduced the possible design of high Numerical Aperture (NA) lens axicon for the generation of longitudinally polarized focal segment with sub wavelength resolution [20].The high NA lens axicon is a system of cemented doublet-lens, where the virtual focal segment created by the aberrated diverging lens can be converted to a real focal segment, of the forward type with a nano scale resolution, by adding a high NA converging lens. The advantage of such a system is that spherical surfaces are routinely produced in any optical workshop, so the lens axicon is easy and inexpensive to manufacture.

2. Tight focusing of radially polarized beam

In this article, we describe a numerical study, based on vector diffraction theory of a radially polarized Bessel Gauss beam that is tightly focused by a combination of binary phase optical element and a high-NA-lens axicon. Here we consider only systems that comprise a diverging lens that has third-order spherical aberration and a perfect high NA converging lens [16]. A schematic diagram of the suggested method is shown in Fig. 1 .

Fig. 1 Schematic diagram of Lens Axicon with binary phase element.

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The analysis was performed on the basis of Richards and Wolf’s vectorial diffraction method [21] widely used for high-NA focusing systems at arbitrary incident polarization [22,23]. In the case of the radial incident polarization, adopting the cylindrical coordinates r, z and the notations of [24,25], radial and longitudinal components of the electric field E_r(r,z) and E_z(r,z) in the vicinity of the focal spot can be written as

\begin{array}{l} E_{r} (r, z) = \int_{0}^{α} \sqrt{\cos (θ)} \sin (2 θ) l (θ) J_{1} (k r \sin (θ)) \exp (i k z \cos (θ)) d θ \\ E_{z} (r, z) = \int_{0}^{α} \sqrt{\cos (θ)} \sin^{2} (θ) l (θ) J_{0} (k r \sin (θ)) \exp (i k z \cos (θ)) d θ, \end{array}

where k = 2π/λ, α = arcsin (NA /n), NA is the numerical aperture and n is the index of refraction between the lens and the sample. J₀(x) and J₁(x) denote Bessel functions of zero and first order and the function

l (θ)

describes amplitude modulation. For illumination by a Bessel-Gaussian beam with its waist in the pupil, this function is given by [24,25],

l (θ) = [\exp [- ξ^{2} {(\frac{\sin (θ)}{\sin (α)})}^{2}] J_{1} (2 ξ (\frac{\sin (θ)}{\sin (α)}))],

where, ξ is the parameter that denoted the ratio of pupil diameter to the beam diameter.

We perform the integration numerically using parameters λ = 405 nm, NA = 0.95 and ξ = 1.From the Fig. 2(a) it is observed that though the longitudinal field is high, the parasitic radial intensity is about 30% of the total intensity which leads to the broadening of the total intensity and the resultant FWHM of the total intensity is 0.68 λ which is in agreement with [26]. Plot of the total intensity distribution in yz plane depicted in Fig. 2(b) shows that the DOF is less than 1λ. Hence, in order to increase the DOF, a five belt π-phase binary optical element placed on the aperture of a high NA (0.95) aplanatic focusing lens is suggested in [26].This system produces a sub wavelength super resolution (0.43 λ) longitudinally polarized light beam which propagates over a long distance of (z = 4 λ) without divergence.

Fig. 2 (a) Intensity profile of the radial component (E_r ²), longitudinal component (E_z ²), and the total field (E_r ² + E_z ²) of the focal plane of the high NA lens for radial polarized Bessel-Gaussian beam. (b) Contour plot for the total intensity distribution.

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However it is observed that though the spot size remains constant within these regions it diverges very rapidly outside this region. This is shown in Fig. 3(a) which is Fig. 4(a) of [26] in large z-domain. It is observed that this effect is due to the drastic increase of the radial component of the total intensity which has been shifted from the focus by the binary optical phase element and is shown in Fig. 3(b).

Fig. 3 Contour plots for the electric field density distributions in the yz-plane for lens after additional phase modulation. (a) Total energy density distribution. (b). Radial component.

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Fig. 4 Intensity profile of the radial component(E_r ²), longitudinal component(E_z ²), and the total field (E_r ² + E_z ²) of the focal plane of the high NA lens axicon for radial polarized Bessel-Gaussian beam (b) Contour plot for the total intensity distribution.

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3. High NA lens axicon

In order to have a longitudinally polarized beam of good quality and with better depth of focus, one should suppress the radial field component for a long propagation distance. We show that this is possible in lens axicon by making a doublet consisting of an aberrated diverging lens and a perfect high NA converging lens. The intensity distribution of the lens axicon is evaluated by replacing the function $l (θ)$ with the function $l (θ) A (θ)$ where $A (θ)$ is the non paraxial transmittance function of the thin aberrated diverging lens [20],

A (θ) = \exp (i k {(β \frac{\sin (θ)}{\sin (α)})}^{4} + \frac{1}{2 f} {(\frac{\sin (θ)}{\sin (α)})}^{2}),

where k = 2π/λ, f is the focal length and β is the aberration coefficient. In our calculation we take f = 18.4mm, β = 6.667x10⁻⁵ mm⁻³. This results in an equiconcave diverging lens which is simple to manufacture [16]. The focal distribution of the lens axicon is calculated by including the transmission function of the aberrated diverging lens on the aperture of the high NA focusing lens. The intensity profile of the radial component, the longitudinal component and the total field of the longitudinally polarized beam in the focal cross section are shown in Fig. 4(a).

