Rotational Doppler Effect of Vector Beams

Abstract

The optical rotational Doppler effect occurs when vortex beams are scattered by rotating objects and demonstrate the Doppler frequency shifts in scattered beams, which are associated with the optical angular momentum of vortex beams and the relative rotating angular velocity. Here, we investigate the rotational Doppler effect of a rotating vector beam in its tight focusing and find that similar Doppler frequency shifts arise when the polarization order and vortex charge satisfy some specific relations. We note that the vortex charge is the indispensable parameter in the rotational Doppler effect of the vector vortex beam. Nonetheless, the sign and magnitude of the frequency shift are only determined by the angular frequency of the rotating beam and are irrelevant to the vortex charge of the beam. In addition, the on-axis energy flow and spin angular momentum in the focal plane can also be well modulated, accompanying the rotational Doppler effect of the vector vortex beam. Our results may be applied to an optical micro-manipulation, especially for some kinds of microparticles which are sensitive to frequency changes.

1. Introduction

The Doppler effect is a classic wave phenomenon and displays a frequency shift when the relative motion occurs between the wave source and the observer. The frequency shift is proportional to the unshifted sources’ frequency and the relative velocity, and it has been extensively applied in detecting the translational velocity of surfaces and fluids [1,2]. A similar frequency shift also can be obtained when vortex beams are scattered by rotating objects [3,4,5,6,7,8,9,10,11,12,13,14], and this is associated with the optical angular momentum of vortex beams and the relative rotating angular velocity. This is called the rotational Doppler effect of the optical beam. In recent years, more and more research has studied the rotational Doppler effect because of its potential applications in the field of angular velocity measurement [3,6,7,12,13,14], fluid flow vorticity measurement [15,16], optical sensors [17], nonlinear optical systems [18,19], etc. The study of the rotational Doppler effect has also been extended to different waves, such as partially coherent beams [20] and sound waves [21]. Meanwhile, because the rotation of the beam has a close relation with the optical angular momentum [22,23], the Doppler frequency shift also can be found in some effects induced by the spin–orbit coupling of light [24,25]. All these studies indicate that the intrinsic structures of the beams are closely related to the rotational Doppler effect.

The vector beam is a kind of structured optical beam with the spatially variant states of polarization, such as radially (or azimuthally) polarized beam [26], hybridly polarized beams [27,28], full Poincare beams [29,30,31], etc. It is known that the polarization distribution of vector beams determines their spin angular momentum [32,33], and the curl of the polarization distribution can induce a kind of intrinsic orbital angular momentum of optical beams [28]. It is known that the Doppler frequency shift is determined by the optical angular momentum of the beams and the relative rotation [3,6]. Then, what are the manifestations of the rotational Doppler effect of vector beams? Can the polarization distribution of vector beams affect the rotational Doppler effect?

Here, we investigate the tight focusing of rotating monochromatic radially polarized vortex beams and find that the frequency shifts in the field components occur on the optical axis in the focal plane when the polarization order and vortex charge satisfy some specific relations. The sign and magnitude of the frequency shift are determined by the angular frequency of the rotating beam about its optical axis. It can be considered a manifestation of the rotational Doppler effect of vector vortex beams. Just like the traditional Doppler effect of vortex beams, we note that the vortex charge is the indispensable parameter in the rotational Doppler effect of vector beams. In addition, by changing the polarization order and vortex charge of the incident beams, the on-axis energy flow and spin angular momentum can be well modulated. Namely, the multi-modulation of the frequency shift, energy flow, and spin angular momentum can be realized in the tight focusing of vector vortex beams. Our results provide a novel technique for modulating the focal field and may be applied to the field of optical micro-manipulation, especially for some kinds of microparticles which are sensitive to frequency changes.

