1. Introduction

It has long been recognized, dating back to Poynting [1] and Beth [2], that circularly polarized light carries spin angular mometum (SAM) associated with the spin of photons. Much later, in 1992 Allen et al. [3] made the breakthrough observation that Laguerre-Gaussian (LG) modes, and more generally light beams with an azimuthal phase dependence of exp(imϕ) [m = 0,±1,…], possess orbital angular momentum (OAM) in the overall propagating direction (z-direction). Following Allen’s discovery, the OAM of light has rapidly developed into an exciting research area [4,5]. Besides bringing new insight into our fundamental understanding of light, the optical OAM has found potential applications in optical tweezers and spanners [6,7], optical communications [8,9], and quantum information processing [10,11].

Since Allen’s poineering work, most studies in this area have focused on traveling beams, and much less attention has been paid to the OAM of evanescent waves. Evanescent optical fields can be employed to actively manipulate small particles near a surface [12,13] or to provide optomechanical coupling between light and mechanical objects via gradient forces [14,15], and the added dimension of OAM is expected to introduce new aspects to these applications. While evanescent light with azimuthal phase dependence has been investigated in the context of surface optical vortex [16,17], the focus was mainly on its phase and intensity properties. A quantitative study on the evanescent optical OAM is still lacking.

In this paper, we present a detailed analysis of the optical OAM in evanescent Bessel fields, which have transverse (x–y plane) spatial profiles identical to those of traveling Bessel beams, but decay exponentially in the z-direction [18–20]. We derive an analytic formula for the OAM density of the evanescent Bessel wave, and show that the paraxial part of the OAM vanishes when integrated over the azimuthal angle ϕ, which is in stark contrast to the traveling-wave case. Even more remarkably, for an evanescent mth order Bessel field, the OAM per unit energy is ${\left(1+\frac{{\kappa }^2}{k^2}\right)}^{-1}\frac{m}{\omega }$, reduced by a factor of $\left(1+\frac{{\kappa }^2}{k^2}\right)$ from the standard result $\frac{m}{\omega }$ for the corresponding traveling beam (here κ is the amplitude decay rate of the evanescent light, and ω, k are the angular frequency and the wave number, respectively).

2. Linear momentum of evanecsent fields

It is well known that an evanescent field does not carry linear momentum in the z-direction (along which the field decays). However, if the azimuthal component of the linear momentum exists, then the field possesses angular momentum in the z-direction. From now on, unless explicitly stated otherwise, “angular momentum” refers to its z-component. In this section, we derive a general formula for the linear momentum of evanescent light, which will not only suggest some remarkable difference between the angular momenta of the evenescent and the traveling beams, but also serve as a basis for further study on the OAM of evanescent Bessel waves in the next section.

We choose to work in the Lorentz gauge and consider a monochromatic field polarized in the x-y plane [21]

With A(r) = A(r) (αe_x +βe_y) and A (r) being a solution of Eq. (2), it is easy to show that both the vector and the scalar potentials obey the wave equations (i.e., the Maxwell’s equations in the Lorentz gauge)

Thus the assumption on the polarization of the vector potential in Eq. (1) is entirely consistent.

For an evanescent wave, we take

Substituting Eq. (3) into Eq. (4) yields

The linear momentum density of the field is determined by

Note that we have not made the paraxial approximation, and thus the non-paraxial terms (i.e., products containing high-order derivatives or more than one first-order derivatives) are retained in Eq. (8). This is necessary because for the evanescent field, the variation in the transverse plane is usually comparable to or even larger than that in the z-direction. As expected, the integral of p_z (z component of the linear momentum) over x and y vanishes due to the total-derivative feature of each term in p_z. However, our main interest is in the transverse (x-y plane) part p_⊥, particularly the azimuthal component pϕ, which is related to the angular momentum density by j_z = ρ pϕ, with ρ as the radius of the polar coordinates.

To conclude this section, we make a comparison between the linear momentum density p in Eq. (8) and that of the traveling field A_tra (r,t) = u(x,y)e^ikz (αe_x + βe_y)e^−iωt [the subscript “tra” means “traveling”]

3. OAM of Evanescent Bessel Waves

Equation (8), obtained without specifying the transverse spatial profile u(x,y) in Eq. (3), gives the linear momentum density p of an evanescent field, which in turn determines the angular momentum density (in the z-direction) via j_z = ρ pϕ. In this section, we specifically investigate the angular momentum of the evanescent Bessel wave, with u(x,y) in Eq. (3) as (converted to the polar-coordinate system)

We mainly focus on the OAM, although the SAM is also briefly discussed at the end of the section. For convenience, we will frequently adopt the simplified notation a(ρ) = J_m (μρ) in the analysis.

