# Lasing with conical diffraction feature in the KGd(WO_{4})_{2}:Nd biaxial crystal

## Abstract

With an experimental set-up designed to record simultaneously the far-field and the near-field patterns, we got lasing with feature of conical diffraction in the biaxial Nd^{3+}-doped KGd(WO_{4})_{2} crystal. The key-point is that the lasing direction is not single and is constituted by an angular distribution including the optical axis. Very slight changes of crystal orientation leads to crescent shape 1068-nm light distributions in the near-field. The beam launched towards the biaxial crystal is mainly linear polarized with its intensity in agreement with the Nd fluorescence angular distribution. A theoretical background is provided, including the monoclinic and triclinic symmetries and laser amplification including elliptical modes and cavity round trip.

## 1 Introduction

Pleochroism in biaxial anisotropic crystals was recognized more than a century ago. This phenomenon is the result of the polarization dependence of the light absorption due to dopants or various colour centres. For propagation directions close to the optical axes, the plane-wave modes (far-field detection) which propagates unchanged are elliptically polarized [1, 2, 3], and we expect also this behaviour for stimulated emission. This latter case is studied in [4] for the Nd^{3+}-doped KGd(WO_{4})_{2} (KGW) biaxial laser crystal. The anisotropy of the absorption (or amplification) splits an optical axis in two new ones, each propagating unchanged a right or left circularly polarized light. Launching the inverse circular polarization, i.e., left or right, respectively, leads to propagation of the singular Voigt wave [3, 4, 5]. Its existence in a medium depends on a key parameter [6]. This wave could find applications in optical sensing [7], for example, from a porous biaxial dielectric.

On another hand in the near field, the wave propagation along an optical axis of a transparent biaxial crystal leads to internal conical refraction [8, 9]: the focused flux of light energy is split into a cone and finally rings instead of being doubly refracted as in the other wave directions. Several opticians studied experimentally this phenomenon (Poggendorff rings, Raman spot and so on) in the two last centuries, and its theoretical description was obtained [10, 11, 12, 13, 14]. Nowadays a few applications have emerged: polarization demultiplexing and multiplexing [15] for free-space optical communication, optical trapping of microspheres [16], enhanced resolution microscopy by subwavelength localization [17, 18], two-photon polymerization [19], and polarimetry [20, 21]. The agreement between the theoretical description and experimental data has been extensively published recently in the far field [22, 23] as well as imaging the near field [24, 25, 26, 27, 28, 29].

Applications of optical axis-oriented biaxial-doped crystals exist also in the field of lasers. A KGW:Yb^{3+} laser was built with a polarization state selected in an arbitrarily direction without any additional cavity component [30], simply by moving the pump area inside the gain medium. A high-quality Gaussian beam was obtained [31, 32] emerging from the output plane mirror after being generated along the optical axis of a KGW:Nd^{3+} crystal. In contrast, the light energy output was found distributed inside a crescent-shaped area in [33, 34] which is more similar with the characteristic focused conical refraction pattern. So, the conical refraction/diffraction is clearly more complicated when it occurs inside a laser cavity. This is also what we have experienced in the present work as we show below, due to the fact that the experimenter does not control the generated beam going through the crystal: the latter results in complex interactions with the anisotropic stimulated emission gain and is in fact a property of the system itself. For this reason we could speak of “self-conical diffraction”. In this context, the present study includes both far-field and near-field complementary behaviours, the far field revealing the optical axis position inside the laser angular distribution as well as the polarization distribution.

This work is organized as follow. Section 2 is devoted to a theoretical background including the case of monoclinic and triclinic symmetries and laser amplification including elliptical modes and cavity round trip (chirality is also included despite not necessary here). The experimental method in Sect. 3 explains how different conical diffraction patterns can be obtained from very slightly different KGW:Nd^{3+} laser crystal orientations inside the cavity, all with a propagation direction distribution of lasing including the optical axis direction. Section 4 is devoted to results and modelling.

