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Abstract
The optical spin-orbit interaction (SOI) caused by momentum-dependent Pancharatnam–Berry phase (PB) provides new opportunities in the development of spin-optical devices, but the relatively low conversion efficiency limits its application. Here, through rigorous full-wave analyses on it in a parity-time (PT) symmetric system with thickness less than a wavelength, we find that the conversion efficiency of the SOI can be enhanced in both transmission and reflection in a wide range of incidence angles. When the parameters of the PT symmetric system meet the requirement of coherent perfect absorbers-laser mode, the effective anisotropy between the TM and TE components (e.g., a difference of their Fresnel coefficients) within the beam will be amplified dramatically, which results in significantly enhanced conversion efficiency of SOIs (up to 106). These findings offer an effective way to modulate the SOIs with an ultra-thin PT symmetric system, and may exhibit applications in spin-orbit optical devices.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The optical spin-orbit interaction (SOI) refers to the conversion or coupling between the spin angular momentum (SAM) and orbital angular momentum (OAM) of light [1,2]. There are two kinds of OAMs [3], one is the intrinsic OAM, which has a helical wave-front, the other is extrinsic OAM, which presents an offset trajectory of the beam with respect to the reference point. The conversion from SAM to intrinsic OAM results in an optical vortex [4–7]. The conversion from SAM to extrinsic OAM manifests the generation of photonic spin Hall effect [8–11]. The SOIs widely exist in the systems of interface reflection and refraction, inhomogeneous anisotropic medium, strong focusing, particle scattering and evanescent wave [2,11–15]. The SOIs and their related phenomena play important roles in the fields of optics, nano-photonics, and plasmonic, show great application potential in precision measurement, edge detection, and biosensing [16–23].
The SOIs can originate from the inhomogeneous anisotropy of external materials (for example, Q-plate [5]) or the “effective” anisotropy of TM- and TE-wave components of each plane waves within a beam [24], dictated by the coordinate- and momentum-dependent Pancharatnam-Berry (PB) phase [25], respectively. Intriguingly, we found that the SOIs based momentum-dependent PB phase can produce both optical vortex and spin-Hall effect, and also a topological phase transition between them [26,27]. The conversion efficiencies (η) of the SOIs based the momentum-dependent PB phase in traditional materials are generally very low. It can be enhanced in microwave frequencies with a metamaterial slab [27], but its efficiency is still relatively low (less than 10%) and the fabrication of the metamaterial is extremely complicated. Meanwhile, It is found that enhancing the anisotropy of Fresnel coefficients or the difference of Fresnel coefficients (DFC) between TM- and TE-waves is the key to increase η [27]. Due to the anisotropy of o- and e-wave components of different plane waves within the beam, a uniaxial crystal is used to improve DFC [28–30], which is very bulk and not suitable for miniaturization. The surface plasmon resonance [31,32] and the Tamm state [33,34] can amplify the DFC to some extent and have the limited enhancement of η (<1). Recently, parity-time (PT) symmetric systems in optics have attract a lot of attention because of their intriguing properties, such as the exceptional points [35–37]and the coherent perfect absorbers-laser (CPA-L) mode [38–41]. This provides new opportunities to enhance the DFC and then improve the η dramatically.
Here, we uncover the details of the enhancement of the SOIs based the momentum-dependent PB phase in a PT symmetric system with thickness thinner than a wavelength. Based on calculations with a rigorous full-wave theory in circular-polarization (CP) bases and the Berry phase concept, it is found that: (1) The η of the reflection (${\eta _r}$) depends on the incident side (loss or gain), but that of the transmission (${\eta _t}$) is independent of the incident side properties; (2) Varying parameters of PT symmetric system, both ${\eta _t}$ and ${\eta _r}$ much greater than 1 can be obtained in a wide range of incident angle (0 to 90°); (3) The maximum ${\eta _t}$ and ${\eta _r}$ are related to the CPA-L mode in this PT symmetric system and up to 106.
2. Theory and model
Generally, PT symmetric system consists of gain and loss layer with equal thickness ($l$). Their refractive indexes meet ${n_{gain}} = {n_{loss}}^\ast $, as shown in Fig. 1. Considering that the circularly polarized Gaussian beam is incident on the PT symmetric system placed in the air, the beam can be decomposed into many plane waves, and the propagation direction of different plane waves is slightly different from that of the center plane wave. Set (x, y, and z) as the laboratory Cartesian coordinate system, $({{\textrm{x}^a},{\textrm{y}^a},{\textrm{z}^a}} )$ as the local coordinate system, and $a \in \{{i,r,t} \}$ represents the incident, reflected and transmitted beam respectively. As shown in Fig. 1(b), ${\textrm{z}^a}$ is parallel to the propagation direction of the beam and the directions of y and ${\textrm{y}^a}$ are consistent.