2. Basic Theory

Let us consider the description of rotating monochromatic radially polarized vortex beams. The rotation of an optical beam introduces a relative rotation between the incident beams and the observer, and it can be described by a rotation matrix, as follows:

$$R(t)=\left[\begin{array}{cc}\cos (\Omega t)& \sin (\Omega t)\\ -\sin (\Omega t)& \cos (\Omega t)\end{array}\right],$$

(1)

where $\Omega$ is the angular frequency of the rotating beam about its optical axis. The polarization distribution of the radially polarized vortex beam is described by the vectorial coefficient matrix ${[c_x(\varphi ),c_y(\varphi )]}^T={[\cos (m\varphi ),\sin (m\varphi )]}^T$, where $m$ is the polarization order of the vector beam. Then, the rotation of the polarization distribution can be described by $R(t){[c_x(\varphi ),c_y(\varphi )]}^T$, as shown in Figure 1. This means that there is a whole polarization state distribution rotation of the light beam when the rotation matrix is introduced; this leads to the whole rotation of the focal field, but the focal field property cannot be changed.

The tight focusing of vector beams can be analyzed with the Richards–Wolf diffraction integral, which takes the following form [34]:

$$\left[\begin{array}{c}E(r,\phi ,z)\\ H(r,\phi ,z)\end{array}\right]=-{\displaystyle \frac{if}{\lambda }}{\displaystyle {\int }_0^{\alpha }}{\displaystyle {\int }_0^{2\pi }T(\theta )}F(\theta ,\varphi )\left[\begin{array}{c}{P}_E(\theta ,\varphi )\\ P_H(\theta ,\varphi )\end{array}\right]Exp\left\{ik\left[r\sin \theta \cos \left(\varphi -\phi \right)+z\cos \theta \right]\right\}\sin \theta d\theta d\varphi ,$$

(2)

where

$$P_E(\theta ,\varphi )=\left[\begin{array}{cc}A(\theta ,\varphi )& C(\theta ,\varphi )\\ C(\theta ,\varphi )& B(\theta ,\varphi )\\ -D(\theta ,\varphi )& -E(\theta ,\varphi )\end{array}\right]R(t)\left[\begin{array}{c}{c}_x(\varphi )\\ c_y(\varphi )\end{array}\right],P_H(\theta ,\varphi )=\left[\begin{array}{cc}C(\theta ,\varphi )& -A(\theta ,\varphi )\\ B(\theta ,\varphi )& -C(\theta ,\varphi )\\ -E(\theta ,\varphi )& D(\theta ,\varphi )\end{array}\right]R(t)\left[\begin{array}{c}{c}_x(\varphi )\\ c_y(\varphi )\end{array}\right],$$

and

$$A(\theta ,\varphi )=1+{\cos }^2\varphi (\cos \theta -1),B(\theta ,\varphi )=1+{\sin }^2\varphi (\cos \theta -1),$$

$$C(\theta ,\varphi )=\sin \varphi \cos \varphi (\cos \theta -1),D(\theta ,\varphi )=\cos \varphi \sin \theta ,E(\theta ,\varphi )=\sin \varphi \sin \theta ,$$

where $T(\theta )=\sqrt{\cos (\theta )}$ is the apodization function; $F(\theta ,\varphi )$ is the complex amplitude of the incident field. Obviously, the rotation of the radially polarized vortex beam has been introduced into the Richards–Wolf diffraction integral. We consider the amplitude $F(\theta ,\varphi )$ of the incident field to be concentrated in a narrow annular region with central angle ${\theta }_0$ and width $\Delta \theta$, where $\Delta \theta$ is a small parameter. For simplicity, we take the complex amplitude $F({\theta }_0,\varphi )=\exp (il\varphi )$ in our numerical calculation, where $l$ is the vortex charge of the beam.