## 2 Theoretical background

### 2.1 Plane wave refraction

An incident plane wave (**k**_{i} wave-vector) launched on the \(x^{\prime } {\text{O}}y^{\prime }\) entrance face of the crystal generates an inhomogeneous refracted (transmitted) wave with a \({\mathbf{k}}_{\text{t}} = {\mathbf{k}}_{\text{t}}^{\prime } + i{\mathbf{k}}_{\text{t}}^{\prime \prime }\) wave-vector (Fig. 1b). Another cartesian frame (**x**_{1}, **x**_{2}, **x**_{3}) is needed: its third axis **x**_{3} is parallel to the real refracted wave-vector \({\mathbf{k}}_{\text{t}}^{\prime }\) (\(r, \varphi\)) ((\(r, \varphi\)): polar and azimuthal angles), so it is transverse and it coincides locally with the \({\mathbf{e}}_{{{\varvec{\uptheta}} = {\mathbf{r}}}} ,{\mathbf{e}}_{{\varvec{\upphi}}} ,{\mathbf{e}}_{{\mathbf{k}}}\), unit vectors of the spherical coordinates.

The refraction law is given by the equality of the tangential components of the wave-vectors with three results. First \({\mathbf{k}}_{\text{t}}^{\prime }\) is in the incidence plane, secondly the refracted polar angle can be calculated from \(\sin \left( r \right) = \sin \left( i \right)k_{i} /k_{\text{t}}^{\prime }\), and third the \(k_{{{\text{t}}1}}^{\prime \prime }\) transverse component can be calculated from the \(k_{{{\text{t}}3}}^{\prime \prime }\) longitudinal one by: \(k_{{{\text{t}}1}}^{\prime \prime } = k_{{{\text{t}}3}}^{\prime \prime } tg\left( r \right)\). Because the imaginary part is much smaller than the real one and because we will use only incidences close to the normal incidence (\(k_{{{\text{t}}1}}^{\prime \prime } \ll k_{{{\text{t}}3}}^{\prime \prime }\)), we can make two approximations. First we simplify the calculation of the refraction angle by using an average refractive index *n*_{moy} inside the crystal, so: \(k_{\text{t}}^{\prime } = 2\pi n_{\text{moy}} /\lambda\) and secondly we will neglect the transverse component \(k_{{{\text{t}}1}}^{\prime \prime }\) and we will treat the refracted wave as homogeneous.

*r*refraction and

*φ*Euler angles which appear along the following calculation will be obtained from the \(\left( {k_{{ix^{\prime } }} ,k_{{iy^{\prime } }} } \right)\) tangential components by:

The last useful detail is to examine how are the wave-vector **k**_{R} of the wave reflected back on a mirror orthogonal to \(z^{\prime }\) axis (in the next section it will be the output cavity mirror) and the wave-vector **k**_{Rt} of the wave reflected and then transmitted inside the crystal. Inspection of Fig. 1b reveals that the polar angle of **k**_{Rt} is *π* – *r*, while the *φ* azimuthal angle is unchanged.

### 2.2 Propagation of the refracted plane wave in the transverse framework (**x**_{1}, **x**_{2}, **x**_{3})

^{−1}) of a laser material are fully obtained in any direction of propagation for the two modes from the linear permittivity tensor:

*x*=

*N*

_{p},

*y*=

*N*

_{m},

*z*=

*N*

_{g}dielectric frame, where at this step we have written Eq. (3), the real part (related to the refraction indices: \(\varepsilon_{ii}^{\prime } = n_{i}^{2}\)) is diagonal but the imaginary part is not in the monoclinic and triclinic crystal symmetries or in case of gyrotropy.

**D**, it is convenient to work in the (

**x**

_{1},

**x**

_{2},

**x**

_{3}) transverse frame. The following relation occurs between the three components of the electric field \(\widehat{\varvec{E}}\):

*S*

_{1}is the 3-D rotation making the (

*x*,

*y*,

*z*) → (\(x^{\prime } ,y^{\prime } ,z^{\prime }\)) frame transfer and

*S*

_{2}(

*r*,

*φ*) makes the (\(x^{\prime } ,y^{\prime } ,z^{\prime }\)) → (

*x*

_{1},

*x*

_{2},

*x*

_{3}) transfer (see Fig. 1b). The relative position of the two (

*x*,

*y*,

*z*) and (\(x^{\prime } ,y^{\prime } ,z^{\prime }\)) frames can be any, but in the peculiar case of the optical axis-oriented crystal such as Fig. 1a

