Optical needles with arbitrary homogeneous three-dimensional polarization

Li Hang; Ying Wang; Peifeng Chen

doi:10.1364/OE.386204

1. Introduction

Optical needles, also called needles of light [1–3], are optical fields which have large longitudinal full width at half maximum (LFWHM) and small transverse full width at half maximum (TFWHM). They have become one of the most attractive topics because of their wide applications in data storage [4–6], optical trapping [7,8], fluorescent imaging [9,10], optical coherence tomography [11,12] and so on. By tightly focusing, great developments have been achieved for longitudinally and transversely polarized optical needles [1–3,13–19]. However, none of them can generate optical needles with arbitrary polarization. Over the past few years, there are several papers about generating optical fields with three-dimensional polarization [20–22]. But their aims are not needles. In this paper, we propose a method to generate optical needles with arbitrary three-dimensional polarization by focusing complex input fields. The input fields comprise radially polarized beam and azimuthally polarized vortex beams. The radially polarized beam can generate the longitudinally polarized field [23], while the azimuthally polarized beams with vortex phase $\exp (\pm i\varphi )$ can generate left- and right-handed polarized fields. These three components constitute three-dimensional unitary bases. Thus, an arbitrary polarization can be obtained.

2. Theory and configuration

Figure 1 shows how an azimuthally polarized beam with vortex phase $\exp (i\varphi )$ generates a left-handed polarized beam. The dots and crosses are the directions of the electric field when the phases are not considered. The numbers near the dots and crosses are the phases of the electric fields. In Fig. 1(a), the electric field near the focus is parallel to $y$-axis, while in Fig. 1(b) it is parallel to $x$-axis. Since there is a phase retardation of $\pi /2$, the total field is left-handed polarized. Similarly, the focal field will be right-handed polarized when the incident beam is azimuthally polarized with vortex phase $\exp (-i\varphi )$. In fact, this is an example of orbital-to-spin conversion [24,25]. An annular input beam comprised of radially polarized and azimuthally polarized vortex components can be expressed as

(1)$$\begin{aligned} \mathbf{E}_{in}=\left\{ \begin{aligned} & =f\mathbf{e}_r+g_{{-}1}\exp({-}i\varphi)\mathbf{e}_\varphi+g_{1}\exp(i\varphi)\mathbf{e}_\varphi , & |\theta-\theta_0|\leq \Delta\theta/2\\ & =0, & |\theta-\theta_0|> \Delta\theta/2, \end{aligned} \right. \end{aligned}$$

where $\theta _0$ and $\Delta \theta$ are the angular position and width of the incident beam. Such beams can be generated by spatial light modulators [26,27]. For a parabolic mirror in Fig. 1, $\theta _0=90^\circ$ is chosen in this paper. The results for an arbitrary $\theta _0$ are listed in the appendix. $f$, $g_{-1}$ and $g_1$ are complex numbers. According to the Richards-Wolf vector diffraction theory [28], when $\Delta \theta \ll \theta _0$, the electric field near the focus can be expressed as [29]

(2)$$\begin{aligned} \mathbf{E}(r,\phi,z)\approx \frac{A(z)}{\sqrt2} \left[\begin{array}{ccc}-i g_{{-}1}(e^{{-}2i\phi}J_{2}+J_{0})\mathbf{e}_x+ig_{1}(e^{2i\phi}J_{2}+J_{0})\mathbf{e}_x \\ g_{{-}1}(e^{{-}2i\phi}J_{2}-J_{0})\mathbf{e}_{y}+ g_1(e^{2i\phi}J_{2}-J_{0})\mathbf{e}_{y} \\-2 fJ_{0}\mathbf{e}_z \end{array}\right]\\ = A(z) \left[\begin{array}{ccc} i({-}g_{{-}1}J_{0}+g_1e^{2i\phi}J_{2})\mathbf{e}_R \\ i(g_1J_{0}-g_{{-}1}e^{{-}2i\phi}J_{2})\mathbf{e}_{L} \\-\sqrt{2} fJ_{0}\mathbf{e}_z \end{array}\right], \end{aligned}$$

where $A(z)\propto \frac {\sin (kz\sin \frac {\Delta \theta }{2})}{kz}$, $k$ is wave number. $J_n=J_n(kr)$, where $J_n(\cdot )$ denotes the nth-order Bessel function of the first kind. $\mathbf {e}_R=(\mathbf {e}_x-i\mathbf {e}_y)/\sqrt {2}$ and $\mathbf {e}_L=(\mathbf {e}_x+i\mathbf {e}_y)/\sqrt {2}$ represent the unit vector of right- and left-handed circular polarization states, respectively. Obviously, the focal field is a non-diffracting beam. This result is reliable when $|z|<\mathrm {LFWHM}/2$ and the LFWHM is $0.8743\lambda /\Delta \theta$ [29,30]. For example, if $\Delta \theta =0.01$rad, LFWHM is about 87$\lambda$.

