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Coupling losses in laser cavities with a hollow rectangular or planar waveguide
Coupling losses in laser cavities with a hollow rectangular or planar waveguide
V. V. Kubarevროგორ მოგეწონათ ეს წიგნი?
როგორი ხარისხისაა ეს ფაილი?
ჩატვირთეთ, ხარისხის შესაფასებლად
როგორი ხარისხისაა ჩატვირთული ფაილი?
წელი:
1998
ენა:
english
DOI:
10.1070/QE1998v028n05ABEH001237
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PDF, 156 KB
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Home Search Collections Journals About Contact us My IOPscience Coupling losses in laser cavities with a hollow rectangular or planar waveguide This content has been downloaded from IOPscience. Please scroll down to see the full text. 1998 Quantum Electron. 28 406 (http://iopscience.iop.org/10637818/28/5/A10) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 130.237.29.138 This content was downloaded on 08/09/2015 at 15:58 Please note that terms and conditions apply. Quantum Electronics 28 (5) 406 ^ 410 (1998) ß1998 Kvantovaya Elektronika and Turpion Ltd LASER BEAMS. CAVITIES PACS numbers: 42.60.Da; 42.79.Gn Coupling losses in laser cavities with a hollow rectangular or planar waveguide V V Kubarev Abstract. The problem of the coupling losses experienced by the fundamental waveguide mode is solved for a laser cavity formed by two mirrors with a hollow rectangular or planar waveguide between them. The optimal configurations and mirror positions are found for waveguides with different ratios of the sides. Laser cavities supporting a wide range of wavelengths are considered. 1. Introduction Cavities with hollow dielectric or metal waveguides are frequently used in lasers to improve their parameters. In some cases the rectangular geometry of a waveguide (Fig. 1) is optimal. The limit a ! 1 corresponds to a planar waveguide. Since the modes inside a waveguide have plane wavefronts, a trivial waveguide laser cavity is formed if plane mirrors are placed at the ends of the waveguide. In real lasers, because of the need for adjustment, these mirrors are placed at finite distances from the waveguide ends. Sometimes the mirrors have to be at considerable distances from the waveguide ends. For example, in freeelectron lasers such a gap is required for coupling in and out of the electron beam, whereas in conventional lasers it is needed to accommodate special intracavity devices (selection cells, polarisers, 1 2 y y 2b z Ry x 2a x ; z Rx z Figure 1. Geometry used in calculations of the coupling losses: ( 1 ) waveguide; ( 2 ) mirror. V V Kubarev G I Budker Institute of Nuclear Physics (State Scientific Centre of the Russian Federation), Siberian Division of the Russian Academy of Sciences, prospekt akad. Lavrent'eva 11, 630090 Novosibirsk Received 10 October 1997 Kvantovaya Elektronika 25 (5) 419 ^ 423 (1998) Translated by A Tybulewicz etc.) The cavity is then no longer of pure waveguide type, but represents a combination of waveguide and open sections. We shall consider waveguides with transverse halfdimensions a, b 4 l. Then, weakly diverging radiation from the waveguide end can be expanded in terms of normal Gaussian modes travelling in free space, as described in Ref. [1]. The above inequality makes it possible to represent also the radiation inside the waveguide by a sum of specific hollow waveguide modes. The modes in waveguides with circular and rectangular cross sections are described in Refs [2, 3], respectively. We shall consider a laser with a sufficiently long waveguide selecting one fundamental EH11 mode if the waveguide is rectangular or the T E01 (T M01 ) mode in the planar geometry. Such a mode is selected either by the cavity length corresponding to a resonance, as in the case of submillimetre lasers [4, 5], or by mode competition in a homogeneously broadened active medium [6]. Our aim is to calculate the coupling losses when the main waveguide mode is converted into normal Gaussian modes and then, after reflection from a mirror, back into the main waveguide mode. This problem was considered for circular waveguides and partly solved in Ref. [6]. We shall use a similar method to solve the problem of waveguides with rectangular and planar geometries [7]. We shall eliminate a technical error in Ref. [6], which leads to some changes in the optimal cavity configurations and increases the minimal losses. The correct solution of the problem of waveguides with the circular geometry will be reported in a separate paper. 