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Abstract
The optical properties of thin films are generally determined by direct photometric quantities. We show that additional insight into the properties of anisotropic thin films can be obtained by computing the polarization eigenstates and eigenvalues of their Jones matrices. We consider helically structured thin films, which display intriguing optical response, such as the circular Bragg resonance. Using numerical simulations and actual measurements, we show that the eigenvectors are mutually orthogonal in most regions of the wavevector space, except near the circular Bragg and the oblique resonances. Special wavevector values, called exceptional points, are found where the Jones matrix becomes defective and its eigenvectors coalesce. Exceptional points are also found in pairs of wavevector values differing only by a sample rotation by π around the direction normal to the sample; this property is shown to arise from Saxton - de Hoop’s reciprocity principle, which applies to lossy materials and contains time reversal symmetry, which only applies to lossless materials, as a special case.
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1. Introduction
Glancing angle deposition (GLAD) is a thin-film deposition technique that makes it possible to deposit anisotropic thin films [1,2] whose optical properties can be controlled by rotating and changing the inclination of the substrate while the physical vapor deposition process takes place. The slanted columnar crystalline structure resulting from the inclined substrate produces, by a shadowing effect, anisotropic nanoporous morphology [3]. Moreover, the size of the crystalline and porous phases is generally much smaller than the optical wavelength; the material is thus equivalent to an effective anisotropic medium whose optical properties can be controlled by adjusting the deposition conditions [4,5] or by subsequent annealing [6]. Because the structure of the film can be dynamically changed along the growth direction, such films are also called sculptured thin films (STF).
Among the most common STFs structures are zig-zag films, obtained by serial bideposition [7], and helically structured thin films (HSTFs) [8,9], obtained by incrementally rotating the inclined substrate by some small angle around the normal direction to produce chiral optical response. HSTFs have received attention due to the possibility of producing circular dichroism and optical activity of much larger magnitude than what can usually be found in natural materials [10]. When analyzing HSTFs’ optical properties, the emphasis is generally placed on direct photometric measurements, such as the coefficients of the Mueller matrix, which enable the determination of e.g., the magnitude of the linear or circular dichroisms.
Here, we use a different approach and focus our attention on the polarization eigenmodes and eigenvalues of the Jones matrices describing the optical response of HSTFs. The Jones formalism is well adapted to samples where depolarization phenomena are negligible. In such case, Mueller and Jones formalisms are equivalent but the former has redundant matrix coefficients parameters; moreover, the elements of the Jones matrix and vectors are easier to interpret than the coefficients of the Mueller matrix [11]. The amplitudes and relative phases of the Jones matrix, J, can be written as:
Ellipsometric measurements allow one to estimate the amplitude and phase of the elements inside the bracket of Eq. (1) (seven parameters in total), while the general phase factor, φss, is not accessible [11]. The calculation of the eigenvalues and eigenvectors of the Jones matrix enables one to easily identify the well-known phenomenon of circular Bragg resonance (CBR) and the less conspicuous oblique resonance [12] that occurs at twice the wavelength of the former. It also enables one to classify the optical response of such film for a range of wavevectors (k) as homogeneous or inhomogeneous [13], depending on whether the eigenvectors are mutually orthogonal or not. Moreover, HSTFs may exhibit at some values of k vector values a defective Jones matrix, where the two eigenvectors coalesce [14]. Such points are called exceptional points (EPs).
These points are attractive because the eigenvalues and eigenvectors are expected to change as a square-root function at an EP with a change in the film’s permittivity tensor arising from e.g., a change in humidity or temperature. This property arises because the discriminant of the quadratic characteristic polynomial equation, that solves for the eigenvalues and eigenvectors, is zero at an EP. The large slope of the square-root function at the origin can produce a large sensitivity gain. Devices operating at an EP have already found applications in ultrasensitive angular rotation measurements using a ring gyroscope [15,16], for particle detection [17], for measuring the Newtonian constant of gravitation [18], or for obtaining single frequency laser emission [19–21]. In the polarization space, EPs have been demonstrated mostly with metasurfaces, whose fabrication involves time-consuming numerical modeling and expensive nanofabrication techniques [22–25]. Films containing EPs produced by GLAD are a simpler and cheaper alternative to the former methods because they rely on a conventional thin-film deposition technique, i.e., GLAD, that lends itself to their mass production.