For the study of OAM, we only consider linear polarization (so that SAM does not come into play), which is characterized by real αβ* in Eq. (1). Without loss of generality, we take both α and β to be real. Substituting Eq. (10) into Eq. (8) and using the polar-coordinate differential operators in Eq. (25), we derive the azimuthal component of the linear momentum density for the evanescent Bessel wave (see the Appendix for more details)

The first line in the square bracket on the right-hand side of Eq. (11) comes from the paraxial terms in Eq. (8), and the second line from the non-paraxial ones. The OAM density (in the z-direction) is achieved by multiplying p_ϕ with the radius ρ

For comparison purpose, we also calculate the OAM density from Eq. (9) for the traveling Bessel wave, with u(x,y) and α, β the same as in Eq. (8),

The most remarkable feature of the evanescent Bessel wave OAM, according to Eq. (13), is that the paraxial part vanishes when integrated over the azimuthal angle. This is drastically different from the traveling wave OAM in Eq. (14), where the paraxial contribution is typically the most significant.

A result of essential importance in the optical OAM theory is that the OAM per unit energy in a linearly-polarized traveling beam with azimuthal phase dependence exp(imϕ) [but otherwise rotationally symmetric in the transverse plane] obeys the simple relation [23]

Equation (15) is a well-known formula, and we can confirm it specifically for the traveling Bessel beam [following a procedure similar to the derivation of Eq. (19) below] but will not show the details here. However, for an evanescent wave this relation is violated. Particularly, for the evanescent mth order Bessel wave, the OAM-energy ratio is reduced by a factor of $\left(1+\frac{{\kappa }^2}{k^2}\right)$, i.e.,

To prove Eq. (19), we recall the explicit form of a(ρ), i.e., J_m (μρ) with $\mu =\sqrt{k^2+{\kappa }^2}$. We only consider m ≠ 0 because the validity of Eq. (19) is obvious for m = 0. First we insert Eq. (13) into the evanescent-wave counterpart of Eq. (16) and integrate over the azimuthal angle to obtain

We need to point out that for an ideal Bessel function, the integrals in both Eqs. (20) and (21) diverge. Thus in the definition of ${\mathcal{J}}_z$ and ℰ, we have implicitly assumed a cutoff at a sufficiently large radius ρ_cut [i.e., a(ρ > ρ_cut) = 0] such that the essential features of the Bessel distribution are preserved while the integrals are kept finite.

To go further, we note the asymptotic behavior of the Bessel function, i.e., as ρ → ∞

It follows that in the same limit

Since $\frac{m^2}{\rho }{a}^2(\rho )\le \frac{2m^2}{\pi \mu {\rho }^2}$ for ρ → ∞ and $\frac{m^2}{\rho }{a}^2(\rho )\to 0$ for ρ → 0 [due to $J_m(s\to 0)\to {\left(-\frac{s}{2}\right)}^{|m|}$], the integral of $\frac{m^2}{\rho }{a}^2(\rho )$ over ρ is finite. On the other hand, the other two terms in Eq. (22) become periodic (rather than vanish) for ρ → ∞ and the integrals of them actually diverge. As mentioned above, we cut off the integrals at some sufficiently large radius. The cutoff radius ρ_cut should be many periods after the Bessel function has settled into the asymptotic cosine function, such that (i) compared to the other terms, the integrals of $\frac{m^2}{\rho }{a}^2(\rho )$ in ${\mathcal{J}}_z$ and ℰ are negligible, (ii) the integrals of ρa² (ρ) and ρa′² (ρ) are dominated by contributions from the asymptotic region, and (iii) the integrals of ${\sin }^2\left(\mu \rho -\frac{\pi }{4}-\frac{m\pi }{2}\right)$ and ${\cos }^2\left(\mu \rho -\frac{\pi }{4}-\frac{m\pi }{2}\right)$ in the asymptotic region are essentially equivalent. With such arrangement, we simplify Eqs. (20) and (21) into

Thus, the OAM per unit energy of the evenescent Bessel field is reduced by a factor of $\left(1+\frac{{\kappa }^2}{k^2}\right)$ compared to that of the traveling Bessel beam. For κ = 0, which is the “transitional point” between the evanescent and the traveling waves, Eqs. (19) and (15) are identical. As κ gets larger, the OAM-energy ratio becomes smaller, suggesting that an evanescent wave more tightly localized to the surface possesses less OAM. This qualitative difference between the OAM of the evanescent and the traveling fields is one of the main points of this paper.