According to the Richards–Wolf diffraction integral, the components of the tightly focused vector vortex beam are obtained directly as the following expressions.

$$\begin{array}{c}{E}_x(r,\phi ,z)={\displaystyle \frac{1}{2}}\pi (\cos {\theta }_0+1)p(z)\{i^{l-m}{J}_{l-m}(kr\sin {\theta }_0)\exp [i(l-m)\phi ]\exp (-i\Omega t)\hfill \\ +i^{l+m}{J}_{l+m}(kr\sin {\theta }_0)\exp [i(l+m)\phi ]\exp (i\Omega t)\}\hfill \\ +{\displaystyle \frac{1}{2}}\pi (\cos {\theta }_0-1)p(z)\{i^{l-m+2}{J}_{l-m+2}(kr\sin {\theta }_0)\exp [i(l-m+2)\phi ]\exp (-i\Omega t)\hfill \\ +i^{l+m-2}{J}_{l+m-2}(kr\sin {\theta }_0)\exp [i(l+m-2)\phi ]\exp (i\Omega t)\},\hfill \end{array}$$

(3)

$$\begin{array}{c}{E}_y(r,\phi ,z)={\displaystyle \frac{i}{2}}\pi (\cos {\theta }_0+1)p(z)\{i^{l-m}{J}_{l-m}(kr\sin {\theta }_0)\exp [i(l-m)\phi ]\exp (-i\Omega t)\hfill \\ -i^{l+m}{J}_{l+m}(kr\sin {\theta }_0)\exp [i(l+m)\phi ]\exp (i\Omega t)\}\hfill \\ -{\displaystyle \frac{i}{2}}\pi (\cos {\theta }_0-1)p(z)\{i^{l-m+2}{J}_{l-m+2}(kr\sin {\theta }_0)\exp [i(l-m+2)\phi ]\exp (-i\Omega t)\hfill \\ -i^{l+m-2}{J}_{l+m-2}(kr\sin {\theta }_0)\exp [i(l+m-2)\phi ]\exp (i\Omega t)\},\hfill \end{array}$$

(4)

$$\begin{array}{c}{E}_z(r,\phi ,z)=-\pi \sin {\theta }_{0}p(z)\{i^{l-m+1}{J}_{l-m+1}(kr\sin {\theta }_0)\exp [i(l-m+1)\phi ]\exp (-i\Omega t)\hfill \\ +i^{l+m-1}{J}_{l+m-1}(kr\sin {\theta }_0)\exp [i(l+m-1)\phi ]\exp (i\Omega t)\},\end{array}$$

(5)

$$\begin{array}{c}{H}_x(r,\phi ,z)=-{\displaystyle \frac{i}{2}}\pi (\cos {\theta }_0+1)p(z)\{i^{l-m}{J}_{l-m}(kr\sin {\theta }_0)\exp [i(l-m)\phi ]\exp (-i\Omega t)\hfill \\ -i^{l+m}{J}_{l+m}(kr\sin {\theta }_0)\exp [i(l+m)\phi ]\exp (i\Omega t)\}\hfill \\ -{\displaystyle \frac{i}{2}}\pi (\cos {\theta }_0-1)p(z)\{i^{l-m+2}{J}_{l-m+2}(kr\sin {\theta }_0)\exp [i(l-m+2)\phi ]\exp (-i\Omega t)\hfill \\ -i^{l+m-2}{J}_{l+m-2}(kr\sin {\theta }_0)\exp [i(l+m-2)\phi ]\exp (i\Omega t)\},\hfill \end{array}$$

(6)

$$\begin{array}{c}{H}_y(r,\phi ,z)={\displaystyle \frac{1}{2}}\pi (\cos {\theta }_0+1)p(z)\{i^{l-m}{J}_{l-m}(kr\sin {\theta }_0)\exp [i(l-m)\phi ]\exp (-i\Omega t)\hfill \\ +i^{l+m}{J}_{l+m}(kr\sin {\theta }_0)\exp [i(l+m)\phi ]\exp (i\Omega t)\}\hfill \\ -{\displaystyle \frac{1}{2}}\pi (\cos {\theta }_0-1)p(z)\{i^{l-m+2}{J}_{l-m+2}(kr\sin {\theta }_0)\exp [i(l-m+2)\phi ]\exp (-i\Omega t)\hfill \\ +i^{l+m-2}{J}_{l+m-2}(kr\sin {\theta }_0)\exp [i(l+m-2)\phi ]\exp (i\Omega t)\},\hfill \end{array}$$