This manuscript is structured as follows. In section 2, we introduce HSTFs and describe CBR phenomenon and oblique resonance that takes place at twice the wavelength of the former. We characterize the optical response of HSTFs using the eigenvalues and eigenvectors of the Jones matrix. The eigenvector analysis enables us to show that the HSTFs behave as a homogeneous element, i.e., the two eigenvectors are nearly orthogonal, in most of the k-vector space. However, near each resonance, the two eigenvectors are not orthogonal and almost merge in some regions of the k-vector space. We then show the impossibility to classify, by a continuity/proximity rule, the two eigenvectors. In section 3, we trace the origin of this discontinuity to the existence of EPs, where the two eigenvectors coalesce. We show that EPs are found in the vicinity of the circular and oblique Bragg resonances. The elimination of a resonance by tuning the structural parameters of the HSTF is found to suppress EPs that are otherwise present in its vicinity, emphasizing the role played by resonances in the appearance of EPs. EPs are also found to appear in pairs, wherein each observed EP comes along with another EP when reversing the direction of the reflected beam. This phenomenon is shown to originate from a reciprocity principle due to Saxton [26] and de Hoop [27], which applies to lossy materials and contains time-reversal symmetry, which applies only for lossless elements, as a special case. In section 4, we show an experimental example of a HSTF displaying an EP and demonstrating the validity of Saxton-de Hoop’s reciprocity principle. We conclude this paper with a summary of our results.
2. Optical resonances in helically structured thin films (HSTFs)
Some optical properties of materials are determined to a large extent by the symmetry group of the crystalline structure. For instance, chiral materials, i.e., materials that lack improper rotation symmetry elements such as centers of inversion, reflection planes and rotation-reflection axes [28], can exhibit circular dichroism and rotatory power, also known as optical activity. Optical activity and circular dichroism in chiral natural materials, such as quartz, is generally small in comparison to their linear birefringence and is observed mostly in transmission. These optical effects can be significantly enhanced at the optical wavelengths in helically structured stratified media, also known as Reusch piles [29]. Such configuration is obtained by stacking identical thin films with linear birefringent response, whose optical axes are incrementally rotated by some angle Δφ in the normal direction. Although the intrinsic properties of each layer of the material being used are not chiral, the obtained helical structure produces chiral optical properties, such as optical activity. In particular, when circularly-polarized light impinges on a multilayer with the same helical handedness, the CBR phenomenon takes place for the wavelength range [30]:
where Λ is the pitch of the helix and n1 and n2 are the low and high principal refractive indices of individual layers. The CBR phenomenon can be qualitatively understood by considering the elementary process of a circularly polarized light at normal incidence between two layers of identical, anisotropic dielectric materials, whose principal axes are twisted by an angle Δφ around the normal to the interface. In the limit of small anisotropy, the reflected beam keeps a circular polarization state but the direction of rotation of the electric field vector is reversed. Also, the direction of the electric field of the reflected beam is rotated with respect to the incident beam; this rotation amounts, for circularly-polarized light, to a phase shift. The CBR takes place when constructive interference between the reflected partial waves at each interface of the multilayer occurs. This happens if the spatial handedness of the incident circular light matches that of the helical structure of the film and for a resonance wavelength λ0 given by [10]:CBR is exhibited e.g., by the cuticle of some beetles [31] and by cholesteric liquid crystals [32]. HSTFs can also be made by GLAD. The material to be deposited is placed inside a vacuum chamber and is evaporated by a heating source, such as an electron beam. The vapor products are deposited on a substrate inclined at some angle θ. By rotating the sample holder by some angle Δφ, the orientation of the crystalline columns forming at the sample’s surface also rotate by the same angle and a helically-structured film is obtained. Interesting optical properties can be observed when several periods of the film are deposited.