Fig. 1. Schematic diagrams of an azimuthally polarized beam with vortex phase $\exp (i\varphi )$ generating a left-handed polarized beam. (a) in $xOz$ plane; (b) in $yOz$ plane.

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The TFWHM can be obtain by solving the equation $|\mathbf {E}(r,\phi ,z)|^2=|\mathbf {E}(0,0,z)|^2/2$ or

(3)$$\begin{aligned} (\alpha^2+\beta^2+2\gamma^2)J_0^2+(\alpha^2+\beta^2)J_2^2-2\alpha\beta\cos(\delta_{{-}1}-\delta_{1}-2\phi)J_0J_2 \\=(\alpha^2+\beta^2+2\gamma^2)/2, \end{aligned}$$

with

(4)$$\begin{aligned} \alpha & =|g_{{-}1}|/\sqrt{|g_{{-}1}|^2+|g_1|^2+|f|^2}\\ \beta & =|g_1|/\sqrt{|g_{{-}1}|^2+|g_1|^2+|f|^2}\\ \gamma & =|f|/\sqrt{|g_{{-}1}|^2+|g_1|^2+|f|^2}, \end{aligned},$$

where $\delta _{-1}$ and $\delta _{1}$ are the complex arguments of $g_{-1}$ and $g_1$, respectively. Equation 3 shows that TFWHMs will be dependent on $\phi$ unless $\alpha \beta =0$. Because of the interaction of the two azimuthally polarized components, the focal spot is not a circle, but like an ellipse. Hence, we use minimum (when $\delta _{-1}-\delta _{1}-2\phi =\pi$), median (when $\delta _{-1}-\delta _{1}-2\phi =\pi /2$) and maximum (when $\delta _{-1}-\delta _{1}-2\phi =0$) TFWHM to evaluate the size of focal spots. As shown in Fig. 2, their ranges are 0.32 $\sim$ 0.37$\lambda$, 0.36 $\sim$ 0.37$\lambda$ and 0.36 $\sim$ 0.43$\lambda$, respectively. When $\alpha ^2=\beta ^2=0.5$, the influence of the two azimuthally polarized components reaches the maximum, so the focal spot reanches the minimum value of minimum TFWHMs ($0.32\lambda$) and the maximum value of maximum TFWHMs ($0.43\lambda$). Median TFWHMs are only dependent on $\gamma$. The radially polarized component only generate zero-order Bessel component, while the azimuthally polarized components can both generate zero- and second-order Bessel component. Therefore, the median TFWHM increase as $\gamma$ decrease. These results are consistent with the previous paper [29].

Fig. 2. Minimum (a), median (b) and maximum (c) TFWHM as functions of $\alpha$ and $\beta$.

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The electrical field strength at the focus is $\mathbf {E}_0=\mathbf {E}(0,0,0)=-ig_{-1}\mathbf {e}_R+ig_1\mathbf {e}_L-\sqrt {2}f\mathbf {e}_z$ ($A(z)$ will be omitted in the following). To evaluate the polarized homogeneity near the focus quantitatively, we can define polarized homogeneity as

(5)$$\eta=\frac{\int_S|\mathbf{E}^*\cdot \frac{\mathbf{E}_0}{|\mathbf{E}_0|}|^2dS}{ \int_S\mathbf{E}^*\cdot \mathbf{E}dS},$$

where $S$ is the integral area in the plane which is parallel to the focal plane. Considering a circular area $r<R$, the polarized homogeneity can be simplified as

(6)$$\eta=\frac{\int_0^R [(\alpha^2+\beta^2+2\gamma^2)^2J_0^2+2\alpha^2\beta^2J_2^2] rdr}{\int_0^R[(\alpha^2+\beta^2+2\gamma^2)^2J_0^2+(\alpha^2+\beta^2+2\gamma^2)(\alpha^2+\beta^2)J_2^2] rdr.}$$

