1. Introduction

Whispering gallery modes (WGM) also known as morphology dependent resonances (MDR) are well known resonant phenomena that extract portions of the incident electromagnetic field and trap them inside a cavity of revolution. For the purpose of this paper we limit the discussion to the case of the optical part of the electromagnetic spectrum and the cavities of revolution to optical microspheres. Conditions required for WGMs to occur have been studied extensively [1–5] and various applications employing the phenomena have been proposed and demonstrated [6–11]. In those studies emphasis has usually been placed on the effects of the geometry and chemical composition of the microspheres and the physical and chemical properties of the surrounding media on the resonances. Meanwhile the delivery mechanism used to bring the incident electromagnetic field to the microsphere has not been studied in such detail as the other MDR properties.

In this paper, we reports on effects of different wavelengths of light propagating in a single mode fiber coupled to a microsphere through a half-coupler. Also we report on resonances in microspheres that result from using such cylindrical waveguides for the light delivery and the effects of different modes propagating in the waveguide on the resonances. These effects manifest themselves in the coupling of low-order linearly polarized (LP) modes in a cylindrical waveguide into the WGMs of microspheres.

2. Theoretical background

From the mathematical theory of mode propagation in a cylindrical optical fiber using the weakly guiding modes approximation [12, 13], it follows that the longitudinal components of the propagating fields are small compared with the transverse ones. The modes are considered to be linearly polarized in a plane transverse to the direction of propagation [14]. Such modes are called linearly polarized and have designations as LP_νµ modes. Each mode has its own propagation constant β which is usually expressed as a function of the normalized frequency V, $V=\left(2\pi a/\lambda \right)\sqrt{n_1^2-n_2^2}\approx \left(2\sqrt{2}\pi a/\lambda \right)\sqrt{\Delta n{n}_2}$, where a is the fiber core radius, λ is the operating wavelength, $\Delta n=n_1-n_2$ is the difference in refractive indexes of the fiber core and cladding respectively. Furthermore, n₁k₀ ≥ β ≥ n₂k₀, where k₀ = 2π/λ₀ is the propagation constant in vacuum and λ₀ is the wavelength in vacuum. The value of the normalized frequency determines the number of modes the optical fiber can propagate. If V < 2.405 only the lowest mode, LP_01, could propagate, and the fiber is called a single-mode fiber.

The subscript ν in the designation for linearly polarized modes represents the νth-order Bessel function that corresponds to the cutoff condition for the mode and µ represents the number of successive zeros in the corresponding Bessel function [15]. Furthermore, modes with ν = 0 have two orthogonal states of polarization and radial symmetry. Those states are identical, a condition that is called degeneracy. Modes with ν > 0 do not have radial symmetry, and angular distribution of intensities in the transverse plane varies with the angle. The next linearly polarized mode, LP₁₁, is a combination or a superposition of exact modes derived from solutions of Maxwell equations in the cylindrical coordinates [16, 17]. That mode can be constructed, as is shown in [13], choosing sin(νφ) or cos(νφ) in the traverse angular distribution of the field intensity with ν = 1, where φ is the angle, and two orthogonal states of polarization. As a result, the LP₁₁ mode has four possible distributions with π/2 symmetry in the circumferential φ direction [18]. Thus, in addition to being degenerate in terms of the orthogonal polarization, the LP₁₁ mode is also degenerate in the angular φ direction leading to a four-fold degeneracy in cylindrical coordinates [19].

Whispering gallery modes in microspheres are governed by the solutions of Maxwell equations in spherical coordinates in terms of eigenfunctions for the radial, azimuthal, and polar coordinates. These eigenfunctions are associated with the radial (n), polar (l), and azimuthal (m), mode numbers. Furthermore, the properties of the resonances are dependent on the TE and TM orientation of the electromagnetic fields captured by the microspheres [20]. The TE and TM designations indicate an electric (TE), or magnetic (TM) field vectors perpendicular to the plane of incidence. The plane of incidence is defined as the plane that contains the incident, reflected and transmitted propagating light rays. Thus, in the case of microspheres the plane of incidence is the one that passes through the center of a microsphere.

Due to the difference in the characteristic equations for TE and TM modes [21] and their boundary conditions at the interfaces [22], resonances formed by TE and TM polarizations are split forming spacing in their positions with TM resonances following TE resonances with the same l and n [23].

Coupling of light into a microsphere can be accomplished using techniques that involve tapered fibers, half-blocks, focused beams, and others [24–27]. As light enters a microsphere it excites resonances whose properties depend on the geometry and material of the microsphere, the surrounding medium, and the light delivering mechanism. In the case where an optical fiber is side-coupled to a microsphere, different conditions exist for coupling different modes from the fiber into the microsphere.