Since the cutoff of the integrals plays a significant role in the above argument, it is helpful to know more precisely how large the cutoff radius ρ_cut should be. To this end, we numerically plot $\frac{m^2}{\rho }{a}^2(\rho )$, μ²ρa² (ρ), ρa′² (ρ) as functions of ρ in Fig. 1, and the integrals of them versus the upper-bound ρ_up of the integration interval in Fig. 2. The parameters are set as κ = k (or equivalently $\mu =\sqrt{2}k$) and m = 3. From Fig. 1 we see that for ρ > 4λ (λ is the wavelength), $\frac{m^2}{\rho }{a}^2(\rho )$ is essentially zero and μ²ρa² (ρ), ρa′² (ρ) behave asymptotically. Figure 2 shows that as the upper-bound ρ_up exceeds 40λ, ${\displaystyle {\int }_0^{{\rho }_{up}}d\rho \frac{m^2}{\rho }{a}^2}(\rho )$ is less than 1% ${\displaystyle {\int }_0^{{\rho }_{up}}d\rho {\mu }^2\rho a^2}(\rho )$ and ${\displaystyle {\int }_0^{{\rho }_{up}}d\rho \rho {a^{\prime }}^2}(\rho )$, and the difference between the later two integrals is also less than 1%. So in this case we can safely set the cutoff radius at ρ_cut = 40λ. Of course, generally the appropriate ρ_cut depends on the specific values of κ, m, and k.

 

Fig. 1 $\frac{m^2}{\rho }{a}^2$ (ρ) [red dotted], μ²ρa² (ρ) [black solid], ρa′² (ρ) [green dashed] as functions of ρ. Here a(ρ) = J_m (μρ), with $\mu =\sqrt{2}k$ and m = 3.

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Fig. 2 ${\displaystyle {\int }_0^{{\rho }_{up}}d\rho }\frac{m^2}{\rho }{a}^2(\rho )$ [red dotted], ${\displaystyle {\int }_0^{{\rho }_{up}}d\rho }{\mu }^2\rho a^2(\rho )$ [black solid], ${\displaystyle {\int }_0^{{\rho }_{up}}d\rho }\rho {a^{\prime }}^2(\rho )$ [green dashed] versus ρ_up. Here a(ρ) = J_m (μρ), with $\rho \sqrt{2}k$ and m = 3.

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We can draw a similar conclusion for the SAM in a circularly-polarized evanescent 0th-order Bessel field, i.e., with u = J₀ (μρ), $|\alpha |=|\beta |=\frac{1}{\sqrt{2}}$, and αβ^* being imaginary in Eqs. (1) and (3). Based on Eq. (8), we achieve after a fairly lengthy calculation

Following essentially the same steps that led to Eq. (19), we have

Finally, we remark that evanescent Bessel beams may be generated via total internal reflection or surface plasmon resonance excitation, as proposed in [18–20]. In principle, our prediction in this paper can be tested by measuring the torque acting on a mechanical object in the evanescent field and comparing the result with that in the corresponding traveling field.

4. Summary

In summary, there are crucial differences between the OAM of evanescent and traveling light. Paritcularly, we studied the evanescent Bessel field and found that the traditionally dominant paraxial part of the OAM vanishes when averaged over the azimuthal angle. The standard OAM-energy relation, i.e., the OAM per unit energy equals $\frac{m}{\omega }$ in a linearly-polarized traveling wave with azimuthal phase dependence exp (imϕ), is violated in the evanescent wave. For an mth-order evanescent Bessel field, the OAM-energy ratio is instead given by ${\left(1+\frac{{\kappa }^2}{k^2}\right)}^{-1}\frac{m}{\omega }$, which indicates that the OAM decreases as the field is more firmly localized to the surface. Our study gives insight into the distinctive features of the OAM in evanescent light and may provide theoretical support for potential experiments.