Equation (2) shows that the transverse and longitudinal distribution of the electric field are separated, so the polarized homogeneity is independent of $z$. Only when $\alpha =\beta =0$, $\eta =1$. This is because the azimuthally polarized beams with vortex phase $\exp (\pm i\varphi )$ cannot be transformed into left- or right-handed polarized beams completely. According to Eq. (6), the polarized homogeneity is shown in Fig. 3. It is always larger than 0.989 when $R=0.185\lambda$ and larger than 0.977 when $R=0.22\lambda$. It increases as $\gamma$ increases because of the incomplete transformation of the azimuthally polarized components. For a fixed $\gamma$, the maximum polarized homogeneity is at $\alpha =\beta$. This is because the bad transformation of two azimuthally polarized components can cancel partly.

Fig. 3. Polarized homogeneity as functions of $\alpha$ and $\beta$. (a) and (b) correspond to median and maximum TFWHM, respectively.

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Figure 4 shows that the polarized homogeneity decreases as $R$ increases when $R<0.6\lambda$. When $R<$TFWHM$/2$, the polarized homogeneity is larger than 0.99. However, it drops severely when $R>$TFWHM$/2$. This is because the second order Bessel function gradually dominate the field when $R$ increases. In the following, $R=0.22\lambda$ will be used to calculate the polarized homogeneity because it represents the maximum of maximum TFWHM.

Fig. 4. Polarized homogeneity as functions of $R$ when $\alpha =\beta =1/\sqrt {3}$.

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3. Examples

That $\mathbf {E}_0$ is linearly polarized is equivalent to $\mathbf {E}_0\times \mathbf {E}_0^*=\mathbf {0}$. Figure 5 is an example when $g_{-1}=g_1=-f$. The $x$-component of electric field is nearly zero. The $y$-component and $z$-component have a phase retardation of $\pi$. Hence, the focal field is linearly polarized in $yOz$ plane. The minimum, median and maximum TFWHM are 0.34, 0.36 and 0.39$\lambda$, respectively. The polarized homogeneity is 0.991 (R=0.22$\lambda$).

Fig. 5. Intensity and phase (insets) profiles on the focal plane when $g_{-1}=g_1=-f$.

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That $\mathbf {E}_0$ is circularly polarized is equivalent to $|\mathbf {E}_0\times \mathbf {E}_0^*|=\mathbf {E}_0\cdot \mathbf {E}_0^*$. Figure 6 is an example when $g_{-1}=g_1=-if$. Compared with last example, the azimuthally polarized components have phases of $\pi /2$. The intensity profiles in Fig. 6 are the same as Fig. 5, while the phase retardation between y-component and z-component of the focal field is $\pi /2$. Hence, the focal field is circularly polarized in $yOz$ plane. Fig. 7 is an example of elliptical polarization with $f=ig_{-1}=-g_1/\sqrt 3$. In order to display the polarization clearly, we use an orthogonal transformation:

(7)$$\begin{aligned} \left[\begin{array}{ccc}E_{x'} \\ E_{y'} \\E_{z'} \end{array}\right]= \left[\begin{array}{ccc} \frac{\sqrt{3}}{\sqrt{10}}\ \frac{-2}{\sqrt{10}} \frac{\sqrt{3}}{\sqrt{10}} \\ \frac{1}{\sqrt{2}} \ \frac{0}{\sqrt{2}} \ \frac{-1}{\sqrt{2}} \\\frac{1}{\sqrt{5}}\ \frac{\sqrt{3}}{\sqrt{5}}\ \frac{1}{\sqrt{5}} \end{array}\right]\left[\begin{array}{ccc}E_x \\ E_y \\E_z \end{array}\right]. \end{aligned}$$

Fig. 6. Intensity and phase (insets) profiles on the focal plane when $g_{-1}=g_1=-if$.

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Fig. 7. Intensity and phase (insets) profiles on the focal plane when $f=ig_{-1}=-g_1/\sqrt 3$.

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As shown in Fig. 7, $|E_{z'}|^2$ is nearly zero. The phase retardation between $x'$-component and $y'$-component are about $\pi /2$, but their amplitudes are different. So the focal field is elliptically polarized. Figures 8(a) and (b) display the modulus and argument of $E_{x'}/E_{y'}$ on $x-$axis and $y-$axis. They can help to conclude that the polarization is elliptic. The minimum, median and maximum TFWHM are 0.34, 0.37 and 0.40$\lambda$, respectively. The polarized homogeneity is 0.988 (R=0.22$\lambda$).