*S*

_{1}it is simply:

*V*

_{z}being the angle between the optical axis and

**z**, whereas:

**x**

_{1},

**x**

_{2},

**x**

_{3}) frame in such a way that the transverse eigenmodes frame → (

**x**

_{1},

**x**

_{2}) transfer is operated by the matrix:

### 2.3 Far-field pattern

*of the input beam at the entrance face of the crystal is a priori known in the (\(x^{\prime } ,y^{\prime } ,z^{\prime }\)) frame and can be decomposed in plane waves from its 2D Fourier transform \(\widehat{\varvec{E}}\). So its two tangential components are: \(\left[ {\begin{array}{*{20}c} {\widehat{\varvec{E}}_{{x^{\prime } }} \left( {k_{{ix^{\prime } }} ,k_{{iy^{\prime } }} } \right)} \\ {\widehat{\varvec{E}}_{{y^{\prime } }} \left( {k_{{ix^{\prime } }} ,k_{{iy^{\prime } }} } \right)} \\ \end{array} } \right].\) The refraction is such that the tangential electric field is continuous on the air/crystal separation surface, but in the present work we will use an anti-reflection coating so reflexion is neglected and the tangential electric field is transmitted. Then three steps are needed to calculate their spatial evolution through the crystal.*

**E***Step 1*we obtain the \(\widehat{\varvec{E}}\) tangential components in the (**x**_{1},**x**_{2}) transverse frame:where the square matrix is$$\left[ {\begin{array}{*{20}c} {\widehat{\varvec{E}}_{1} \left( {k_{{ix^{\prime } }} ,k_{{iy^{\prime } }} } \right)} \\ {\widehat{\varvec{E}}_{2} \left( {k_{{ix^{\prime } }} ,k_{{iy^{\prime } }} } \right)} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\cos \left( r \right)\cos \left( \varphi \right)} & {\cos \left( r \right)\cos \left( \varphi \right)} \\ { - \sin \left( \varphi \right)} & {\cos \left( \varphi \right)} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\widehat{\varvec{E}}_{{x^{\prime } }} \left( {k_{{ix^{\prime } }} ,k_{{iy^{\prime } }} } \right)} \\ {\widehat{\varvec{E}}_{{y^{\prime } }} \left( {k_{{ix^{\prime } }} ,k_{{iy^{\prime } }} } \right)} \\ \end{array} } \right]$$(13)*S*_{2}(*r*,*φ*) restricted to its tangential part.*Step 2*with the help of the eigenmodes we obtain the \(\widehat{\varvec{E}}\) tangential components after*L*=*e*/cos(*r*) length path crossed inside the crystal (thickness:*e*):and consequently dropping the \((k_{{ix^{\prime } }} ,k_{{iy^{\prime } }} )\) notation: \(\widehat{\varvec{E}}_{3} \left( L \right) = - \frac{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varepsilon }_{31} \widehat{\varvec{E}}_{1} \left( L \right) + \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varepsilon }_{32} \widehat{\varvec{E}}_{2} \left( L \right)}}{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\varepsilon }_{33} }}\)$$\left[ {\begin{array}{*{20}c} {\widehat{\varvec{E}}_{1} \left( {L,k_{{ix^{\prime } }} ,k_{{iy^{\prime } }} } \right)} \\ {\widehat{\varvec{E}}_{2} \left( {L,k_{{ix^{\prime } }} ,k_{{iy^{\prime } }} } \right)} \\ \end{array} } \right] = S_{3} \left[ {\begin{array}{*{20}c} {e^{{ik_{ + } L}} } & 0 \\ 0 & {e^{{ik_{ - } L}} } \\ \end{array} } \right]S_{3}^{ - 1} \left[ {\begin{array}{*{20}c} {\widehat{\varvec{E}}_{1} \left( {k_{{ix^{\prime } }} ,k_{{iy^{\prime } }} } \right)} \\ {\widehat{\varvec{E}}_{2} \left( {k_{{ix^{\prime } }} ,k_{{iy^{\prime } }} } \right)} \\ \end{array} } \right]$$(14)*Step 3*we go back to the (\(x^{\prime } ,y^{\prime } ,z^{\prime }\)) frame with a compressed notation:$$\left[ {\widehat{\varvec{E}}\left( {L,k_{{ix^{\prime } }} ,k_{{iy^{\prime } }} } \right)} \right]_{{x^{\prime } y^{\prime } z^{\prime } }} = S_{2}^{ - 1} \left( {r,\varphi } \right) \left[ {\widehat{\varvec{E}}\left( {L,k_{{ix^{\prime } }} ,k_{{iy^{\prime } }} } \right)} \right]_{x1x2x3}$$(15)

