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Abstract
We report an unexplored magneto-optic effect, triple circular reflection/refraction, for the manufacture of nonreciprocal beam splitters. Such beam splitters split up linearly polarized light into opposite-helicity plus linearly polarized beams all traveling in different directions. We present a mathematical description of the effect and its experimental verification in the visible and near-infrared in bismuth-substituted iron garnets. The phenomenon also provides the existence of quadruple and higher order reflection/refraction modes. The effect in question could produce nonreciprocal photonic devices that operate in classical or quantum regimes such as beam steerers, isolators, and routers.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
We demonstrate that magneto-optic (MO) nonreciprocity induces a novel phenomenon in the form of triple reflection/refraction. The unique nonreciprocal modes that engender these additional propagation channels result from the simultaneous activation of circular and linear magnetic birefringence, enabled by adopting proper geometrical configurations with respect to the magnetization direction. We present experimental verification of the phenomena, the means for their activation and their physical, i.e. polarization state, characteristics.
Double reflection/refraction within circularly-birefringent media such as chiral and MO material have been observed before [1,2]. However, the coupling between circular, elliptical and linearly-polarized reflected and refracted modes, the presence of the latter, the emergence of differences in phase and beam-propagation directions between them, and the impact of nonreciprocity, have not been observed and characterized.
It is usually assumed that the angle of incidence and of reflection at an interface between two media are equal. This is a result of isotropy in the refractive index within the given materials [3]. The variation of all electromagnetic fields at the plane of reflection must satisfy the boundary conditions at all points, at all times. This implies that for a uniform refractive index within the medium of incidence, the angles of incidence and reflection are the same. However, this is not the case in MO media. The refractive index for circularly polarized light propagating parallel to the magnetization direction is different from that of the corresponding reflected beam [1,4–7].
Within MO materials, a circularly polarized beam of any given helicity propagating parallel to the magnetization will travel with a different refractive index from the opposite helicity mode. For MO media, this requires that circularly polarized beams of opposite helicities reflect and refract differently. Moreover, they exchange reflection and refraction propagation directions upon magnetization reversal. The same occurs for light propagating in the backward direction, anti-parallel to the forward magnetization direction. This does not happen in chiral (reciprocal) materials, and is the origin of the nonreciprocal operation in MO media. In addition, normal to the magnetization direction, linear-magnetic birefringence kicks in, also referred to as the Voigt effect, or the Cotton-Mouton effect. The difference in refractive index between the incoming and reflected beams in a beam splitter, gives rise to different reflection angles for incoming beams of opposite helicities.
Two reflected modes may be activated within the MO material by such a beam splitter for any incoming circularly polarized beam, as a result of linear-magnetic birefringence. The reflected beams will not only travel with different indices from that of the incident beam, but they will also travel with different indices from each other, depending on polarization. One of these modes is elliptically polarized. The other is linearly polarized. Thus, the two reflected beams will themselves travel in different directions from each other, since they possess different refractive indices, as discussed below. Hence, in contradistinction to the case of double reflection, where each circularly polarized incoming mode is reflected in a different direction, we find that each circularly polarized incoming mode may be reflected in two different directions, instead of just one.
There are other phenomena that produce beam splitting with double reflection or refraction. One such case is that of the Spin Hall effect of light, where s- and p- polarizations (electric fields parallel or perpendicular to the plane-of-incidence) are displaced from each other upon reflection from a magnetic cobalt film [8]. The effect has been attributed to the complex refractive index of the cobalt [8]. Linearly-birefringent dielectric materials with real refractive indices, such as calcite or lithium niobate, also produce beam-splitting, since different linearly-polarized components travel with different phase speeds. Differences in refractive index between circularly-polarized modes in chiral materials also produce double refraction. Nevertheless, reciprocal beam splitters, whether linearly birefringent or chiral, will not produce a different response for the returning light. All the above cases differ from the beam-splitting functionality of MO materials, with real refraction indices, where the separated beams are nonreciprocally operated upon, and where more than two reflected or refracted beams are generated.
In Section 2, we present the theory behind the latter phenomena. In Section 3, we provide experimental verification and characterization of these effects. Section 4 discusses possible applications of the additional nonreciprocal reflection and refraction channels disclosed here.
