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The polarization characteristics of Whispering-Gallery-Mode (WGM) fiber lasers based on evanescent-wave-coupled gain are investigated. For the laser gain is excited by side-pumping scheme, it is found that the polarization property of lasing emission is simply dependent on the polarized states of the pump beams. The polarization property of lasing emission depends on the propagating situation of the pump beams in an optical fiber if the laser gain is excited by evanescent-wave pumping scheme, that is, if the pump beams within the fiber are meridional beams, the lasing emission is a transverse electric (TE) wave that forms a special radial polarization emission. However, if the pump beams within the fiber are skew beams, both transverse magnetic (TM) and TE waves exist simultaneously in lasing emission that forms a special axially and radially mixed polarization emission. Pumped by skew beams, the wave-number differences between TE and TM waves are also investigated quantitatively, the results demonstrate that the wave-number difference decreases with the increase of the fiber diameter and the refractive index (RI) of the cladding solution. The observed polarization characteristics have been well explained based on lasing radiation mechanism of WGM fiber laser of gain coupled by evanescent wave.
©2013 Optical Society of America
1. Introduction
Evanescent-wave gain coupled circular microcavity lasers, such as cylindrical [1–6], spherical [7,8], capillary [9,10] and fiber knot lasers [11], have generated much interest in recent years, which is mainly due to the fact that the gain media are distributed around their resonators and their potential applications in integrated optics, optoelectronics and optofluidics [12,13]. As the evanescent field of a WGM in a circular cavity extends into the gain medium, the gain can be coupled into the WGM that provides optical feedback required by lasing oscillations. To excite the gain molecules optically around a circular cavity, laser beams are usually pumped from the outside of the gain medium [1–4], this pumping configuration, called side-pumping scheme, suffers a low pump efficiency because the pump energy is absorbed by all excited molecules and only a small number of molecules residing in the WGM evanescent field contributes to the optical gain. To solve this problem, evanescent-wave pumping scheme has been used in circular cavities in recent years, such as bare optical fiber [5, 6], capillary [9], and microfiber knot [11], which reduces lasing threshold from 200 μJ /pulse in the side-pumping scheme [1] to 100 nJ /pulse [11]. The optical gain is produced directly by the pump light in the side-pumping scheme, the polarized state of the pump light determines the polarized states of the excited molecules, and the polarization properties of lasing emission of the WGM laser consequently. However, the optical gain is produced by an evanescent field of the pump light in the evanescent-wave pumping scheme, therefore, it is the polarized state of the evanescent field of the pump light, rather than the pump light itself, that determines the polarized states of the excited molecules, and then the polarization properties of lasing emission of the WGM laser. For the side-pumping scheme, the polarization property of lasing emission is found to dependent on the polarized states of the pumping beams simply. If the laser gain is excited by the evanescent-wave pumping scheme, we find that the polarization property of lasing emission depends on the propagating situation of the pump beams in an optical fiber. That is, the lasing emission is TE wave if the pump beams in the fiber are meridional beams [14], while both TE and TM waves coexist in lasing emission if pump beams in the fiber are skew beams [14], and the wave-number difference between TE and TM waves decreases with the increase of the fiber diameter and the RI of the cladding solution. The explanations for observed phenomenon are supplied in the present paper based on our experimental results.
2. Experimental setup
The experimental setup is shown in Fig. 1 schematically. The laser beam (532.0 nm, fundamental mode – TEM00), generated by a frequency doubled and Q-switched Nd:YAG laser, were used as a pump light with a fixed pulse energy (~1.5 mJ) and frequency (10 Hz). The required pump energy was obtained by varying the polarization direction of a polarizer (P1), the other polarizer (P2) was used to determine the polarized state of the pump light. A bare fiber (F1, RI = 1.458) was inserted into a long glass capillary (C), the open space between F1 and C was filled with pure water, ethanol and ethylene-glycol mixed solution of the Rhodamine 6G dye with a concentration of 4 × 10−3 Mol/L. The RI of the mixed solution, acting as the cladding solution, was varied from 1.333 to 1.430 (measured by an Abbe refractometer) by adjusting the volume ratio of the three solutions.
Fig. 1 Schematic illustration of the experimental setup. P1, P2 and P3: polarizers. L1, L2, L3: lenses. F1: bare quartz fiber. C: glass capillary. F2: detecting fiber. M1, M2 and M3: mirrors. LC: cylindrical lens. EP: evanescent field of pump light. EW: evanescent field of WGM. LW: WGM laser radiation.