The two first components are the tangential ((\(x^{\prime } , \, y^{\prime }\)) frame) wanted far field from which we can obtain the wave intensity by the hermitian scalar product after eventually multiplication by the 2 × 2 Jones matrix representing any polarizer (linear or circular) located at the exit face of the crystal.

### 2.4 Near-field pattern

*iFT*(transverse \(x^{\prime } y^{\prime }\) components in a compressed notation):

The intensity of the near-field transmitted wave is obtained from the hermitian scalar product, whereas the phase wave front is obtained as the imaginary part of the field components.

### 2.5 The case of the KGW:Nd symmetry

This KGW:Nd crystal belongs to the C2/c space group, and it is monoclinic and centrosymmetric. The twofold **b** crystallographic axis coincides with the **Np** = **x** (=**b**) principal axis of the dielectric frame. The two other **Nm** = **y** and **Ng** = **z** principal axes lie in the **a**–**c** crystallographic plane with a known orientation [35], but this latter has no role in the present optical study.

To obtain the real part of the tensor (3), we chose to recalculate the *n*_{p} and *n*_{g} refractive indices values (and keeping *n*_{m} = 1.986 [35]) in order to get simultaneously the *V*_{z} = 46.6° polar angle of the optical axis [36], and anticipating the study given in the next Sect. 3, the first dark ring radius (0.3285°) from far-field conoscopy at 1064 nm and the semi-angle A (0.96°) of the hollow cone from internal conical refraction. This procedure was used because refractive indices previous published values failed to yield the experimental data (this situation was similar in [5]). The result is *n*_{p} = 1.9516 and *n*_{g} = 2.0183.

*θ*

_{0}rotation of the

*x*,

*y*,

*z*frame around the

**x**(=

**b**) axis as it is allowed by the monoclinic symmetry. This rotation means that generally absorption and fluorescence extrema do not coincide with the three principal

*x*,

*y*,

*z*axes. A non-diagonal term \(\varepsilon_{yz}^{\prime \prime }\) appears if we express \(\varepsilon^{\prime \prime }\) in the

*xyz*frame from its diagonal expression in the \((\tilde{x},\tilde{y},\tilde{z})\) frame:

We have the relations: \(k_{{\tilde{i}}}^{\prime \prime } = N\sigma_{{\tilde{i}}} = 2\pi /\left( {n_{{\tilde{i}}} \lambda } \right)\varepsilon_{{\tilde{i}\tilde{i}}}^{\prime \prime }\) where *N* is the population inversion for amplification and \(\sigma_{{\tilde{i}}}\) the stimulated emission cross section: *σ*_{p} = 11.93 × 10^{−20} cm^{2}, *σ*_{m} = 32.26 × 10^{−20} cm^{2}, *σ*_{g} = 8.30 × 10^{−20} cm^{2} [37].

*θ*

_{0}angle can be obtained from the formula:

Such a rotation (*θ*_{0}#10°) was reported for KGW:Nd [4], and it was already encountered in the monoclinic YCOB:Nd [38]. Its wavelength dependence imposes it as a fourth spectroscopic parameter for monoclinic crystals [39, 40].

## 3 Experimental methods

The crystal is pumped through a dichroic beam splitter at 810 nm with a fibre-coupled Limo laser diode (0.22 NA and 200 µm diameter). The KGW:Nd crystal has 3 % Nd doping and has 3.15 mm thickness.

Lasing occured at 1068 nm. A polarizer and a power meter can be installed towards the M1 laser output. The M2 laser output is analysed along two optical paths: one for imaging (CCD 2) the far field which is present at the focal plane located at 7.5 cm of an achromatic lens (AC 7.5 cm), the second one for imaging on the CCD 1 the field intensity distribution (near field) located on the M2 mirror by the AC 7.5 cm lens correctly adjusted for a clear image with *G* = 38.6 estimated magnification. On the far-field conoscopy path, two achromatic AC 10-cm and AC 5-cm lenses are installed, first to have two times magnification of the far-field size and second to accommodate a quarter-wave plate (fast axis at 45°) and/or a linear polarizer (both to constitute a circular polarizer).