(7)

$$\begin{array}{c}{H}_z(r,\phi ,z)=i\pi \sin {\theta }_{0}p(z)\{i^{l-m+1}{J}_{l-m+1}(kr\sin {\theta }_0)\exp [i(l-m+1)\phi ]\exp (-i\Omega t)\\ -i^{l+m-1}{J}_{l+m-1}(kr\sin {\theta }_0)\exp [i(l+m-1)\phi ]\exp (i\Omega t)\},\end{array}$$

(8)

where $p(z)=-if\sqrt{\cos {\theta }_0}\sin {\theta }_0\exp (ikz\sin {\theta }_0)/\lambda$, $f$ is the focal length, and $\lambda$ is the incident wavelength. We can see that the focal field is a kind of polychromatic wave field and is the superposition of the wave components with the different vortex charge and frequency shift. The occurrence of a frequency shift can be considered as a manifestation of the rotational Doppler effect of the vector beam. From the calculation process of the focal field components, we know that the vortex charge is the indispensable parameter in the rotational Doppler effect of the vector vortex beam; the Doppler frequency shift will vanish if the vortex charge is zero. Next, we will investigate the rotational Doppler effect of the vector beam and some accompanying modulation effects of on-axis energy flow and spin angular momentum in the focal plane in detail.

3. Analyses and Discussion

Generally, the longitudinal component of a focal field cannot be neglected in the tight focusing of vector beams [26]. In previous works [3,4,5,6,7,8,9,10,11,12,13,14], researchers focused on the rotational Doppler effect of scalar vortex beams, and the frequency shift in the field components is identical. We note that when the polarization order and vortex charge of the incident beam take some specific values, the monochromatic focal field with the Doppler frequency shift can be obtained on the optical axis in the focal plane; a schematic is shown in Figure 2. More interestingly, the Doppler frequency shift in the transverse and longitudinal components of the focal field is different when the polarization order and vortex charge satisfy the different relations, as shown in

Table 1. Frequency shift in focal field components on the optical axis in the focal plane.

	$l-m=-2$	$l-m=-1$	$l-m=0$	$l+m=0$	$l+m=1$	$l+m=2$
Transverse components $E_x$$,E_y$$,H_x$$,H_y$	$-\Omega$	$0$	$-\Omega$	$\Omega$	$0$	$\Omega$
Longitudinal components $E_z$$,H_z$	$0$	$-\Omega$	$0$	$0$	$\Omega$	$0$

. This shows that the frequency shifts are determined by the angular frequency and the relation of the polarization order and vortex charge of the incident beam. Though the vortex charge of the beams is irrelevant to the magnitude of the Doppler frequency shift, it cannot take the value of zero ($l\ne 0$). This is an interesting phenomenon and is completely distinct from the traditional rotational Doppler effect of the scalar vortex beam.

According to the Richards–Wolf diffraction integral, we know that if the vortex charge is zero, the Doppler frequency shift vanishes in the focal field. This means that the vortex charge is an indispensable parameter in the rotational Doppler effect of vector beams. We know that when the polarization order and vortex charge satisfy some specific relations, the frequency shift in the monochromatic transverse (or longitudinal) component of the focal field occurs on the optical axis in the focal plane. For instance, when $l-m=-2$, according to the expressions of the focal field components (Equations (3)–(8)), the electric and magnetic field components on the optical axis in the focal plane can be obtained directly,

$$E_x{(r=0,\phi ,z)}_{l-m=-2}={\displaystyle \frac{1}{2}}\pi (\cos {\theta }_0-1)p(z)\exp (-i\Omega t),$$

(9)

$$E_y{(r=0,\phi ,z)}_{l-m=-2}=-{\displaystyle \frac{i}{2}}\pi (\cos {\theta }_0-1)p(z)\exp (-i\Omega t),$$