2. Mode expansion According to Krammer [3], the intensity ( Ex or Hy ) of the field of the EH11 mode at the emitting end of a rectangular waveguide is proportional to: Ex / F x, y, 0 Fx xFy y ( px py aÿ1=2 cos bÿ1=2 cos 2a 2b 0 inside waveguide , outside waveguide . (1) The field intensities in a onedimensional planar waveguide are proportional to Fy ( y). We shall expand this field as a Fourier series in terms of normal Gaussian modes, which are found in Ref. [1] for a symmetric square beam. The smallest number of modes is needed in such an expansion if the distribution described by expression (1), which is asymmetric under the substitution of the variables x $ y, is expanded also in terms of asymmetric (relative to this substitution) normal Gaussian modes. Generalisation of the treatment given in Ref. [1] to an asymmetric rectangular geometry makes it easy to show that the Coupling losses in waveguide laser cavities 407 eigenfunctions of the wave equation for the electromagnetic field in free space are then Cmn x, y, z, t Cxm x, zCyn y, z expi Ot ÿ kz Km ox0 ox z 1=2 Hm p 2 ( x 2 exp ÿ iFxm z ox z ox z x ) p y oy0 1=2 x2 ÿ ik Hn 2 Kn 2rx z oy z oy z ( ) y 2 y2 exp ÿ iFyn z ÿ ik expi Ot ÿ kz : (2) 2ry z oy z The eigenfunctions of a planar waveguide are Cyn ( y, z) expi(Ot ÿ kz). In the above expression the coefficients Km and Kn perform normalisation; Hm and Hn are the Hermite polynomials; the quantities dependent on z are ox z ox0 1 z=bx0 2 1=2 ; (3) oy z oy0 1 z=by0 2 1=2 ; rx z z1 bx0 =z2 ; ry z z1 by0 =z2 ; ÿ Fxm z m 12 arctan z=bx0 ; ÿ Fyn z n 12 arctan z=by0 ; (6) bx0 po2x0 =l ; (7) by0 po2y0 =l : (4) (5) The physical meaning of the quantities occurring in the above expressions is as follows: ox ( z) is the characteristic beam radius along the x direction; ox0 is the same radius at the beam waist (at z 0); rx ( z) is the wavefront radius in the y 0 plane; bx0 is the distance along the z axis from the beam waist to a point where rx ( z) is minimal; Fxm ( z) is an additional phase shift. The quantities with the index y have similar physical meanings. We shall now omit an unimportant, for the problem in hand, phase factor expi(Ot ÿ kz), where O is the radiation frequency and k is the wave number. The dependences on x and y permit separation of the wave equation into two parts which apply to the waveguide and to free space, respectively. This property is used in deriving expressions (1) and (2) where these variables are separated, and the functions Fx and Fy , and Cxm and Cyn are identical. Therefore, for brevity we shall write down, whenever possible, only the functions of the variable x and we shall bear in mind that the functions of the variable y are obtained by the elementary substitutions x $ y, a $ b, o0x $ o0y . We shall assume that the waists of all the normal Gaussian modes ( z 0, rx ry 1) are located at the waveguide end where the radiation emerging from the waveguide also has a plane wavefront and we shall use the notation Cxm ( x, 0) Cxm ( x). Expansion of expression (1) as a Fourier series in terms of these modes gives X Fx x Am Cxm x , (8) m Am 1 ÿ1 Fx xCxm dx Cxm x Km Hm p x 2 o0x a Fx xCxm dx , (9) x 2 exp ÿ , o0x (10) ÿa Km 1=4 2 2m m!o0x ÿ1=2 : p (11) The coefficients in expression (11) correspond to the normalisation 1 ÿ1 Cxm xCxk x dx dmk , (12) where dmk is the Kronecker delta; m, k 0, 2, 4, . . . . We shall find the functions Cxm and Cyn by selecting certain characteristic beam radii o0x and o0y , which satisfy solely the condition o0x , o0y 4 l. However, there are certain specific values of o0x and o0y for which the problem is simplified maximally and the physical size is represented better. Since the cosinusoidal distribution of expression (1) is close to the zeroth Gaussian mode, we shall select such values of o0x and o0y that the coefficient A0 in expression (8) is maximal. The functions Fx ( x) and Fy ( y) are symmetric under the substitutions x $ y, a $ b. We shall retain this symmetry