2.1 Circular and oblique Bragg resonances
An eight-period, right-handed HSTF made of TiO2 on a glass substrate is grown using the conditions shown in Table 1. Scanning electron microscope images of the cross-section of this sample can be found in the Supplemental of Ref. [14], as well as a description of the GLAD system and ellipsometric set-up used to characterize the optical properties. The phenomenon of CBR can best be observed by looking at the m03 Mueller coefficient. It is shown in Fig. 1 for incidence angle of θinc = 10°, 30° and 50° for two opposite values of azimuthal incidence angle φazi = 0° and 180°. This parameter is closely related to the circular dichroism of our film by the relation:
where Rrl is the power fraction of incident right-polarized reflected into left-polarized light, etc. A peak of CBR is seen at λ=490 nm, Fig. 1; it shifts towards lower wavelengths as the incidence angle increases. It also retains its magnitude and does not change sign as the sample is rotated by 180°. Another peak is also seen at twice the wavelength of CBR. It has a different character from the first one: it changes sign when the sample is rotated by 180°, which indicates that this peak does not originate from a chiral phenomenon. Its magnitude, negligible near normal incidence, increases with incidence angle, but remains smaller than CBR’s peak reflectance.
Fig. 1. Experimental m03 Mueller coefficients at a) θinc = 10°, b) 30° and c) 50° angles of incidence for φazi = 0° and 180° with the sample described in Table 1. Adapted with permission from the Supplemental of Ref. [14]. © The Optical Society.
Table 1. Deposition conditions and sample parameters
The mechanism underlying the oblique Bragg resonance is illustrated in Fig. 2. The index ellipsoid of the refractive index tensor is assumed to have the same orientation as the growth direction of the crystalline columns. Hence, the inclined index ellipsoid rotates with the z coordinate perpendicular to the surface. At normal incidence, light feels an index modulation with a period p equal to half the period of the helical pitch, of the HSTF, i.e., p=Λ/2. This creates a Bragg resonance at the same wavelength as the CBR. However, for inclined wavevectors, the period p of index modulation is equal to Λ, and a Bragg resonance takes place at λ=2Λn. Note that this phenomenon is not linked to the chirality of the HSTF.
Fig. 2. Illustration of the index modulation caused by the inclined columnar structure in HSTFs at (a) normal and (b) oblique incidence. The index modulation seen by light has periods p=Λ/2 and p=Λ respectively. The latter causes a Bragg resonance at λ=2λc.
Numerical simulations are used to calculate the reflectance of a free-standing HSTF with similar properties. The simulation parameters are shown in Table 2. The principal indices of the index ellipsoid are na, nb and nc, and there is an antireflection coating made of an isotropic layer of index nAR and suitable thickness d in order to accentuate the CBR. The principal refractive indices and column angle are reasonable values for TiO2 deposited by GLAD in the visible/near-IR range. No dispersion was included in our numerical simulations. The optical properties of the multilayer are calculated with the software Optikan III that implements Berreman’s model [33].
Table 2. Simulation parameters of the HSTF
The simulated total reflectance for totally depolarized (natural) light is shown in Fig. 3 for a film with thirty periods. A large number of periods was chosen to better emphasize the two resonances. One can clearly see the CBR near λ=500 nm. At normal incidence (Fig. 3(a)), no other reflectance peak is observed, but one can clearly see a resonance in reflectance at twice the wavelength of CBR for oblique incidence. It also has equal magnitude for two orthogonal polarization states of the incident light, thus it is a non-polarization-selective resonance. Hence, the overall reflectance can approach unity at high incidence angles for a large number of periods, Fig. 3(a). This contrasts with the CBR, whose total reflectance of depolarized light reaches its maximum at about 50% because only the circular polarization with the same handedness as that of the helical thin undergoes resonance in reflection. In Fig. 3(b), the reflectance is also shown for the case where the inclination of the crystalline columns was set at 0°; in such condition, the oblique resonance does not take place. Indeed, the index ellipsoid is oriented perpendicular to the z axis, and consequently no index modulation with period p=Λ exists that would create a resonance at λ=2λ0.
Fig. 3. Calculated total reflectance for a) for φazi = 0° with thirty periods of the simulated parameters shown in Table 2; b) Same condition as a) except that the angle of crystalline columns is θcol = 0° instead of 50°: the oblique resonance does not take place.