2. Theory
The primary theory underlying the additional reflection channels can be easily described by Fig. 1. In it, there is shown the proposed simplest case: when the incident light propagates along the direction of magnetization. Within the MO material, there exists an added mode as a result of the linear-magnetic birefringence for propagation nearly perpendicular to the direction of magnetization. The incident electromagnetic wave is shown as a superposition of left- and right-circularly-polarized optical modes. Within the MO medium, these polarization states propagate with different phase velocities parallel to the magnetization direction. They are then refracted out of the material or internally reflected. This is shown schematically in Fig. 1(a), whereas in Fig. 1(b) the magnetization direction is flipped, switching the refractive indices between opposite helicity modes.
Fig. 1. (a) Beam splitting effect for input light propagating along the magnetization direction with optical components having refractive indices ${n_ + }$ and ${n_ - }$. The beam is a superposition of left and right circularly polarized light. Following the phase matching condition at the interface between the media, the reflected beams propagate at different angles hence ${\theta _{i, \pm }} \ne {\theta _{r, \pm }}$. The transmitted beams also follow this same principle. (b) The same circular polarizations (relative to the wave vector) are shown propagating opposite to the magnetization direction in (a). However, in both cases there are the added modes for reflected light with electric field components in the direction of magnetization. These added modes occur for both initial circular polarizations.
In both cases the circular polarization states propagating parallel (a) and antiparallel (b) to the magnetization direction have different refractive indices within the MO medium and obey separate phase-matching conditions at the interface with the adjacent medium. This, in turn, requires that the circular polarization states refract at different unique angles, separating the two states in different propagation directions. This is also true for the case of internal reflection, since the refractive index for light propagating nearly normal to the magnetization has a different refractive index from that of the incident beam. Both incident polarization states satisfy separate boundary conditions. However, the reflected light will contain terms with electric field components in the direction of magnetization. This, in turn, will require a second separation to occur creating a quadruple reflection. The second separation will be between the elliptical polarization (${E_{x,y}}$) mode and the linear polarization (${E_z}$) mode which travel at different speeds within the MO material. Chiral materials have also been shown to exhibit double refraction and reflection for circularly polarized light in the reciprocal regime [2]. However, a chiral material lacks the unique mode for electric fields along the magnetization direction and will not have an additional reflection/refraction. These individual modes may be solved for and generalized with the application of Maxwell’s equations, as shown in the next section.
2.1 Maxwell equations analysis
For MO materials with magnetization along the z-direction, the dielectric permittivity tensor is as shown in Eq. (1) [9–11]. We assume that the electromagnetic wave propagates in the xz-plane. The off-diagonal component g is known as the MO gyrotropy parameter. $i = \sqrt { - 1} $.
Provided the application of the MO permittivity tensor to the Maxwell equations, a modified inhomogeneous wave equation may be derived. This wave equation, shown here as Eq. (2), differs from the non-magnetic case by an additional z-dependent rate of change term proportional to the gyrotropy, within the MO material. An absence of free electrical charges is assumed, which is a good approximation for MO materials at room temperature.
This inhomogeneous wave equation may be solved using the perpendicularity between the wave vector $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} $ of the electromagnetic wave and the displacement vector $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over D} = \; \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over \epsilon } \; \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} $. To solve the inhomogeneous wave equation, we introduce the assumption that the wave vector for an arbitrary direction between the x- and z-axes in the MO medium is given by a linear combination of the wave vectors along each of these axes, as shown in Eq. (3). Here we define the angular dependent terms of the wave vector as sines and cosines of the angle ${\theta _{\vec{M}, \pm }}$, the angle between the direction of propagation and the direction of magnetization. With this nomenclature, the angular dependent terms will range from 0 to 1 oppositely, enforcing the axial propagation limits. This construction will allow the definition of a general wave vector for all propagation within the xz-plane. The specified axial refractive indices, shown as Eq. (4) and Eq. (5) for elliptical polarizations (${E_{x,y}}$) and the Eq. (6) for linear polarization (${E_z}$), are found by solving the inhomogeneous wave equation.