in the normal Gaussian modes when seeking the optimal solution of the problem, i.e. we shall select such values of o0x and o0y for which o0x =a o0y =b. We shall find o0x from the equation 1=4 a qA0 2 px aÿ1=2 cos qo0x p 2a ÿa ( ) q x 2 ÿ1=2 o0x exp ÿ 0, (13) qo0x o0x the numerical solution of which gives o0x 0:70324895a. It follows from expressions (9) ^ (11) that qA0 =qo0x A2 ÿ1 2ÿ1=2 o0x . Therefore, A2 0 when A0 has its maximum value. The fraction of the power of the original radiation emerging from the waveguide end, represented by the first j harmonics of the expansion in terms of each transverse coordinate, is !2 j X 2 Am : (14) Bj m0 Table 1 gives the values of Am and Bj . We can see that the zeroth Gaussian mode carries about 99% of the power emerging from the waveguide and the first 64 terms contain more than 99.9%. Such a number of the expansion terms ensures that the calculated losses are accurate to within 0:1%. The dominant role of the zeroth mode in the optimal mode expansion makes it possible to determine directly the nearoptimal mirror configurations for a range of distances z ensuring that the coupling losses are nearminimal: the mirror surface should coincide with the wavefront of this mode Table 1. m, j Am Bj 0 2 4 6 8 10 12 14 0.994641 1.73 10ÿ6 ÿ0.089058 0.035870 0.009085 ÿ0.023540 0.017857 0.005534 0.98931 0.98931 0.99449 0.99706 0.99722 0.99833 0.99897 0.99903 408 V V Kubarev (as is true also of the other modes). We shall call such a mirror adaptive. If this problem were solved by the method of calculating the Fresnel ^ Kirchhoff integral, it would have required laborious twodimensional calculations and very long computer times. c (%) a=b 1 6 1.5 2 4 3 4 5 6 10 3. Calculation of the losses 2 It is convenient to divide the problem into two parts. In the first part we shall calculate the coupling losses for an idealised adaptive (in general, toroidal) mirror. In the second part we shall consider the coupling losses for an arbitrary toroidal mirror located at different distances from the waveguide end (this case is of much greater practical value). The mirror size will always be assumed to be sufficiently large, because the aperture losses on the mirrors should be much less than the coupling losses. We shall also assume that the radiation emerging outside the waveguide aperture 2a 2b in the z 0 plane is lost completely. 3.1 Adaptive toroidal mirror Since the wavefronts of normal Gaussian modes coincide with the surface of an adaptive toroidal mirror, the amplitude distributions of these modes on the waveguide end are, after reflection from the mirror, identical with the initial emittedradiation distributions. The coupling losses then arise because the phase velocities of various modes are different functions of z, so that the modes reflected from the mirror return to the waveguide end with different phases. The coupling losses c ( z) can be described quantitatively in terms of the overlap integrals: c z 1 ÿ a ÿa Fx xFx0 x, z dx b ÿb Fy yFy0 y, z dy 2 , (15) where Fx0 and Fy0 are functions of the field which returns to the waveguide end after reflection from the mirror: X Fx0 x; z Am Cxm x exp2iFxm z : (16) m Substitution of expression (16) in expression (15), followed by elementary transformations, gives X A2m A2q cosf2Fxm z ÿ Fxq zg c z 1 ÿ m;q X n;p A2n A2p cosf2Fyn z ÿ Fyp zg , (17) where the indices m, n, q, and p are even integers from zero to 14. It follows readily from expression (15) that the coupling losses of the planar cy ( z) and square cr ( z) waveguides with the same transverse size are related by cy ( z) 1 ÿ 1ÿ cr ( z)1=2 . We shall bear in mind this relationship and consider later the general case of rectangular geometry. Fig. 2 shows the dependences of the coupling losses, calculated on the basis of expression (17) for waveguides with different ratios of the sides, on a convenient dimensionless parameter z=by0. We note that in this problem the losses are invariant under the transformation z=by0 $ (a=b)2 =( z=by0 ). Therefore, if the logarithmic scale is adopted for a horizontal axis (abscissa) in Fig. 2, all the curves representing the losses are symmetric relative to the points z=by0 a=b. The losses in a square waveguide (a b) are