2.2 Classification of the eigenstates of the Jones matrix
To get a deeper insight into the optical response of HSTFs, it is instructive to calculate the eigenvalues and eigenvectors of the Jones matrix in reflection for different k-vector values. The eigenvalue equation is expressed as:
The eigenstates can be found by solving the characteristic equation:
The zj are then converted into (X, Y, Z) coordinates on the Poincaré sphere [34]. These are shown in reflection as a function of wavelength λ and incidence angle θinc for orientation of the plane of incidence φazi = 0 in Fig. 4 for a 12-period stack of the material described in Table 2. The strong circular dichroism of HSTF at the CBR is clearly visible from the eigenvalues, shown in Fig. 4(a), 4(e), and from the Poincaré coordinates (X, Y, Z) (0,0, ±1), which indicate that the eigenstates are circularly polarized. Another interesting feature is the eigenstates of the oblique resonance: their eigenvalues have equal magnitude, and the eigenstates have diagonal and anti-diagonal polarization states respectively, i.e., (X, Y, Z) (0, ±1, 0), Cf. Figure 4(c) and 4(g).
Fig. 4. From top to bottom: calculated modulus square of the eigenvalue and (X,Y,Z) coordinates on the Poincaré sphere of the (a-d) 1st and (e-h) 2nd eigenvectors for 12 periods of the materials described by the parameters in Table 2 as a function of wavelength λ and incidence angle θinc for orientation of the plane of incidence φazi = 0. White dots indicate discontinuities from one θinc value to the next.
Plotting the eigenvectors and eigenvalues in the k-vector spaces, as shown in Fig. 4, requires a sorting procedure. Here, we use a proximity rule such that, for a fixed value of the incidence angle, the eigenstates coordinates (and eigenvalues) belonging to the first or second mode remain continuous as λ is varied. Moreover, we define the first (i.e. index 1) eigenvalue/eigenvector as the one for which CBR takes place. However, when doing so, we find jumps between rows at points indicated by white dots in Fig. 4. The same issue takes place when the classification is carried out at a fixed wavelength by applying a proximity/continuity rule between consecutive θinc values instead: jumps now take place between columns at the same (λ, θinc) locations. Therefore, it appears that it is not possible to classify the modes 1 and 2 in a unique and unambiguous way.
Plotting the map of the proximity of the eigenvectors, as obtained from the scalar product:
provides insight about the origin of these discontinuities, as shown in Fig. 5. Since the principal axes of the refractive index tensor of the individual layers have different orientations with respect to the plane of incidence of light, the Jones matrix need not be diagonal in the (TM) p- and (TE) s-direction coordinate system. Nevertheless, for most values of (λ, θinc), the eigenvectors are diametrically opposed on the Poincaré sphere, i.e., ${\vec{R}_1}\cdot {\vec{R}_2} ={-} 1$. When this happens, the corresponding Jones vectors are mutually orthogonal, i.e.,Yet, there are a few regions where the proximity factor ${\vec{R}_1}\cdot {\vec{R}_2}$ reaches a value very close to one, which means that the proximity of the two eigenstates is high. A zoom in those regions is shown in Fig. 5(b), 5(c) and 5(d). The points of maximum proximity, indicated by a white dot, exactly match the discontinuities found in the mapping of the eigenvalues and eigenstates. The characteristics of the different discontinuity points found in this particular HSTF are summarized in Table 3.
Table 3. Location in ($\lambda,\;{{\theta }_{\textrm{inc}}}$) space, proximity value of the eigenstates, modulus square of the eigenvalue (reflectance), and Poincaré coordinates of the points of discontinuity for the example of the HSTF.
Fig. 5. Calculated proximity function ${\vec{R}_1}\cdot {\vec{R}_2} = {X_1}{X_2} + {Y_1}{Y_2} + {Z_1}{Z_2}$ of the two eigenstates as a function of wavelength λ and incidence angle θinc for orientation of the plane of incidence φazi = 0. a) full map. The regions of high proximity b-d are zoomed in b) c) d).
3. Exceptional points in HSTFs
3.1 Existence of exceptional points
The values of the proximity factor ${\vec{R}_1}\cdot {\vec{R}_2}$ shown in Table 3 are very close to one in all five cases of discontinuity. This strongly suggests that the Jones matrix becomes defective at those points. This happens when the the discriminant of Eq. (7) is zero, i.e.,
Then, the two eigenvectors of a Jones matrix coalesce and the Jones matrix cannot be diagonalized. Such particular eigenstate is called an exceptional point [35]. It is impossible to classify in a unique way the eigenstates based on a proximity rule near an EP. We show in the Supplemental of Ref. [14] that this arises from the fact that, when a closed-path trajectory is made around an EP in a two-parameter space, one polarization state is converted into another and it takes two cycles in order to recover the initial eigenstate. This is a topological property of the branch point singularity at an EP: it does not depend on the shape of the closed circuit, the only thing that matters being whether the trajectory contains an EP or not.