For the elliptical polarization (${E_{x,y}}$) modes, the axial refractive index for propagation along the z-axis (${\nu _z}$) is the standard refractive index for circular polarization in xy-plane, ${n_ + }$ and ${n_ - }$, as expected. While for propagation along the x-axis, the axial refractive index (${\nu _x}$) becomes the geometric average of the circular polarization refractive indices. Notice that although the displacement vector is perpendicular to $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over k} $ in this case, the electric field vector $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over E} $, in general, is not. Hence it will acquire circulating components in the x- and y-directions. However, there will not be any electric field component in the z-direction for this mode. The linear polarization (${E_z}$) mode is linearly polarized along the z-direction, with no x- and y-components. In this case, the refractive index simply acquires the non-gyrotropic value $\sqrt \epsilon $. This is only the case when the propagating light is linearly polarized in the same direction of the magnetization. The two modes described above may exist at the same time and may originate from the same source. However, these two modes will follow separate beam paths and travel at different phase speeds. This will result in separate propagation directions for internal reflection within the MO medium, as previously shown in Fig. 1. The general expression for the refractive indices of the reflected light may now be found using these axial components. Following the previous definitions, a general definition of the wave vectors may be expressed for all directions of propagation and modes within MO material, as shown in Eq. (7) and Eq. (8).
Using this same nomenclature and the phase matching condition, the general refractive index for any internally reflected light may be expressed as shown in Eq. (9) and Eq. (10).
The nomenclature for positive and negative cases represents the reflected refractive indices for initial incident superposition of the left and right circular polarized states. ${\emptyset _{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over M} }}$ is the angle between the magnetization direction and the normal to the plane-of-incidence for the beam splitter. There is also the term $\varDelta \theta $, which represents the angle of beam separation for the reflected beams, ($\varDelta \theta = |{{\theta_{{E_{x,y}}, + }} - {\theta_{{E_{x,y}}, - }}} |\; $). The beam separation can thus be found by solving for $\varDelta \theta $. Solutions for both ${n_{r, \pm }}$ and $\varDelta \theta $ can be simultaneously found using numerical methods. The refractive indices shown in Eq. (9) and Eq. (10) are the general refractive indices for light of any polarization propagating in any direction with the MO media. The result is that elliptically polarized light of opposite helicity will reflect and refract with separate directions of propagation. While at the same time there is a propagation of a linear $({{E_z}} )$ mode that wil reflect/refract in a different direction, creating triple reflection/refraction while allowing for quadruple and higher order reflection/refraction modes. The mathematical description of the phenomenon is discussed in more detail in the Supplementary Information section.
3. Experimental verification
For verification of these additional modes of reflection/refraction, we separately launched left- and right-circularly polarized laser light to a latching Faraday rotator bismuth-substituted iron garnet (BiIG) sample, then measured the angular difference of the resulting transmitted beams. Iron garnets are a commonly used MO material [9–12], and the theoretical beam separation for both transmission and reflection can be calculated. More generally the values of $\sqrt \varepsilon \approx 2.57$ and $g \approx 0.037$ may be used for this material for the propagating wavelength of 532 nm. Similarly, the values of $\sqrt \varepsilon \approx 2.31$ and $g \approx 0.002$ for the propagating wavelength of 1550 nm. The elemental per-formula-unit composition of the sample in these experiments was Bi1.22(Eu0.45Ho0.55)1.78Fe4.044Ga0.956O12. This is a ferrimagnetic material [12], that maintains its magnetization state after poling and does not require a constantly applied external magnetic field to preserve it [13–15]. Different bismuth substitution levels may be used to adjust the Faraday rotation [9–12]. The BiIG sample was processed using precision polishing to produce one surface normal to the internal magnetic field and another surface $30^\circ $ off the normal, ${\emptyset _{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over M} }} = 30^\circ $. The design of the experiment can be easily described in Fig. 2.
Fig. 2. Experimental design to verify ${E_z}$ mode and generalized refractive index for ${E_{x,y}}$ mode. The internal magnetic field is static and nonchanging as the sample is rotated. This will create internally refracted beams to have electric field components in the z-direction, forcing separate propagation directions for the unique ${E_z}$ modes and the circular polarization states.