3. Experimental

The experimental setup is shown in Fig. 1 . It consists of a laser controller, a tunable laser, a half-block coupler with a microsphere positioned on the top of it, linear polarizer, and optical fibers. The laser is tuned by applying a time dependent electrical current from the laser controller the tunable laser. In response, the tunable laser emits light whose wavelength varies in time according to the current applied to the laser diode junction. Thus, both the wavelength of the emitted light and the laser current have an identical time dependency.

 

Fig. 1 Schematics of experimental setup.

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The laser light is coupled into a commercial grade step-index single-mode fiber specified for 1300/1550 nm wavelengths.

A commercially available glass microsphere of about 454-µm in diameter is placed on the half-block coupler which allows certain wavelength to be coupled into the microsphere [28, 29]. Complete information about the microsphere could be found in [30]. The half-block coupler is a structure that has a fiber embedded in a glass block in such a manner that a portion of the fiber cylindrical surface is exposed and polished. This arrangement with the half-block coupler permits placing the microsphere close to the core of the light carrying fiber and at the same time improves mechanical stability of the experimental setup.

The photodetector detects the light and, when the microsphere is absent, displays a continuous spectrum across all wavelength emitted by the laser over the given period of time. The reinserted microsphere extracts certain wavelengths, and the extracted portions of the signal do not propagate toward the photodetector. The extracted portions of spectrum are the resonant wavelengths coupled into the microsphere. They appear as sharp narrow dips in the spectrum transmitted through the optical fiber.

A linear polarizer was incorporated into the path of the light. This was done by placing the polarizer between two fibers as shown in Fig. 1 and aligning the fiber ends to achieve the maximum coupling of light from one fiber to another through the polarizer. The portion of fiber between the linear polarizer and the half-block was kept straight to prevent an accidental change in the state of polarization of light delivered to the microsphere. The polarizer was oriented such that the angle was read starting from a value 0 in the vertical plane and rotated about its optical axis in the XY plane (see Fig. 1).

The experiment was conducted at two operational wavelengths, 1300 and 1550 nm, generated by two different diodes. The resonances were observed and recorded every 10 angular degrees of incremental change in the polarizer orientation. The intensity of the whispering gallery modes resonances changed as the polarizer was rotated and, with the rotation of polarizer, the depth of the dips in the transmission spectrum of the fiber varied periodically. Examples of spectra with resonances obtained at the operating wavelength of 1550 nm and two polarization angles, 0 and 60 angular degrees, are shown in Fig. 2 .

 

Fig. 2 Resonant spectra obtained using the 1550 nm operating wavelength and two angular positions of the polarizer: (a) 0 degrees and (b) 60 degrees.

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In Fig. 2 two groups of resonances are clearly visible. One is located in the wavelength range from 1548.9 and 1549.0 nm and the other one in the range between 1549.8 and 1549.9 nm. The presence of several resonances in each group is attributed to imperfections in the geometry of microspheres used. As the polarizer is rotated the relative depth in the resonance signals also changes. The changes appeared to happen synchronously within the same group and with some delay between the grouped resonances.

Repeating the experiment using wavelengths near 1300 nm showed similar results with two closely located bands of resonances (Fig. 3 ). The responses of those resonances to changes in the angle of the polarizer also appeared to be similar to those observed at 1550 nm. In this case two groups of resonances also appeared, one in the wavelength range close to 1303.75 nm and the other one the vicinity of 1303.95 nm.

 

Fig. 3 Resonant spectra obtained using the 1300 nm operating wavelength and two angular positions of the polarizer: (a) 0 degrees and (b) 60 degrees.

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4. Results and discussion

Relative intensities of WGM resonances obtained at each of the operational wavelengths as a function of rotational position of the polarizer are plotted in Fig. 4 . Figure 4 contains two parts, Fig. 4(a) and Fig. 4(b), each of them representing variations in intensity of resonances at two operational wavelengths, 1500 nm and 1300 nm, respectively. Figures 4(a) and 4(b) have two differently colored plots, one color is blue and the other one is red. The blue and red color plots correspond to resonances that appear in both, Fig. 2 and Fig. 3, in the shorter and longer wavelength regions correspondingly. All plots in Fig. 4 appear to be near-sinusoidal profile. Moreover, blue and red color plots in both, Fig. 4(a) and Fig. 4(b), appeared to be out of phase with each other.

 

Fig. 4 Resonance peak intensity as a function of polarization angle for two operational wavelengths: (a) 1550 nm and (b) 1300 nm.

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From the plots, it becomes apparent that the resonances in a microsphere obtained at different wavelengths exhibit different properties. Both plots in Fig. 4(a), blue and red, while having near-sinusoidal profiles, also appear to exhibit a 180 degrees periodicity. Moreover, these plots appear to have a 90 degrees shift with respect to each other. Both plots in Fig. 4(b), blue and red, also have near-sinusoidal profiles, however with a 90 degrees periodicity. Moreover, plots in Fig. 4(b), in addition to being out-of phase, appear to be shifted with respect to each other in the angular domain by about 45 degrees [31].