The evolution of eigenstates by proximity on a closed loop is illustrated in Fig. 6 for two cases about an EP. When the closed trajectory encloses the EP, Fig. 6(a)–(b), the eigenvectors are swapped after one rotation, whereas, when the closed trajectory does not enclose the EP, Fig. 6(c)–(d), each eigenstate remains the same after one revolution. Hence, the impossibility to obtain unambiguous tagging of each eigenstate by a continuity rule can be traced to the existence of EPs. The swapping of eigenstates when encircling an EP proves the existence of a branch-point singularity at an EP. An experimental example of eigenstate swapping will be shown in section 4. It is important to note that what we call exceptional points are actually exceptional lines in the 3D (λ, θinc, φazi) parametric space. The continuity of EPs coordinates in different slices of (θinc, φazi) differing by a small Δλ increment is easily verified numerically. The 1D (i.e., curvilinear) topological structure of EPs is essential for the notion of encirclement in a 3D space to make any sense.
Fig. 6. Calculated evolution of the eigenstates on closed circuits enclosing a discontinuity point (a,b) or not (c,d).
3.2 Connection between EPs and Bragg resonances
As seen in Table 3, the EPs obtained are all located either near the CBR or near the oblique resonance. This raises the question whether the existence of an EP is linked to the existence of a resonance nearby. Using numerical simulations, it is possible to remove either resonance with a minor modification of the structure of the HTSF. For instance, changing the rotation angle Δφ between successive layers from 20° to 90° without changing the helical period Λ enables one to eliminate the CBR, while keeping the oblique resonance. As seen in Fig. 7(a), the regions of high-proximity eigenstates near the circular Brag resonance disappear together with the EPs nearby, while some EPs are still found near the oblique resonance. Next, changing the inclination of the crystalline columns from θcol = 50° to θcol = 0° without changing other simulation parameters enables one to discard the oblique resonance while retaining the circular resonance. The opposite situation now takes place, Fig. 7(b): the regions of high proximity eigenstates that formerly existed near the oblique resonance have disappeared together with the EPs nearby, while an EP is still found near the CBR. This suggests that resonances play a role in the appearance of EPs in HSTFs.
Fig. 7. Calculated proximity map calculated or the parameters of Table 2 except that a) Δφ=90° and b) θcol = 0°. White dots indicate confirmed EPs.
Another interesting feature is the fact, found both numerically and confirmed experimentally (cf. section 4), that EPs have a circular polarization state (Z ≈ ±1) near the oblique resonance. Similarly, EPs have a diagonal polarization state (Y ≈± 1) near the CBR. The reason for the particular eigenstate of the EPs near each resonance is unknown but it does bear similarities with EPs found in absorbing biaxial crystals, described by Pancharatnam in his classical paper [36]. Briefly, Pancharatnam calls singular axes these special propagation directions that have degenerate eigenstates. He shows that the singular axes in absorbing biaxial crystals arises from the fact that the optic axes (OA) have degenerate real parts of refractive index (i.e., show no birefringence) but do show diattenuation, which defines {H, V} rectilinear eigenstates in the direction of the OA. When slightly tilting the k vector away from the OA perpendicular to the plane defined by the two OA, there is no appreciable change of diattenuation but birefringence with diagonal/antidiagonal, {D, D’}, polarization eigenstates adds up to the diattenuation and makes the eigenstates non orthogonal. At some specific point, there exist the right proportions of {H, V} diattenuation and {D, D’} birefringence that create a degenerate circularly polarized eigenstate: the singular axis.
3.3 EPs and reciprocity
It is possible to study EPs at a specific wavelength λ as a function of the orientation of the incident k-vector, which can be parametrized by (θinc, φazi). For instance, at λ=980.6 nm, near the oblique resonance, four EPs are identified by encirclement, Fig. 8(a), and a similar situation is found at λ=400 nm, near the CBR, Fig. 8(b). In general, for each EP observed at (θinc, φazi), another EP is found at (θinc, φazi+π). Note that the incident k-vector for (θinc, φazi+π) corresponds to the inversion of the reflected beam for (θinc, φazi).