Fig. 1. The SOIs of light in parity-time symmetric system. (a) Schematic diagram of optical vortex produced by circularly polarized light passes through the PT symmetric system. The ${n_{gain}}$, ${n_{loss}}$ refer to the refractive index (real part:$n$, imaginary part:$\Delta n$) of the gain and loss medium with thickness of l, respectively. (b) Schematic diagram of optical spin-Hall effect, and the laboratory coordinates and local coordinates. The ${y^i}$, ${y^t}$ and ${y^r}$ are consistent with the y direction.
In the CP basis (spin basis), at a transverse reference plane ${\textrm{z}^a} = {\textrm{d}^a}$, it has ${\boldsymbol{k}^a} \cdot {{\mathbf{\cal R}}^a} = \boldsymbol{k}_ \bot ^a \cdot {\mathbf{\cal R}}_ \bot ^a + \boldsymbol{k}_z^a \cdot {\mathbf{\cal R}}_z^a$ and the light beams can be written as:
Considering that the Gaussian beam is finite in width, in the recent work [27], a full-wave theory has been established to connect the reflected/transmitted and incident beam, i.e.,
For a left-handed CP Gaussian incident beam with the half-width of the beam waist ${\omega _0}$ in its waist plane:${ {\boldsymbol{E}_ \bot^i({{\mathbf{\cal R}}_ \bot^i} )} |_{{z^i} = {d^i}}} = {e^{ - {{({\mathrm{{\cal R}}_ \bot^i/{\omega_0}} )}^2}}}\hat{\boldsymbol{V}}_ + ^i,$ the angular spectrum is:
Now, the conversion efficiency of SOI can be defined as:
where $P_\sigma ^a = \,\int_{ - \infty }^\infty {\hat{\boldsymbol{z}} \cdot Re({\boldsymbol{E}{{_\sigma^a}^\ast } \times \boldsymbol{H}_\sigma^a} )d{\boldsymbol{k}^a}}$. The Eq. (9) can be approximated as [27]:3. Results
3.1 Difference of Fresnel coefficients between TM- and TE-waves (DFC)
The modulus of DFC of transmission ($|{{D_t}} |= |{{t_{TM}} - {t_{TE}}} |$) and reflection ($|{{D_r}} |= |{{r_{TM}} - {r_{TE}}} |$) is firstly analyzed which is the key to enhance the η. The DFC can be modulated with refractive index, incident angle and direction (loss or gain side). The patterns of $|{{D_t}} |$ are the same whether the light incident from the loss side (Inc.-) or from the gain side (Inc.+), as shown in Fig. 2(a-c). However, the $|{{D_r}} |$ depends on the incident side, as shown in Fig. 2(d-f) and Fig. 2(g-i). This is typical in reciprocal systems [44,45]: the transmission is the same from both sides, but the reflection depends on the side of incidence.
Fig. 2. The maps of modulus of DFC of transmission ($|{{D_t}} |$) and reflection ($|{{D_r}} |$) as a function of the incident angle (0-90°) and the $\Delta n$ (0.6-1.1). The n is 3.0(a, d, and g), 3.25(b, e, and h) and 3.5(c, f, and i). The Inc.- and Inc.+ represent the light incident from the loss side and the gain side, respectively. Here, the thickness (l) is 0.2λ.
In the maps with n equal to 3.25 and 3.5, the maximum $|{{D_t}} |$ and $|{{D_r}} |$ can up to 103. More interestingly, these maximum $|{{D_t}} |$ and $|{{D_r}} |$ locate at the same position in Fig. 2(b, e) and the maximum $|{{D_t}} |$ and $|{{D_r}} |$ also locate at the same position in Fig. 2(c, f). Although the $|{{D_r}} |$ depends on the incident side, the locations of the maximum $|{{D_r}} |$ are the same for Inc.- and Inc.+. The incident angle (${\theta _{max}}$) and other parameters of these maximum points are shown in Table 1. The ${\theta _{max}}$ of n equal to 3.25 and 3.5 is 1 and 67.07°, respectively.
Table 1. PT symmetric system parameters extracting from Fig. 2.