Two setup branches were arranged to realize the side-pumping and evanescent-wave pumping schemes as shown in Fig. 1. For the side-pumping scheme, three mirrors (M1, M2 and M3) were used to guide the pumping beam onto a cylindrical lens (LC, focal length = 30 mm), which compressed the beam onto the bare fiber (F1, diameter of 125 μm) with a rectangle spot of ~4 × 0.3 mm2. For the evanescent-wave pumping scheme, two lenses L1 and L2, were employed to compress the size of the pump beam, the pump beam was longitudinally coupled into F1 (diameter of 94.7 μm) along the fiber axis by a lens (L3, focal length = 75 mm) with a conical angle of 2θi = 7.6°. The beam would propagate within F1 by total internal reflection (TIR) if the entrance angle θi was smaller than the critical angle θic, which was 15.9° in our experiments. To make sure the pump beams within F1 were all meridional beams, the incident direction of the pump beams was set strictly to be along the axial direction of F1. The pump beams would be skew beams within F1 by tilting F1 to an angle ~10° relative to the Z-axis as shown in Fig. 1. The WGM lasing emission (Lw) from the rim of F1 was recorded by a spectrometer (Spectrapro 500i) with an ICCD detector (PI-Max 1024RB) via an optical fiber F2, the spectrometer had a 0.05 nm (~2 cm−1) spectral resolution when a grating of density 2400 g/mm was used. The intensity of Lw also would be detected by a photo detector (PD, DSi200) after the lasing emission passing through an analyzer P3 positioned on the Y-Z plane, the polarization of the Lw was checked by rotating P3. The angle between P3’s polarization direction and the Z-axis was defined as Φ, which is zero if P3 was set along the Z-axis direction.
3. Experimental results and discussion
3.1 Side-pumping scheme
The side-pumping scheme was basically the same as the one Moon et al used in [1]. But three polarized states of the pump beams (A: polarizing along the Y-axis, B: along the midway in Y-Z plane, and C: along the Z-axis), as shown in Fig. 2, were used to excite dye molecules directly. The excited molecules residing in the evanescent field (Ew) of a WGM contributed to an effective optical gain, supporting by the WGM, lasing oscillation occurred in the circular cavity built in the cross section of fiber F1, and WGM lasing emission (Lw) was received by the detective fiber F2.
Fig. 2 Experimental setup of side-pumping scheme. D1: diameter of bare quartz fiber. D2: inner diameter of the glass capillary, the open space between the quartz fiber and the capillary is filled with dye solution (RI = 1.362). Three polarized states of the pump light are labeled as (A), (B) and (C). The drawing is not on scale.
Pumped by three polarized states of laser beams A, B and C, the acquired WGM lasing spectra are shown in Figs. 3(a) - 3(c), respectively. A wavelength shift occurs obviously between the spectra Figs. 3(a) and 3(c) indicated by a pair of dash lines. Due to the polarized state B is a combination of the state A and B, the spectrum Fig. 3(b) seems to be a simple overlap of the spectra Figs. 3(a) and 3(c). To check the polarization property of the lasing emission (Lw) for different polarized states of the pump beams, Lw’ s intensity was recorded by the PD together with the polarizer P3. A long wavelength pass filter was placed between the used PD and the polarizer P3 in order to avoid the pump light entering directly the detector PD. The background subtracted and normalized signals were shown in Fig. 4.
Fig. 3 The lasing spectra acquired by the side-pumping scheme. (a): the TE-wave spectrum pumped by the polarized state A. (b): the TE&TM mixed wave spectrum pumped by the polarized state B, and (c): the TM-wave spectrum pumped by the polarized state C. Two groups of dash lines indicate peak positions, and the spectra are up shift for clarity.
Fig. 4 Polarization-analysis results of lasing emission. Diamond points with error bar: the intensity of the lasing emission pumped by the polarized states C, the solid curve is drawn by cos2Φ. Square points with error bar: the intensity of the lasing emission pumped by the polarized states A, the solid curve is drawn by sin2Φ.
For the polarized state A, as shown in Fig. 4 by the square points with error bar, the Lw’s intensity reaches its maximum when P3 is set vertically to the F1’s axis (Φ = 90° and 270°), while it reaches its minimum when P3’s polarization direction is set along the F1’s axis (Φ = 0°,180° and 360°), the solid curve is drawn according to IL = sin2Φ. The results indicate that the lasing emission was a TE wave (electric vector is vertical to the Z-axis), and the direction of electric vector of lasing emission is the same as that of the pump light. For the polarized state C, as shown in Fig. 4 by the diamond points with error bar (the solid curve is drawn by IL = cos2Φ), a π/2 phase shift exists between the two curves, which means that the lasing emission is a TM wave (electric vector is along the Z-axis), and the direction of electric vector of the lasing emission is also the same as that of the pump light.