This symmetry takes its origin from the Lorentz lemma, which can be stated as follows [37]. Consider light scattering from a material having a linear, time-invariant, local response and containing no current source. Moreover, the material is assumed to be reciprocal, i.e., non-magnetic, which implies that the permittivity and permeability tensors are symmetric. Let $\{{{{\vec{E}}_1},{{\vec{H}}_1}} \}$ and $\{{{{\vec{E}}_2},{{\vec{H}}_2}} \}$ be two electromagnetic vector fields created by two different distributions of current sources ; then :
Fig. 8. Calculated proximity map shown in the (φazi, θinc) at (a) λ=980.6 nm and (b) λ= 400.0 nm near the oblique and circular resonances. In each map EPs are found by pairs differing by Δφazi = 180°. EP encirclements are shown for the former and EPs are identified by white circles in Fig. 8(b).
The scattering amplitude for a scattered wave in the direction $\vec{k}^{\prime}$ whose polarization state is projected on a Jones vector $\vec{v}$, arising from the incident light in the direction $\vec{k}$ polarized with vector $\vec{u}$ is equal to the amplitude of a scattered wave in the direction $- \vec{k}$, whose polarization state is projected on Jones vector ${\vec{u}^\ast }$, arising from an incident light propagating in the direction $- \vec{k}^{\prime}$polarized with a vector ${\vec{v}^\ast }$.
Here, the symbol * denotes the complex conjugate and represents the time-reversal operator of a polarization state: the ellipse of polarization remains unchanged but the sense of rotation of the electric vector is reversed. This theorem applies to anisotropic and inhomogeneous materials. Importantly, it is also valid for absorbing materials. Saxton-de Hoop’s principle can be expressed as :
where suffixes T and $^\dagger$ denote the transpose and the transpose conjugate, and M and Mr are the Jones matrices in the direct and reverse direction of the k-vector respectively. As pointed out by Potton [38], Eq. (12) implies that:It is instructive to compare Saxton-de Hoop’s principle with time-reversal symmetry, illustrated in Fig. 8(b), where the transmitted and reflected states, $\vec{v}$ and $\vec{w}$ are expressed as:
and where R and T are reflection and transmission Jones matrices of the multilayer stack. Upon reversal of the reflected and transmitted waves, the resulting combined Jones state $\vec{x}$ is given according to Eq. (13) by:By introducing Eqs. (14)–(15) into Eq. (16), we have:
Now, for a lossless material, we have:
Introduction of Eqs. (14)–(15) into Eq. (18) leads to :
Equation (20) expresses time-reversal symmetry illustrated in Fig. 9(b). Saxton-de Hoop’s reciprocity principle thus contains time-reversal symmetry, which still applies to inhomogeneous and anisotropic materials, but is restricted to lossless media. Time reversality takes place when the time-reversed sequence of events of a physical process is also physically possible. When absorption takes place, time reversal leads to light amplification, which is not possible in thermodynamic equilibrium.
Now, in ellipsometry, Jones matrices are expressed in the {p, s} coordinate system and the p axis is reversed in reflection. Consequently, the reciprocal Jones matrix in reflection is given in [39]:
where Hence, if the reflection Jones matrix M for a given k-vector (λ, θinc, φazi) is :Our HSTFs satisfy all the conditions required for Saxton-de Hoop’s principle to be valid and numerical simulations confirm the validity of Eqs. (23)–(24).
Now, according to Eqs. (10), (23)–(24), the existence of an EP at (λ, θinc, φazi) implies the existence of an EP at (λ, θinc, φazi+π). Also, according to Eqs. (7) and (10), the z coordinate at an EP is given by:
and that of the EP for the reversed propagation is given by:Hence, the polarization states of these two EPs differ in general. For instance, for an EP with right circular polarization, the Jones matrix in the basis of p and s polarizations is given by:
Fig. 9. Illustration of a) Saxton-de Hoop in reflection and b) time-reversal symmetry for a multilayer structure. a) $P_{\vec{v}} \square \equiv \vec{v}^{\dagger} \square^{\vec{v}}$ and $P_{\vec{u}^*} \square \equiv \vec{u}^T \square \vec{u}^*$ are elliptical polarizers that project the incident light of arbitrary state $\square$ on states $\vec{v}$ and ${\vec{u}^\ast} $ respectively.