(10)

$$E_z{(r=0,\phi ,z)}_{l-m=-2}=0,$$

(11)

$$H_x{(r=0,\phi ,z)}_{l-m=-2}=-{\displaystyle \frac{i}{2}}\pi (\cos {\theta }_0-1)p(z)\exp (-i\Omega t),$$

(12)

$$H_y{(r=0,\phi ,z)}_{l-m=-2}=-{\displaystyle \frac{1}{2}}\pi (\cos {\theta }_0-1)p(z)\exp (-i\Omega t),$$

(13)

$$H_z{(r=0,\phi ,z)}_{l-m=-2}=0.$$

(14)

This shows that the electric and magnetic fields have the same frequency shift, which is described by the term $\exp (-i\Omega t)$. A similar phenomenon can be achieved when the polarization order and vortex charge satisfy some specific relations, as shown in Table 1. Yet, when the polarization order and vortex charge satisfy both relations $l-m=-2$ and $l+m=0$ ($l=-1$, $m=1$), or $l+m=2$ and $l-m=0$ ($l=1$, $m=1$), the transverse components on the optical axis in the focal plane are still polychromatic and a superposition of two wave components with the different frequency shift.

It is also known that when the polarization order (or vortex charge) takes some specific value [35,36,37,38], the on-axis negative energy flow can be observed in the focal plane of the tightly focused radially polarized (or vortex) beams. Does the rotational Doppler effect of vector beams affect the evolution of the on-axis negative energy flow in the focal region? Generally, because of the rotational symmetry of the polarization distribution of radially polarized vortex beam, the rotation of the vector beam does not affect the evolution of the focal field. Yet, the different polarization order ($m$) and vortex charge ($l$) determine the occurrence of the Doppler frequency shift and the on-axis negative energy flow. Here, we focus on the evolution of the longitudinal component of the energy flow in the focal plane. According to the definition of the time-averaged energy flow density $S=(c/8\pi )\mathrm{Re}(E^{*}\times H)$, the longitudinal component of the energy flow density in the focal plane can be obtained easily. Figure 3 and Figure 4a show the longitudinal component of the energy flow density in the focal plane when the polarization order ($m$) and vortex charge ($l$) satisfy the relations $l-m=-2$ and $l+m=2$, respectively. The calculation parameters are taken as ${\theta }_0={80}^{\circ }$, the focal length is as $f=120\lambda$, and $\lambda$ is the wavelength of incident light. The same calculation parameters are used throughout this paper, and for the sake of simplicity, we take $\lambda =1$ in the numerical simulation. We know that the same on-axis negative energy flow can be obtained in both cases. According to the results shown in Table 1, when the polarization order ($m$) and vortex charge ($l$) satisfy the relations $l-m=-2$ and $l+m=2$, the frequency shifts in the transverse components of the on-axis focal field are different: one is $-\Omega$, and the other is $\Omega$. This shows that the same on-axis negative energy flow can possess a different frequency shift when the polarization order ($m$) and vortex charge ($l$) satisfy the different relations.

When the polarization order ($m$) and vortex charge ($l$) do not satisfy the relation $l-m=-2$ (or $l+m=2$), the on-axis energy flow in the focal plane is positive or zero, as shown in Figure 4b. We note that $l=-1$ and $m=1$ ($l=1$ and $m=1$) satisfy both relations $l-m=-2$ and $l+m=0$ ($l+m=2$ and $l-m=0$), meaning that the on-axis energy flow is positive. This indicates that by changing the polarization order ($m$) and vortex charge ($l$), different on-axis energy flow (negative, positive, or zero) can be obtained, and a different Doppler frequency shift can be obtained on the optical axis in the focal plane simultaneously. This provides a potential technique to modulate some optical micro-manipulation effects, which are sensitive to frequency changes.