The phenomenon observed in Fig. 4 is associated with changes in the modal structure of light propagating in a single mode fiber under different wavelengths.

At sufficiently long wavelengths, in our case about 1550 nm as shown in Fig. 4(a), the normalized frequency V is below the single-mode cut-off value of 2.405 permitting the propagation of only the fundamental LP₀₁ mode. The intensity distribution of that mode has a nearly Gaussian profile with the tails extending into the cladding. A microsphere placed on the half-block permits coupling of that portion of the fundamental mode into the microsphere producing resonances. The appearance of two groups of resonances is due to the splitting of the incident LP₀₁ mode propagating in the fiber into the TM and TE components by the multiple reflections inside the microsphere.

The resonances depicted in Fig. 4(a), blue and red, exhibit near-sinusoidal profiles, a 180 degrees periodicity, and a 90 degrees shift with respect to each other. These properties of resonances are consistent with those of original TE and TM orientations of the mutually orthogonal components of the incident fundamental LP₀₁ mode. Also, it follows from the plots in Fig. 4(a) and mutual strength of resonances at 0 and 60 degrees of the angular position of the polarizer (see Fig. 2) that the blue and red colors correspond respectively to the TM and TE orientations of the fundamental mode. The wavelength separation between the two groups of resonances presented in Fig. 2 being of about 1 nm suggests that the resonances observed at 1500 nm belong to two different groups of resonances with an adjacent l numbers.

The operation in the shorter wavelength range near 1300 nm resulted in a higher value of V and a more compact intensity profile for the fundamental LP₀₁ mode. In this case less power is propagated into the cladding and more power is concentrated around the longitudinal axis of the fiber core. As the wavelength decrease continues, the value of V becomes very close to the cutoff value for the next mode, LP₁₁. That mode starts appearing close to the core-cladding interface. The peak intensity of the LP₁₁ mode moves towards the center of the fiber core with the continuing decrease of the operating wavelength [32, 33].

Plots in Fig. 4(b) exhibit near sinusoidal profiles with out-of-phase behavior between two groups of resonances and are characteristic of the behavior of two orthogonally polarized components of the incident fiber mode. However, the presence of the 45 degree shift between these two groups and the periodicity of 90 degrees is indicative of the presence of an additional set of linearly polarized components in the transverse plane of the fiber [18, 34]. Thus, the pattern presented in Fig. 4(b) is consistent with the behavior of the LP₁₁ mode and resonances observed in the microsphere at the wavelength around 1300 nm are generated by the coupling of the second-lowest mode, LP₁₁, propagating in the fiber. Moreover, the blue and red color plots in Fig. 4(b) and resonant spectra in Fig. 3 suggest that the blue and red colors represent respectively the TE and TM orientations of the LP₁₁ mode. Thus, the TM resonances follow the TE resonances as theoretically predicted in [23].

Furthermore, the experimental results presented in Figs. 3(b) and 4(b) did not show any manifestations of polarization rotation of the LP₁₁ mode due to spin-orbit interactions [35, 36] and the optical Magnus effect studied in both multimode and few-mode step-index fibers [37, 38].

5. Conclusion

The results presented in this paper have shown a wavelength-dependent coupling of two low-order linearly polarized modes, LP₀₁ and LP₁₁, propagating in a cylindrical fiber and coupling into resonances in a microsphere.

Light propagating in the fiber with the wavelength of 1550 nm and coupled into a microsphere via a half-coupler generates resonances in the microsphere that have properties similar to polarization properties of the LP₀₁ mode in the fiber. On the other hand, light with the wavelength of 1300 nm, under similar conditions, generates resonances with properties similar to the LP₁₁ mode in the fiber. A linear polarizer incorporated in the fiber is employed to recognize unique properties of the linearly polarized modes and assist with the analysis. Furthermore, the switch from resonances generated by the LP₀₁ mode in the fiber to those generated by the LP₁₁ mode is accomplished by switching the operating wavelength from 1550 to 1300 nm.

The presence of certain LP modes in the optical fiber depends on the value of the normalized frequency V. Any variations in the fiber core diameter, the core and cladding refractive indexes, as well as the operating wavelength which affect the value of V. Those variations will also affect the appearance of the fiber modes. While in our experiment we changed the value of V by changing the operating wavelength, changes in any other fiber parameters will lead to similar effects on V and the LP modes.

We demonstrated and correlated the appearance of resonances in a microsphere to linearly polarized modes in an optical fiber. In the process we also demonstrated that the changes in the resonances in a microsphere can be generated by external factors such as the modal distribution in the optical fiber used to bring optical signal to the microsphere.

While the spin-orbit interactions of a photon in an optical fiber and the associate optical Magnus effect were not in the scope of the presented work, investigations of these phenomena should be a topic for a further study.