Hence, the reversed EP has the same diagonal polarization state, also in accordance with simulation results.
4. Experimental example
The ellipsometric setup used to estimate the elements of the Jones Matrix is outlined in the Supplemental of Ref. [14]. The elements of the Jones matrix for the sample obtained in the conditions shown in Table 1 were estimated by generalized ellipsometry [40]. The map of the proximity of the two eigenstates is shown as a function of λ and θinc for φazi = 0o, Fig. 10(a). Eigenvectors are nearly orthogonal, i.e., ${\vec{R}_1}\cdot {\vec{R}_2} \approx{-} 1$, for most values of λ and θinc, but higher values of the scalar product are found near the CBR and near the oblique Bragg resonance at high incidence angles. A scan of the scalar product at θinc = 69° indicates that the two eigenvectors almost coincide near λ=810 nm, Fig. 10(b). The existence of an EP is confirmed by the swapping of the eigenstates after one revolution on a circuit enclosing the high proximity region, cf. inset of Fig. 10(a).
Fig. 10. a) Experimental proximity map obtained at φazi = 0° showing the swapping of eigenstates around the high proximity point located at (λ,θinc)≈ (810 nm, 69°) in inset. b) proximity profile measured at θinc =69°.
The evolution of the eigenstates and eigenvalues in the neighborhood of an EP is shown in Fig. 11. A scan of the X, Y, Z coordinate at θinc = 69° clearly shows the quasi-coincidence of X, Y, Z, cf. Figure 11(a), as well as the coincidence of the modulus square of the eigenvalue (i.e., the reflectance), cf. Figure 11(d), near the EP at λ=810 nm.
Fig. 11. a-c) Measured values of the eigenstate coordinates on the Poincaré sphere and d) square modulus of the eigenvalues measured at θinc =69°.
Saxton-de Hoop’s reciprocity principle is confirmed by the existence of EPs at similar (λ, θinc) values for φazi values differing by π, as shown on the proximity map in Fig. 12(a) and (b). The opposite chirality of EPs for these two φazi values, predicted by eq. (25, 26), is also confirmed, Fig. 12 (c) and (d). The validity of Saxton-de Hoop’s reciprocity principle also proves that any optical activity and circular dichroism observed in HSTFs produced by GLAD has no magnetic origin.
Fig. 12. Experimental comparison of the proximity maps for a) φazi = 0° and b) 180° showing EPs at around the same (λ,θinc) coordinates. c-d) The respective z Poincaré coordinate of the eigenstate is shown, indicating the circular character (z=±1) at the EP with opposite chirality, as predicted by Saxton-de Hoop’s reciprocity principle. Adapted with permission from the Supplemental of Ref. [14]. © The Optical Society.
5. Conclusion
In summary, HSTFs made by GLAD are found to display two kinds of resonance: the well-known CBR and the oblique resonance at twice the wavelength of the former. Calculating the polarization eigenstates and eigenvalues of the Jones matrix of those films gives insight about the nature of these resonances. In most regions of the k-vector space, the eigenmodes are orthogonal but there are regions near the two resonances, where the polarization eigenstates almost coincide. It is not possible to classify the eigenmodes by a proximity rule in a unique manner; the origin of this phenomenon is traced to the existence of EP where the eigenstates coalesce and the Jones matrix becomes defective. The branch-point singularity nature of these points has for corollary the exchange of eigenstate when tracking each one on a closed loop surrounding an EP. We also show that these points appear in pairs, i.e., each EP at (λ, θinc, φazi) has an EP counterpart at (λ, θinc, φazi+π), which corresponds to the reversal of the reflected beam onto itself. This phenomenon takes its origin in Saxton-de Hoop’s reciprocity principle. The latter englobes, but it is more general than, time-reversal symmetry. Experimentally-obtained HSTFs were found to respect this reciprocity principle.
Funding
Natural Sciences and Engineering Research Council of Canada (5215).
Acknowledgments
We thank Pierre St-Onge for his technical support.
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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