We conclude from above experimental results that the polarization property of lasing emission is simply dependent on the polarized state of the pump light in the side-pumping scheme. Moon et al also reported similar phenomenon [1, 15], in which a TM-WGM lasing emission was recorded when the pump beams were polarized along the axis of a capillary. In fact, the polarization state of lasing photon is determined by the vibration state of an excited dye molecule, since the vibration state is decided by the electric vector of the pump light, eventually the direction of electric vector of the pump light determines the polarization state of the lasing emission.
3.2 Evanescent-wave pumping scheme by meridional beams
3.2.1 Polarization analysis of WGM lasing emission
When pump beams are set strictly along the axial direction of F1 (Z-axis), the beams propagating in F1 are meridional beams [14] which travel within the fiber by TIR. As shown in Fig. 1, the evanescent field (Ep) of the pump beams, rather than the pump beams themselves, excites the dye molecules in the dye solution that acts as a cladding layer. Photons in evanescent field (EW) of a WGM stimulate the excited molecules, and then form lasing oscillations in the circular cavity built in a cross section of fiber F1.
Pumped by meridional beams in evanescent-wave pumping scheme, the acquired lasing spectrum is shown in the Fig. 5(a). The spectrum contains a series of peaks whose spacing is about ~0.78nm, the value corresponds to a theoretical mode spacing (△λ ≈λ2/2πan1~0.75 nm) for the used fiber F1 of diameter 2a = 94.7 µm and n1 = 1.458. Similar to the arrangement in the section 3.1, the polarization analysis of the lasing emission indicates that (1) the lasing emission is TE-WGM wave, which means that the electric vector of the lasing emission is always vertical to the fiber F1’ axis; (2) the polarization property of the lasing emission is independent on the polarized state of the pumping light. Modes of the lasing spectrum in Fig. 5(a) are assigned (the assignment procedure is described in detail in the section 3.4.1), the achieved results, shown in Fig. 5(b), also support the spectrum in Fig. 5(a) from a TE wave lasing emission.
Fig. 5 The lasing spectrum acquired by the evanescent-pumping scheme (meridian beam pump) in (a). The mode assignment of the lasing peaks in (b), which indicates that the acquired lasing spectrum is from a TE-WGM wave emission, and the mode order and the number pair (l, n) are (1, 744 to 750).
3.2.2 Radial polarization laser formed by TE-WGM lasing emission
When meridional beams propagate within the fiber F1 by TIR, as shown in Fig. 6(a), the beam distribution in every cross section intersecting the centerline of fiber F1 is the same, since the pump laser is in fundamental mode (TEM00). Therefore, the pump beams in XZ plane can be taken as an example for discussing the polarized state of the pump beams (Fig. 6(b)). When a pump beam travels to the interface of F1 with an entrance angle θ larger than the critical angle θc, the beam will experience a Goos-Hänchen shift dg [16] that is about a wavelength of the pump light λp, and then it is totally reflected into the fiber by the interface. The pump beam tunnels the fiber out a distance of dp that is also a length of λp, and then an evanescent field of the pump beam Ep is formed outside F1. Ep is a traveling wave along the Z-axis, it cannot travel along the X-axis [16], therefore, there is no electric field of Ep existing in the Z-axis, and this characteristic is independent on the polarized states of the pump beams. For the evanescent-wave pumped laser, since no electric field of Ep exists in the Z-axis, it is impossible to produce stimulated photons polarizing in the direction of Z-axis. After dye gain is coupled into F1, the WGM lasing oscillation and emission, of course, will lack the polarization element of Z-axis, so the lasing radiation belongs to TE wave, whose electric field is strictly vertical to the axis of F1.
Fig. 6 Diagram for the formation of the radial polarization laser. The meridional pumping beams propagating in a fiber in (a). The polarized situation of the pumping meridian beams on the fiber’s interface in (b). The radiation from the radial polarization laser in (c).
WGM within a circular cavity is a kind of “Quasi-normal Mode” [17], which means that part of light energy in a WGM leaking tangentially out of the surface of the cavity in the way of evanescent-wave and forms the lasing emission as shown in the solid arrows of Fig. 6(c). As both lasing emission and its electric vector are vertical to the axis of F1, based on a simple geometrical consideration, all electric vectors of the lasing radiations that emit from the same big circle of F1, as shown in the dash arrows of Fig. 6(c), cross at the centre of the big circle. Therefore, if the pump beams are meridional beams, the lasing emission from evanescent-wave pumped WGM fibre laser is a special radial polarization emission [18,19], whose emitting direction is vertical to the axis of F1.