The Fresnel coefficients varying with the incident angle are explored in these PT symmetric system (#PT1, #PT2 and #PT3 in Table 1). For simplicity, the Fresnel coefficients of transmission are mainly discussed. The modulus ($|{{t_{TM}}} |$, $|{{t_{TE}}} |$) and phase (${\psi _{{t_{TM}}}}$, ${\psi _{{t_{TE}}}}$) of Fresnel coefficients in different PT symmetric system are shown in Fig. 3. For #PT1, with the increase of the incident angle, the $|{{t_{TM}}} |$ has a maximum value of 1.0689 located at 54.7° (Fig. 3(a), black), but the $|{{t_{TE}}} |$ decreases and is always less than 1 (Fig. 3(a), red). For #PT2, the maximum $|{{t_{TM}}} |$ and $|{{t_{TE}}} |$ are larger than 103 near the incident angle of 1.2 ° as shown in Fig. 3(b). For #PT3, the maximum $|{{t_{TM}}} |$ is about 102 at the incident angle of 67 ° (Fig. 3(c), black). However, the $|{{t_{TE}}} |$ decreases with the increase of the incident angle (Fig. 3(c), red).
Fig. 3. The Fresnel coefficients of transmission in different PT symmetric systems. (a, b, c) modulus and (d, e, f) phase. The parameters of #PT1, #PT2 and #PT3 as shown in Table 1.
The ${\psi _{{t_{TM}}}}$ and ${\psi _{{t_{TE}}}}$ of #PT1 change continuously with incident angle in the range of 0-90°. The ${\psi _{{t_{TM}}}}$ of #PT2 jumps from -3.1 to 2.9 rad near the incident angle of 1.1 °. For #PT3, the ${\psi _{{t_{TE}}}}$ almost keep constant, but the ${\psi _{{t_{TM}}}}$ has an abrupt change at the incident angle of 67 °.
These incident angles of the maximum modulus and abrupt phase of Fresnel coefficients are shown in Table 1 as $\theta |t |$ and $\theta \psi $, respectively. The maximum $|{{t_{TM}}} |$ and the abrupt change of ${\psi _{{t_{TM}}}}$ almost appear at the same incident angle as the Fresnel coefficient much larger than 1.
3.2 Wave scattering
In a PT symmetric system, novel optical phenomena are usually related to the scattering eigenvalues. So, the relationship between these maximum (or abrupt changed) Fresnel coefficients and the scattering eigenvalues in PT symmetric system is explored.
For this two-port PT-symmetric structure, the scattering eigenvalues are defined as ${S_ \pm } = t \pm \sqrt {{r_ + }{r_ - }} $, where t is the Fresnel coefficient of the transmitted light, ${r_ + }\textrm{, }{r_ - }$ is Fresnel coefficient of reflected beam propagating to the gain and loss sides, respectively [46]. According to the generalized conservation relation, it has $|{T - 1} |= \sqrt {{r_ + }{r_ + }^\mathrm{\ast }{r_ - }{r_ - }^\mathrm{\ast }} $, where $T = t{t^\mathrm{\ast }}$ is the transmittance [47]. When $T < 1, |{{S_ + }} |= |{{S_ - }} |$, the PT symmetric system keeps in the exact PT symmetric phase. When $T > 1, |{{S_ + }} |\ne |{{S_ - }} |$, the PT-symmetric system exhibits the broken PT symmetric phase. When $T = 1$, ${r_ + }{r_ - } = 0$, the phase transition points, the two eigenvalues coalesce to the same value, denoted as the exceptional points [36]. There is another important mode of scattering eigenvalue, that is, coherent perfect absorbers (CPA) mode in PT symmetric system. This mode is accompanied with laser mode. Therefore, it is usually called coherent perfect absorbers-laser mode (CPA-L) [48,49].
The modulus of scattering eigenvalues ($|S |$) of TE- and TM-waves are shown in Fig. 4. The $|{S_ +^{TE}} |$ and $|{S_ -^{TE}} |$ of #PT1 are equal to 1 in Fig. 4(a), but two special exceptional points (EPs) [30] locate at incident angles of 30.3° and 69.7° in Fig. 4(b). The EPs and CPA-L mode of #PT2 locates at incident angle of 62.6° and 1.2° in Fig. 4(c), respectively. Moreover, the EP and CPA-L mode locates at 73.1° and 1.6 ° in Fig. 4(d), respectively. The $|{S_ +^{TE}} |$ and $|{S_ -^{TE}} |$ are equal to 1 in Fig. 4(e), but two EPs locate at incident angles of 51.1° and 76.3° in Fig. 4(f). There is no CPA-L mode for TE-waves and one CPA-L mode for TM-waves at incident angle of 67° for #PT3.
Fig. 4. The modulus of scattering eigenvalues ($|S |$) of TM- and TE-wave in different PT symmetric system. (a, b) #PT1, (c, d) #PT2, (e, f) #PT3.