This sample and the corresponding experimental beam splitting are shown in Fig. 3, where the results for incident left and right circular polarizations are overlapped together to showcase the beam separation. Here the two refracted beams experience beam broadening due to the bandwidth of the applied laser light. This bandwidth in propagation wavelength will cause a difference in the refractive indices in the MO media creating a spread of refracted angles. Also, present within these images is the added ${E_z}$ mode, which will increase the beam broadening. This is inherently due to the ${E_z}$ mode refracting out of the material at an angle between the two circular polarization states. The application of bandpass filters and tuned etalons show that the beam broadening is proportional to the laser bandwidth. Hence, this beam broadening is independent of the beam splitting phenomenon but has an added wavelength dependent application. As shown in Fig. 3, there are two cases: propagation parallel (a, b) and reverse parallel (c, d) to the internal magnetization. The images in Fig. 3 show the difference of transmission angles for left and right circular polarizations incident on the flat side of the BiIG material. This difference is exhibited as a translational shift of the entire beam pattern. Here the different-polarization beams propagate through the BiIG material with different (measured) refractive indices and thus follow unique beam paths. These paths and refractive indices are switched upon flipping the direction of the material magnetization. The respective beams then refract out of the material at different angles.
Fig. 3. Application of 532 nm laser light to the BiIG sample and resulting refracted beams onto a screen. The refracted beams (a) and (b) for propagation parallel to magnetization and (c) and (d) for propagation reverse-parallel are overlaid images for left- and right-circular polarization, respectively. These images are taken separately with the application of opposite circular polarized laser light. The beam translational shifts are shown here for left- (a), (c) and right- (b), (d) circular polarizations. In this example the sample, whose photographic image is included in the figure, has been rotated $22^\circ $ away from the incident laser beam path.
3.1 Methods
The samples were magnetically poled before the optical measurements. Collimated 532 nm and 1550 nm light was directed to the flat side of the BiIG sample. We used polarizers and quarter-wave plates to produce circularly polarized beams of both helicities. After passing through the beam splitter, the light was projected onto rigidly held screens 50 cm away from the BiIG sample, enough to produce distinct images for both polarizations. A near-infrared (IR) detection card was used for the 1550 nm light. Additionally, photographic images of the spots were also taken to cross-correlate with these measurements. An IR camera was used to image the infrared light. The separation between the spots was measured to submillimeter resolution.
3.2 Results
The linearly polarized ${E_z}$ mode refracts out of the material at an angle between the two circular polarization states and causes the beam to broaden. However, the broadening of the beam due to the ${E_z}$ mode will allow for the simple verification of this mode by inserting a polarizer directly after the sample. This simple verification is possible since the ${E_z}$ mode will transmit a linear polarization while the ${E_{xy}}$ modes will transmit elliptical polarizations. The individual results for transmission of linearly and elliptically polarized light are well known, and it may be assumed that the linearly and elliptical polarizations interact at the material interface separately since they acquire separate phases. However, it may be shown with the application of the electromagnetic boundary conditions that even when considered simultaneously the linearly polarized light will transmit purely linearly polarized light, shown in more detail within the supplementary information section.
Hence, the proposed simple verification previously described, shown in Fig. 4, is more than adequate. It is then simple to see that when the polarizer is set to only allow light in the y-direction to pass through, in images (a) and (b), the ${E_z}$ mode no longer appears. Alternatively, the images (c) and (d) show that when the polarizer is set to only allow light in the xz-direction both ${E_z}$ and the ${E_{x,y}}$ modes are present, and the beam is greatly broadened when compared to (a) and (b). The two different initial polarization states shown as (a) and (c) correspond to left-circular polarized light while (b) and (d) correspond to right-circular polarized light. There have been previous results that unknowingly utilized the removal of the linear $({{E_z}} )$ mode in order to enhance the observable separation of the two elliptical $({{E_{xy}}} )$ modes [4,5]. However, the existence of the ${E_z}$ mode was not considered and the results were analyzed using polarizers for pre- and post-selection. Here we show that the removal of the linear $({{E_z}} )$ mode is responsible for the enhancement of double reflection/refraction beam separation within the MO material.
Fig. 4. Resulting images for both initial polarization states with a polarizer immediately after the sample. Here in the images (a) and (b) correspond to the polarizer set to only allow light in the y-direction (perpendicular to the plane of propagation). While the images (c) and (d) correspond to the polarizer set to only allow light in the xz-direction (in the plane of propagation). The images (a) and (c) correspond to left-circular polarized light while (b) and (d) correspond to right-circular polarized light.