3.3 Evanescent-wave pumping scheme by skew beams
3.3.1 Polarization analysis of WGM lasing emission
When pump beams are deviated an angle ~10° from the axial direction of F1, as shown in Fig. 1, the beams propagating in F1 are skew beams which travel within the fiber by TIR. Lasing spectrum without the polarizer P3 is shown in Fig. 7(b), the spectrum consists of two groups of spectral lines and each group is of the same wavelength interval. If the polarization direction of the polarizer P3 is set vertical to the axial direction of F1 by inserting the P3 between fibers F1 and F2, the acquired lasing spectrum is shown in Fig. 7(a), which indicates that one group of the spectral lines (long wavelength part) in Fig. 7(b) vanishes completely and the spectrum in Fig. 7(a) is from TE-wave lasing emission. The acquired lasing spectrum is shown in Fig. 7(c) if the polarization direction of the P3 is set along the axial direction of F1, which indicates that the other group of spectral lines (short wavelength part) in Fig. 7(b) vanishes completely and the spectrum in Fig. 7(c) is from TM-wave lasing emission. Figure 7 suggests clearly that both TE and TM waves coexist in the lasing emission if pump beams propagating in the fiber F1 are skew beams. Similar to the situation pumped by meridian beams (Section 3.2.1), the polarization property demonstrated by Fig. 7 is independent on the polarized state of the pump light.
Fig. 7 The experimental results acquired by the evanescent-pumping scheme (skew-beam pumping). The spectrum without the polarizer (P3) is shown in (b). When the P3 is set vertical to Z-axis, the spectrum is shown in (a). When the P3 is set along Z-axis, the spectrum is shown in (c). Two dash lines indicate peak positions.
3.3.2 Axial & radial mixed polarization laser formed by TM & TE-WGM lasing emission
When pump beams are deviated from the optical fibre’s axis, as shown in Fig. 8(a), the beams propagate as skew beams in the fibre F1 along the Z-axis. To analyse the property of the beams in the fibre, one of the beams with wave vector is taken as an analytic sample. The vector can be decomposed into and , as shown in Fig. 8(a) in dotted arrow, is a component that expresses a light beam propagating in the XY plane by TIR; while is a component that expresses a light beam propagating along the Z-axis. For the beam expressed by , both electric vectors (the vector parallel to Z-axis and the vector vertical to Z-axis) exist simultaneously in the fibre F1. These two kinds of electric vectors also coexist in the evanescent field of pump beams when the beam experiences TIR on the fibre’s interface as shown in Fig. 8(b). Because the dye molecules in the evanescent field of the pump beams are excited by two different electric vectors, two kinds of stimulated photons, polarizing in the Z-axis and in the XY plane, exist simultaneously in the evanescent field of WGMs of the circular cavity built in the optical fibre F1. The WGM lasing oscillation and emission, of course, contains both TE wave and TM wave after the dye gain is coupled into the circular cavity. Combining with the analysis at the end part of this text 3.2.2, as shown in Fig. 8(c), it can be concluded that (1) the electric field of TM wave is parallel to the Z-axis, the lasing emission of TM wave is a special axial polarization emission [18]; (2) the electric field of TE wave crosses at the centre of the circular cavity, the lasing emission of TE wave is a special radial polarization emission. Therefore, when pump beams deviate from the axis of the optical fibre F1 and propagate as skew beams within the fibre, the lasing emission from the evanescent-wave pumped WGM fibre laser is an axial and a radial mixed polarization emission.
Fig. 8 Diagram for the formation of mixed polarization laser. Propagation of the skew beams in a fiber in (a). The polarized situation of the pump beams on the fiber’s interface in (b). Axial and radial mixed polarization laser in (c).