Fig. 8. The modulus and argument of $E_{x'}/E_{y'}$ on $x-$axis and $y-$axis when $f=ig_{-1}=-g_1/\sqrt 3$.

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4. Conclusion

In conclusion, we presented an approach to generate arbitrarily 3D polarized optical needles by tightly focusing vector beams. The size of the focal spots is 0.32 $\sim$ 0.44$\lambda$. The polarized homogeneity is always large than 0.97. Even though a parabolic mirror is used in this paper, with a little modification, the results are also suitable for any aplanatic focusing systems. With a 4$\pi$ confocal system, this method can be also used to generate optical chains with arbitrary 3D polarization.

The authors declare no conflicts of interest.

Appendices

For an arbitrary $\theta _0$, the focal field is

(8)$$\begin{aligned} \mathbf{E}(r,\phi,z)\approx A(z) \left[\begin{array}{ccc} i(\cos \theta_0fe^{i\phi}J_1-g_{{-}1}J_{0}+g_1e^{2i\phi}J_{2})\mathbf{e}_R \\ i(\cos\theta_0fe^{{-}i\phi}J_1+g_1J_{0}-g_{{-}1}e^{{-}2i\phi}J_{2})\mathbf{e}_{L} \\-\sqrt{2} \sin\theta_0fJ_{0}\mathbf{e}_z \end{array}\right], \end{aligned}$$

with

(9)$$A(z)\propto \frac{\sin(kz\sin\theta_0\sin \frac{\Delta\theta}{2})\exp(ikz\cos\theta_0\cos\frac{\Delta\theta}{2})}{kz},$$

where $J_n=J_n(kr\sin \theta _0)$. Its LFWHM is $\frac {0.8743\lambda }{\sin \theta _0\Delta \theta }$ [29,30]. $\mathbf {E}_0=-ig_{-1}\mathbf {e}_R+ig_1\mathbf {e}_L-\sqrt {2}\sin \theta _0\mathbf {e}_z$ and the polarized homogeneity is

(10)$$\eta=\frac{\int_0^R [(\alpha^2+\beta^2+2\sin^2\theta_0\gamma^2)^2J_0^2+\cos^2\theta_0\gamma^2(\alpha^2+\beta^2)J_1^2+2\alpha^2\beta^2J_2^2] rdr}{(\alpha^2+\beta^2+2\gamma^2)\int_0^R[(\alpha^2+\beta^2+2\sin^2\theta_0\gamma^2)J_0^2+2\cos^2\theta_0\gamma^2J_1^2+(\alpha^2+\beta^2)J_2^2] rdr}$$

When $\theta _0\neq \pi /2$, the incident radially polarized beam cannot be transformed into longitudinally polarized beam completely. Hence, the polarized homogeneity $\eta$ is always less than 1.

If a confocal system is used (Fig. 9), this method can generate optical chains. Figure 10 and 11 are an example when $\mathbf {E}_0 \propto (i\mathbf {e}_x-\mathbf {e}_y+\mathbf {e}_z)$. Figure 10(a)$\sim$(c) are intensity and phase profiles of three Cartesian components of the focal field, in which we use an orthogonal transformation:

(11)$$\begin{aligned} \left[\begin{array}{ccc}E_{x'} \\ E_{y'} \\E_{z'} \end{array}\right]= \left[\begin{array}{ccc} 0\ \frac{-1}{\sqrt{2}} \frac{1}{\sqrt{2}} \\ 1 \quad 0 \quad 0 \\0\ \frac{1}{\sqrt{2}}\ \frac{1}{\sqrt{2}} \end{array}\right]\left[\begin{array}{ccc}E_x \\ E_y \\E_z \end{array}\right]. \end{aligned}$$

Fig. 10(d)$\sim$(f) are total intensity profiles in three orthogonal planes. The FWHM is about 0.44$\lambda$, which is samller than the reults in [20–22]. Figure 11 shows an intensity profiles of the optical chain in $xOz$ plane. More details about optical chains can be seen in [29].

Fig. 9. A confocal system. Reprinted with permission from Ref. [29], OSA Publishing.

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Fig. 10. Intensity and phase (insets) profiles near the focus.

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Fig. 11. Intensity profiles of an optical chain in $xOz$ plane.

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Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Optical needles with arbitrary homogeneous three-dimensional polarization

Abstract

1. Introduction

2. Theory and configuration

3. Examples

4. Conclusion

Appendices

Disclosures

References

Cited By

Figures (11)

Equations (11)

Optics Express