The KGW:Nd sample is inserted in a water-cooled copper holder equipped for rotation around with three orthogonal axis (one longitudinal and two transverse). The longitudinal axis rotation was used to orient horizontally the *N*_{p}–*N*_{g} principal plane from 1064 nm far-field conoscopy with a similar procedure than [4]. This orientation (see Fig. 1a) was keeping fixe all along the experiment.

Lasing was studied after the thermal stabilization was reached.

## 4 Results and modelling

The crystal orientations corresponding to a2 and a5 pictures in Fig. 4 have their far-field patterns in various linear and circular polarizations represented in Fig. 3 upper and lower parts, respectively.

*α*= 129°, 104°, 85°, 74°, 68°, 52° and 20° (Fig. 4b) in comparison with \(x^{\prime }\)-axis, so the input electric field in Eq. (13) was:

The calculated near-field patterns (pictures c1–c7 in Fig. 4) are in reasonable agreement with the experimental data (pictures a1–a7 in Fig. 4). Despite the fact that a full circle is not obtained due to the strongly linear polarization of the bean passing through the crystal, it is clear that the doughnut-like distribution that we can guess here is due to the value close to 1 of the ratio of the conical refraction ring radius and the laser waist, and this distribution structure would disappear increasing the laser waist by shortening the M1 arm. An agreement is also seen in Fig. 3: the far-field patterns b2–e2 are similar to b1–e1 and the b4–e4 are similar to b3–e3.

We have also calculated the electric field of the wave reflected back by the M2 output plane mirror (where the wave front is calculated to be plane) and going backwards through the laser crystal a second time. The example of the 74° polarization of the beam on the M1 side is shown in detail in Fig. 4b. Picture b1 represents the beam intensity at the entrance of the laser crystal, picture c4 is the beam on the output M2 mirror, and picture b2 is the reflected beam intensity after a second pass through the laser crystal. We can see that after this round trip the Gaussian beam is reconstituted and more, its integrated intensity is 1.07 time the initial one. This amplification, calculated with 0.2079 % fraction population inversion, is of course what is needed to compensate the transmissions of the two mirrors: 2 % (M1) and 5 % (M2). The fraction population inversion needed to have 7 % round trip amplification for the 7 input polarization angles of Fig. 4b is 0.2885, 0.224, 0.2059, 0.2079, 0.2125, 0.238 and 0.348 % respectively. Because it results in the same pumping rate in all cases, the lower the population inversion, the highest the laser intensity at steady state circulating inside the cavity. This is at least qualitatively in agreement with the laser intensity given in Fig. 4b. Let us add that the corresponding amplification coefficients (cm^{−1}) are directly obtained as the imaginary part of expressions (9, 10) (see also Ref. [4]) after averaging over the propagation directions.

*α*angles launched towards the crystal from the M1 side. The result (Fig. 5d) is in reasonable agreement with the experimental data (Fig. 5a).

## 5 Conclusion

We have used an experimental set-up specially designed to determine simultaneously the far-field and the near-field patterns of lasing in the conditions for conical diffraction in the biaxial KGW:Nd crystal. The key-point is that the lasing direction is not unique and does not reduce to the cavity direction, and it is always constituted by an angular distribution including the optical axis. We checked this with the help of the far-field patterns at 1064 and 1068 nm. Very slight changes of crystal orientation leads to crescent shape 1068-nm light distributions in the near field (output mirror) around the well-known ring characterizing the internal conical refraction. This can be explained because the beam launched towards the biaxial crystal is mainly linear polarized with its intensity closely in agreement with the Nd fluorescence spatial distribution determined in a separate work. In a very particular crystal position, a beam mainly un-polarized was obtained with a near-field distribution almost circular. A theoretical background is provided, including the case of monoclinic and triclinic symmetries and laser amplification including elliptical modes and cavity round trip. The calculation is in reasonable agreement with all the experimental data (near and far fields).

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