It is observed that the parasite radial field intensity is reduced to 5.2% of the total intensity and the FWHM of the spot size is observed as 0.414 λ. The intensity contour plot as shown in Fig. 4(b) depicts that the spot size is constant within certain region. The focal depth extends up to 4λ with uniform intensity but the beam faints rapidly outside this region.

Hence, in order to increase the length of focal segment with uniform intensity, we apply additional phase modulation to the radial polarized Bessel-Gauss input beam using a five belt binary optical phase element. More over we want to optimize the phase element to have high beam quality and high optical efficiency for conversion of the radially polarized beam to a longitudinally polarized beam apart from achieving large uniform focal depth.

The effect of phase modulation on the input radially polarized Bessel-Gauss beam is evaluated by replacing the function $l (θ)$ by $l (θ) T (θ)$ where $T (θ)$ is given by

T (θ) = {\begin{matrix} 1, \begin{matrix} f o r & 0 \leq θ < θ_{1}, θ_{2} \leq θ < θ_{3}, θ_{4} \leq θ < α \end{matrix} \\ - 1 \begin{matrix} \begin{matrix} \begin{matrix} f o r \begin{matrix} θ_{1} \leq θ < θ_{2}, θ_{3} \leq θ < θ_{4} \end{matrix} \end{matrix} \end{matrix} \end{matrix} \end{matrix} .

We choose one structure with random values for θ₁ to θ₄ from all possibilities and simulate their focusing properties by vector diffraction theory. If the structure generates a sub wavelength longitudinally polarized beam with DOF of 6λ and satisfies the limiting conditions of conversion efficiency above 10% and beam quality parameter above 0. 80, it is chosen as the initial structure during the optimization procedures. In the following steps, we continue to vary θ of one chosen zone to generate a flat top on axial total intensity profile until the value of the DOF not getting smaller or the focusing properties do not satisfy the two limiting conditions. The value of the newly chosen zone thickness is used in the next step. Then, we randomly choose the other zone and repeat these procedures to improve the uniformity of the on axial total intensity profile without affecting the two limiting conditions. We repeat these procedures and as an example the set of four angles we optimise for high beam quality, high optical efficiency and uniform intensity longitudinal polarized beam in the extended focal segment of the lens axicon are

{(θ)}_{1}

= 10⁰

{(θ)}_{2}

= 20.5°

{(θ)}_{3}

= 58° and

{(θ)}_{4}

= 65° with corresponding positions r₁ = 0.18,r₂ = 0.36,r₃ = 0.89 and r₄ = 0.92.

The intensity profile of the radial component, the longitudinal component and the total field of the longitudinally polarized beam in the focal cross section are shown in Fig. 5(a) .It is observed that the radial component further reduced to 4.8% and the FWHM of the total intensity spot is reduced to (0.394λ).From the Fig. 5(b) it is observed that the focal depth extends up to 6λ and the intensity remains constant within this region. However, outside this region the intensity starts to decay exponentially with the total intensity spot size being constant. This is in contradiction to the system proposed in [26] where the total intensity spot size grows rapidly outside the non diffractive region as shown in Fig. 3(a).This is because the binary optical element and the high NA lens axicon combination of the proposed system diffracts much of the radial component away from the beam center along the radial axis rather than shifting them to the longitudinal axis and is shown in Fig. 6(a) .

Fig. 5 Intensity profile of the radial component (E_r ²), longitudinal component (E_z ²), and the total field (E_r ² + E_z ²) of the focal plane of the high NA lens axicon for radial polarized Bessel-Gaussian beam with additional phase modulation. (b) Contour plot for the total intensity distribution.

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Fig. 6 Contour plots for the electric field density distributions in the yz-plane for lens axicon after additional phase modulation. (a). Radial component. (b). Longitudinal component.

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It is also observed that the longitudinal component is strongly concentrated in the beam center and suffers a feeble diffraction and is shown in Fig. 6(b).We have calculated the beam quality parameter

(η = j_{z} / (j_{z} + j_{r})) . where ϕ_{z} = 2 π {\int_{0}^{r_{0}} | E_{z} (r, 0) |}^{2} r d r and ϕ_{r} = 2 π {\int_{0}^{r_{0}} | E_{r} (r, 0) |}^{2} r d r .

Here r₀ is the first zero point of the radial electric density and is calculated as 0.607λ.We found the beam quality of the high NA lens axicon with phase modulation is 0.85 and its corresponding conversion efficiency is around 15%. Hence the formation of sub wavelength longitudinally polarized light with large depth of focus and high beam quality is possible with the combination of binary phase element and a high NA lens axicon.

4. Conclusion

Based on vector diffraction theory, tight focusing of a radially polarized Bessel Gaussian beam by a high NA lens axicon is studied. The intensity distribution in the focal region is illustrated by numerical calculations. Further improvement is achieved after introduction of a specially designed additional binary phase element. It is observed that, a sub-wavelength and super-resolution longitudinally polarized non-diffracting beam with a DOF around (6λ) is achieved. We expect such a beam with small spot size and long depth of focus can be widely used in application such as data storage, biomedical imaging, laser drilling, and machining.

Acknowledgement

The authors are grateful to the reviewers for their helpful comments. The authors are also thankful to Prof. S. Kaliappan, Vice chancellor, Anna University Tirunelveli for his continuous motivation to carry out this research.

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Improvement of lens axicon’s performance for longitudinally polarized beam generation by adding a dedicated phase transmittance

Abstract

1. Introduction

2. Tight focusing of radially polarized beam

3. High NA lens axicon

4. Conclusion

Acknowledgement

References and links

Cited By

Figures (6)

Equations (5)

Optics Express