Furthermore, an in-depth verification of the generalized refractive index may be achieved by analyzing the beam separation and different angles of refraction. With the application of the experimental configuration shown in Fig. 2 and Fig. 3, experimental results for the angle of beam separation can be produced to compare with theory. First, the sample was rotated to find the maximum observable beam separation for the two different light sources refracted out of the $30^\circ $ polished surface. As predicted by theory this angle of maximal beam separation approaches the angle of total internal reflection. The angle of maximal observable beam separation for the 532 nm light source was found to occur at ${\sim} 19^\circ $ which corresponds to a beam separation of ${\sim} 2.38^\circ $. Similarly, the angle of maximal observable beam separation for the 1550 nm light source occurs at ${\sim} 10.25^\circ $ which corresponds to a beam separation of ${\sim} 0.44^\circ $.
As the sample was rotated, the internal angle of incidence would be altered, thus varying the refracted angles and the beam separation. These results can then be compared for verification with the proposed general refractive indices, shown in Fig. 5.
Fig. 5. Comparison of theoretical and experimental results for 532 nm and 1550 nm light as a BiIG sample with a flat and $30^\circ $ polished side is rotated away from the incoming beam path. In all cases the peak beam separation occurs when the angle of incidence within the magneto-optical material approaches the angle of total internal reflection at the output surface. The error bars correspond to the standard deviation of the beam separation measurements and the precision of the rotating stage used.
Significant amplification of the beam separation is achieved by operating close to total internal reflection [6]. At that point the angular beam separation is directly proportional to the refractive index difference between the beams in the material and inversely proportional to the cosine of the refraction angle. As the latter approaches 90°, the angular separation diverges. This accounts for the bandwidth induced beam spread as well as the increase in beam peak angular separation. The theoretical results shown in Fig. 5 are the expected beam separation results. The experimental data points for both 532 nm and 1550 nm wavelengths correspond to the angle of maximal observable beam separation and the next 16 degrees following. The comparison of the theoretical and experimental results in Fig. 5, shows that the observed beam separations follow the theoretically predicted pattern.
4. Conclusion
We have experimentally verified the unique modes and general refractive indices for nonreciprocal MO material. A theoretical formulation for this type of beam separation was derived and presented, including general refractive indices formulation for all modes of propagation within the MO material. The experimental verification for the generalized refractive indices were conducted at two different wavelengths, in both cases matching the expected theoretical values for beam separation. A simple experimental verification for the unique linear polarization mode was presented while clarifying previously published results. We also presented the nonreciprocal multiple beam reflection/refraction phenomena, including a unique design for elliptical and linear polarized beam splitters.
The shown triple and quadruple reflection/refraction will occur within the material as many instances as the beam is internally reflected. This is due to the fact that at every instance, regardless of the incident mode within the material, there are two modes reflected and transmitted. Meaning that if either a ${E_{x,y}}$ or ${E_z}$ mode is internally reflected both ${E_{x,y}}$ and ${E_z}$ modes will be present after reflection. Theoretically this may occur an infinite number of times leading to the creation of a diffraction grating -like effect [7]. Such effects may be precisely calculated with the application of the previously shown generalizations of the modes within the MO material and their unique refractive indices.
The solution for the well-defined modes may be applied to create unique polarizing beam splitters. Due to the nature of the double reflection/refraction phenomenon simple circular polarizing beam splitters may be manufactured. However, the added ${E_z}$ mode for MO material provides a linear polarization component to the proposed beam splitters. This added mode will allow for the creation of dual linear and circular polarizing beam splitters. Also, the MO material creates a nonreciprocal response, producing different beam paths depending on propagation direction relative to the magnetization.
Beam splitters are an integral part in many optical applications such as polarization imaging [16,17], free-space optical networks [18], read-write MO data storage [19], and quantum information processing [20,21]. Various kinds of beam splitters have been proposed so far based on bulk optics and advanced photonic structures [22]. Exploring the proposed unique and nonreciprocal beam splitters in the quantum regime may potentially lead to novel quantum phenomena useful for emerging quantum technologies [23]. Aside from their potential application, there have been recent examples of applications of double reflection/refraction phenomena including the measurement of small changes in magnetic fields and microscopy [24–27].
Acknowledgements
ML and SN gratefully acknowledge support from Photonica, Inc. and the Henes Center for Quantum Phenomena. The authors thank Brockway Photography for images of our samples. The samples used in these experiments were obtained from II-VI, Inc.
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.
Supplemental document
See Supplement 1 for supporting content.
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