These incident angles of EPs and CPA-L mode are shown in Table 1 as ${\theta _{EP}}$ and ${\theta _{CPA - L}}$, respectively. By comparing these incident angles in Table 1, it can be found that these maximum $|{{t_{TM}}} |$ and the abrupt change of ${\psi _{{t_{TM}}}}$ are mainly related to the CPA-L modes, not to the EPs. So, we propose that when PT symmetric system meets the requirements of CPA-L mode, the Fresnel coefficients will be amplified dramatically and present the abrupt change of phase, which results in enhancing the DFC. Interestingly, in a PT symmetric cavity with Epsilon-near-zero material, similar results have been reported, it is also found that Fresnel coefficient can be far greater than 1 and strong oscillations at the CPA-laser mode [40], which can also enhance the DFC.
3.3 Conversion efficiency(η) of SOIs
The ${\eta _t}$ and ${\eta _r}$ shown in Fig. 5 were calculated with the full-wave theory to clarity the relationships between η and CPA-L mode in PT symmetric systems. The ${\eta _t}$ is independent of the properties of the incident side, as shown in Fig. 5(a-c). However, the ${\eta _r}$ of loss side is different from that of gain side. The area of green region in Fig. 5(g-i) is larger than that in Fig. 5(d-f). It means the improvement of the ${\eta _t}$ can be more easily realized in gain side with the n varying from 3.0 to 3.5. The enhanced SOIs can be realized in a wide range of incident angle (0 to 90°) with the n of 3.25, as shown in Fig. 5(b, e, and h). The maximum η is about 106 when the n equal to 3.25 and 3.5 with appropriate $\Delta n$ (0.8605 and 1.055, respectively). It is also found that the incident angles of the maximum η are close to that of CPA-L modes in Table 1.
Fig. 5. The maps of ${\eta _t}$ and ${\eta _r}$ as a function of the incident angle (0-90°) and refractive index. The n is 3.0(a, d, g), 3.25(b, e, h) and 3.5(c, f, i) and the $\Delta n$ vary from 0.6 to 1.1. The Inc.- and Inc.+ represent the light incident from the loss side and the gain side, respectively. The thickness is 0.2λ.
The remarkable enhancing of ${\eta _t}$ at the small incident angle appears only in a very narrow band of $\Delta n$ in Fig. 5(b). It means the ${\eta _t}$ is very sensitive to the parameters of PT symmetric system at the small incident angle. So, the ${\eta _t}$ was carefully calculated at the incident angle of 1° and the n of 3. Figure 6(a) shows the map of ${\eta _t}$ versus the thickness (0-λ) and the $\Delta n$ (0-2). There are six local maximum points of ${\eta _t}$ and their parameters are shown in Table 2. Figure 6(b) shows six local maximum points (red) of the $|{{t_{TM}}} |$ which parameters are consistent with that of the maximum ${\eta _t}$. In Fig. 6(c), there are six lines of abrupt change of ${\psi _{{t_{TM}}}}$ depending on the thickness and the $\Delta n$. The end (red) of lines of abrupt change of ${\psi _{{t_{TM}}}}$ are close to the position of local maximum ${\eta _t}$. Figure 6(d) shows the $|{{S_ \pm }} |$ of TM-wave of PT symmetric system with the Δn in Table 2. It is clarity that the thickness in Table 2 meet the requirement of the CPA-L models.
Fig. 6. The properties of transmitted light when incident angle and n equal to 1° and 3, respectively. (a) the ${\eta _t}$, (b) the modulus and (c) phase of Fresnel coefficients, (d) modulus of scattering eigenvalues ($|{{S_ \pm }} |$) of TM-wave.
Table 2. The PT symmetric system parameters of maximum ${\boldsymbol{\eta }_{\boldsymbol{t}}}$ in Fig. 6.
These results indicate that the ${\eta _t}$ can be far greater than 1 and easily modulated with the thickness and $\Delta n$ at the small incident angle. Meanwhile, it further confirms that the CPA-L model is necessary to make the η increase sharply in this PT symmetric system.
4. Conclusion
We have employed the full-wave theory to describe the beam propagation in the PT symmetric system with thickness thinner than a wavelength. We found that the ${\eta _r}$ and ${\eta _t}$ can be modulated by many parameters, such as thickness, refractive index, and incident angle. Both ${\eta _t}$ and ${\eta _r}$ much greater than 1 can be obtained in a wide range of incident angles (0 to 90°). The origin of the enhanced SOI is confirmed: as this PT symmetric system meets the requirement of the CPA-L mode, the DFC will be amplified dramatically, which results in significantly enhancing the η (up to 106). Our findings provide an effective way to modulate the optical SOI with an ultra-thin PT symmetric system, and promise application potential in spin-orbit optical components.
Funding
National Natural Science Foundation of China (11874142, 12174091); Hengyang Normal University (18D28).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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