3.4 Wave-number difference between adjacent TE- and TM-WGMs
3.4.1 Mode assignment of WGMs
Figure 7(b) shows that there is a wavelength difference between adjacent TE- and TM- WGMs when pumped by skew beams in the evanescent-wave pumping scheme. In order to find the law governed the wavelength difference, mode assignment should be carried out in advance. A resonant wavelength of a WGM is assigned by a pair of numbers (l, n) [20], where l is the radial mode order, and n is the angular mode number of a WGM. The resonant wavelength assigned by (l, n) is expressed as, which satisfies with the asymptotic formula in cylindrical cavity [21] as follow
Where n1 and n2 are the index for the cavity (n1 = 1.458) and the medium around the cavity, respectively; m = n1/n2; P = n1/n2 = m for TM wave, while P = n2/n1 = 1/m for TE wave; al is the root of Airy function, the first two roots of Airy function are a1 = 2.338107 and a2 = 4.087949. Mode assignment of WGMs can be achieved according to Eq. (1). For example, based on the experimental spectral data in Fig. 5(a), mode numbers of l, n and polarization state are fitted by Eq. (1), the fitted results are shown in Fig. 5(b) which indicates that the spectrum is from a TE wave lasing emission, and the number pair of (l, n) are (1, 744 to 750). The average wavelength deviations () between the fitted () and experimental data () are smaller than 0.06 nm (shown in Fig. 5(b)).3.4.2 The wave-number difference varied with diameter of fiber F1
The wave-number difference between adjacent TE- and TM-WGMs can be expressed as bellow for the same numbers of (l, n) based on the Eq. (1),
Equation (2) indicates that is a function of fiber radius (a) and the RI of the cladding solution (n2).For the lasers with a fixed RI of cladding solution and different fiber diameters, let and Eq. (2) can be simplified as
Equation (3) indicates that is proportional to the fiber radius (a) inversely. To verify the Eq. (3), experiments are designed by fixed RI of cladding solution (n2 = 1.386) and varied fiber diameters from 2a = 92.0 μm to 252.0 μm. The acquired lasing spectra are shown in Fig. 9.
Fig. 9 The lasing spectra varied with the fiber diameter, where the refractive index of the cladding solution is fixed at 1.386, and the fiber diameters are varied from (a) 2 a = 252.0, (b) 2 a = 192.6, (c) 2 a = 147.2, (d) 2 a = 110.1 to (e) 2a = 92.0 μm.
Mode assignments have been achieved by using the method in the section 3.4.1, which are marked by arrows in the Fig. 9. The measured and calculated wave-number differences are listed in Table 1, and the measured values are basically agreement with theoretic values in the table. Since the theoretic value (a = 252.0 μm) = 2.18 cm−1, which is very close to the 2 cm−1 (spectral resolution used in our spectrometer), two adjacent peaks of TE- and TM-WGMs in Fig. 9(a) cannot be resolved.
Table 1. Measured and calculated wave-number differences with different fiber diameters
3.4.3 The wave-number difference varied with RI of cladding solution
For the lasers with a fixed fiber diameter and different RIs of cladding solution, let Eq. (2) can be simplified as
To verify the Eq. (4), experiments are designed by fixed fiber diameter at 2a = 97.3 μm and varied RI of cladding solution from n2 = 1.354 to 1.395. The acquired lasing spectra are shown in Fig. 10. The measured and calculated wave-number differences are listed in Table 2.
Fig. 10 The lasing spectra varied with the refractive index of the cladding solution (n2) for a fiber of diameter 97.3 μm. The refractive indexes of the cladding solution are (a) n2 = 1.354, (b) n2 = 1.364, (c) n2 = 1.376, (d) n2 = 1.386 and (e) n2 = 1.395, respectively.
Table 2. Measured and calculated wave-number differences with different RI of cladding solution
It is clear that the wave-number difference decreases with the increase of the RI of cladding solution, which is basically consistent with the theoretic expectation from Eq. (4).
4. Summary
The polarization characteristics of WGM fiber lasers pumped and gain coupled by evanescent waves are investigated in this paper. The polarization property of the lasing emission is simply dependent on the polarized states of the pump beams if the lasing gain is excited directly by side-pumping beams. If the lasing gain is excited by an evanescent-wave, the polarization property of lasing emission is dependent on the propagating situation of the pump beam in an optical fiber, that is, the lasing emission is TE-WGM wave that forms a radial polarization emission when the pump beams within the fiber are meridional beams. However, both TM and TE waves coexist in the lasing emission that forms radial and axial mixed polarization emission when pump beams within the fiber are skew beams. The wave-number differences between TE and TM waves, when pumped by skew beams, are also investigated quantitatively, the results show that the wave-number difference decreases with the increase of the fiber diameter and the refractive index of the cladding solution. The observed polarization characteristics are explained based on lasing radiation mechanism of WGM fiber laser of gain coupled by an evanescent wave.
Acknowledgment
This work was supported by the National Science Foundation of China (Grant NO. 11164033), the Applied Basic Research Foundation of Yunnan Province (Grant NO. 2011FA006), the Foundation of Educational Department of Yunnan Province (Grant NO. k1050667) and the Items on Research Team of Science and Technology in Yunnan (IRTSTYN) province.
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