The conversion of the orbital angular momentum to the spin angular momentum is an important issue in the tight focusing of light beams [39,40]. In the field of microparticle manipulation, the optical angular momentum of a light beam can transfer to the microparticle, leading the microparticle to spin around its own axis or orbit around the axis of the focal field [41,42]. It is known that the spin angular momentum can be transferred into the orbital angular momentum in the tight focusing of the circularly polarized beams [39,40]. Because the incident radially polarized vortex beam possesses the intrinsic orbital angular momentum, the rotational Doppler effect should be closely related to the conversion of the orbital to the spin angular momentum on the optical axis in the focal region.

According to the expressions of the focal field components and the definition of the time-averaged spin angular momentum density $J_s=\mathrm{Im}\left(\epsilon E^{*}\times E+\mu H^{*}\times H\right)$, the spin angular momentum distribution in the focal plane can be obtained easily. From the results shown in Table 1, when the polarization order ($m$) and vortex charge ($l$) satisfy the relations $l-m=-1$ and $l+m=1$, we know that the Doppler frequency shift just exists in the longitudinal component of the focal field, and then the transverse spin angular momentum is zero on the optical axis in the focal plane. In order to realize the multi-modulation of the frequency shift, energy flow, and spin angular momentum in the tight focusing of vector vortex beams, we focus on purely the evolution of the longitudinal spin angular momentum on the optical axis in the focal plane.

When the polarization order ($m$) and vortex charge ($l$) satisfy the relations $l-m=-2$ and $l+m=2$ (or $l-m=0$ and $l+m=0$), the on-axis energy flow is negative (or positive), and the Doppler frequency shifts in the transverse components of the focal field are different. Figure 5 shows that when the polarization order ($m$) and vortex charge ($l$) satisfy some specific relations, the on-axis spin angular momentum density can also be negative or positive. In order to ensure the occurrence of the Doppler frequency shift, the vortex charge ($l$) cannot take the value of zero ($l\ne 0$); the incident vector vortex beam possesses the intrinsic orbital angular momentum. One knows that the optical angular momentum is conserved in the tight focusing of the light beam. This means that the conversion of orbital to spin angular momentum on the optical axis in the focal plane can be controlled by changing the polarization order ($m$) and vortex charge ($l$) of incident beams. This implies that the multi-modulation of the Doppler frequency shift, on-axis energy flow, and spin angular momentum can be realized in the tight focusing of vector vortex beams.

4. Conclusions

We have investigated the rotational Doppler effect of a vector vortex beam in terms of its tight focusing. Generally, because of the rotational symmetry of the beams, the polarization distribution rotation of the radially polarized vortex beam does not affect the intensity distribution in the focal plane. Yet, we found that when the polarization order ($m$) and vortex charge ($l$) satisfy some specific relations, a Doppler frequency shift in the field components occurs, and this is determined by the angular frequency of the rotating beam. This is a manifestation of the rotational Doppler effect of the vector beam. We noted that though the vortex charge is irrelevant to the magnitude of the Doppler frequency shift, it is still an indispensable parameter because the rotational Doppler effect will vanish if the vortex charge is zero. We know that the polarization order ($m$) and vortex charge ($l$) do not satisfy the relations shown in Table 1; the field components will display a superposition form with two different frequency shifts ($\exp (i\Omega t)$ and $\exp (-i\Omega t)$), which can be considered the noise of the Doppler frequency shift. Simultaneously, the on-axis energy flow and spin angular momentum can also be well modulated by accompanying the occurrence of the Doppler frequency shift. Namely, by changing the polarization order and vortex charge, the multi-modulation of the Doppler frequency shift, on-axis energy flow, and spin angular momentum can be realized in the tight focusing of vector vortex beams. Compared with the frequency of visible light, though the Doppler frequency shift $\Omega$ is of a very small quantity, the joint modulation of the frequency shift, the on-axis energy flow, and spin angular momentum has some potential applications in the field of optical detection; this provides a technique to potentially improve some optical micro-manipulation effects, especially for frequency-sensitive particles.