similar, as expected, to the corresponding losses in a circular waveguide 0 100, 200, ? ÿ2 10 ÿ1 10 1 10 z by 0 Figure 2. Dependences of the coupling losses, calculated for an adaptive mirror and various rectangular waveguides, on the normalised distance between the mirror and the waveguide. [6]. For very large values of the ratios of the sides, a=b 5 100, the loss functions represent two halved loss distributions for a square waveguide located symmetrically relative to the point z=by0 a=b. In the range of z=by0 shown in Fig. 2 these losses are identical with those in a planar waveguide (a=b 1). A similar influence of the phase shift between the normal Gaussian modes on the nature of the curves in Fig. 2 is described in Ref. [7]. 3.2 Arbitrary toroidal mirror In the general case of a toroidal mirror with arbitrary radii of curvature Rx and Ry , located at an arbitrary distance from the waveguide end, each normal Gaussian mode returns after reflection by a mirror back to the same waveguide end, but it now has a different amplitude distribution of the field over the waveguide cross section and a modified (compared with that emitted) wavefront. The coupling losses are still described by expression (15), but the functions Fx0 and Fy0 are different. The squares of the moduli of the overlap integrals in expression (15) are known to be maximal and equal to unity for Fx Fx0 and Fy Fy0 . Since the main zeroth mode, containing a large fraction of the radiation power, is reflected by an adaptive mirror into itself, such a mirror represents a good (although, as shown below, not always the best) approximation to the ideal. We then find that Fx and Fx0 , and Fy and Fy0 are pairwise similar, and the coupling losses are relatively low (not exceeding 7%). If Fx 6 Fx0 and Fy 6 Fy0 , it follows from the Cauchy ^ Schwarz inequality that the squares of the moduli of the overlap integrals are less than unity and that they decrease with increase in the differences between these functions. It is therefore clear that in the case of a toroidal mirror which differs considerably from an adaptive mirror the coupling losses will be very large since the zeroth mode is reflected by such a mirror onto the waveguide end in a strongly modified form. The functions Fx0 and Fy0 of the field returning to the waveguide are of the form X p x o0x 1=2 Am Km Hm 2 0 Fx0 o0x ox m 2 x x2 0 exp ÿ i F ÿ k , (18) xm 2rx0 ox0 where (o0x =ox0 )1=2 takes account of the change in the field because of a change in the characteristic transverse size of the beam; ox0 is the characteristic size of the field returning 0 is the phase shift on the beam axis to the waveguide end; Fxm caused by the difference between the phase velocities of the Coupling losses in waveguide laser cavities 409 modes; rx0 is the radius of curvature of the returned field. If 0 we use expressions (3) ^ (7), we can calculate the phases Fxm 0 and Fym : 1 z 0 arctan F xm z m 2 bx0 0 bx0 1 bx0 bx 0 ÿ 0 arctan , arctan z 0 2 r rx0 b rx x x (19) where bx0 p(ox0 )2 =l. The returnedbeam parameters bx0 , by0 , rx0 , ry0 are defined in terms of the emittedbeam parameters bx0 , by0 , and fx Rx =2, fy Ry =2: " #2 " #2 z fx ÿ z ÿ b2x0 bx0 fx2 z fx fx ÿ z2 b2x0 fx ÿ z2 b2x0 , (20) rx0 z fx ÿ z ÿ b2x0 z fx fx ÿ z2 b2x0 " z fx bx0 z fx ÿ z ÿ b2x0 fx ÿ z2 b2x0 #2 " fx2 bx0 2 bx0 fx2 fx ÿ z2 b2x0 c (%) 100 R by 0 1 1.5 2 10 2.3 2.9 2.9 1.8 2.3 8.125 5.2 5.2 2.0 1 10ÿ2 8.125 10ÿ1 1 z by 0 10 Figure 3. Dependences of the coupling losses on the normalised distance, calculated for a square waveguide (a b) and spherical mirrors with different normalised radii of curvature (continuous curves) and for the same waveguide and an adaptive mirror (chain curve). #2 : (21) fx ÿ z b2x0 There is an error in the calculation of the mode phases in Ref. [6], which served as our guide in the present investigation. This error is responsible for breakdown of the analogy between this stage of the calculations reported here and those given in Ref. [6]. In particular, the conclusion of Abrams [6] that an adaptive mirror always ensures minimal losses for a given z is incorrect. By way of example, Fig. 3 gives the coupling losses for a square waveguide and various spherical mirrors. We can see that, if z=by0 > 3, the minimal losses for a given z are obtained for a mirror with a smaller radius of curvature than an adaptive mirror and the losses are less than for an adaptive mirror. It can be shown that the coupling losses become smaller than for an adaptive mirror because of the main effect in the formation of a minimum (matching of the wavefront profiles of a Gaussian mode and of the mirror), which is a positive reduction to zero (by the aperture) of the coupling losses (fraction of the radiation outside the waveguide aperture in the z 0 plane) at z R. The ratio of the adaptive losses to the minimal losses for certain values of R=by0 exceeds unity considerably (for example, this ratio is 2 for R=by0 8:125) and the difference between the absolute values of the two sets of losses can reach 2% (R=by0 5:2). However, it should be noted that an adaptive mirror is optimal for z=by0 < 1 and that in some cases it can be regarded as a reasonable first approximation to the optimal mirror. The practical value of this approximation is that the parameters of an adaptive mirror can be determined directly from the formulas in expression (4) for any ratio a=b and no numerical calculations are then needed (however, the losses remain unknown). The loss curve for R=by0 1 in Fig. 3 is needed to deal with a cavity of practically any simple waveguide laser in which plane mirrors are located, because of the need for alignment, at a finite distance from the waveguide ends. We can see that in this case the coupling losses do not exceed 1% if z 4 0:07by0. It is worth mentioning a feature of a cavity with a degenerate (a b) square waveguide which is important for lasers emitting a tunable wavelength, for example, freeelectron lasers. Let us assume that at some distance z from the waveguide end there is a mirror which ensures minimal losses for this value of z at some wavelength. If this wavelength is varied, the losses begin to rise (in accordance with Fig. 3). However, if we move the mirror, we can find such a value of z for which this mirror is still optimal, but for a different value of R=by0 l. If R=by0 5 2, then throughout the range of variation of l the losses can increase to the maximum adaptive losses, i.e. to 7.2%. We can easily see that this method of reducing the losses associated with the variation of l is not always effective for a rectangular waveguide with a 6 b. Fig. 4 illustrates the coupling losses for a waveguide characterised by a=b 5 and containing toroidal mirrors which are adaptive for (z=by0 ) 25, 5, or 1 at a certain wavelength l0 , and the changes in these losses for other values of l. We can see that now, in contrast to a square waveguide, the losses cannot always be minimised for other values of z. An increase in the losses because of variation of l is related to the misc (%) 100 1.5 l l0 1:5ÿ1 1 10 2 1.5 1 10ÿ2 2 1.5 ± 1 2±1 1 10ÿ1 1 4 1 1 10 z by 0 Figure 4. Dependences of the coupling losses on the normalised distance, calculated for a rectangular waveguide with a=b 5 and toroidal mirrors which are adaptive for l=l0 1 and ( z=by0 ) 25 (continuous curves), 5 (dashed curves), and 1 (chain curves), plotted for a range of wavelengths l, and the corresponding dependences for an adaptive mirror (chain curve with two dots) and a plane mirror (dotted curve). 410 match between the mirror shape and the wavefronts of the normal Gaussian modes, which cannot be made small by changing one parameter z in two independent formulas in expression (4) when a 6 b. This property of a rectangular waveguide distinguishes it fundamentally from a degenerate square waveguide and also from a circular waveguide. This is illustrated most strikingly by the curves in Fig. 4 applicable to an adaptive mirror when ( z=by0 ) 25 and a change in l increases strongly the losses; the losses rise significantly also for ( z=by0 ) 5. However, in some specific cases the coupling losses are close to the losses for an adaptive mirror in a very wide range of l. The best result is obtained for an adaptive mirror ( z=by0 ) 1, when the wavelength is increased. A small displacement of the mirror can ensure that the losses do not exceed 6% when the wavelength is altered by a factor of 4. The situation improves greatly when the ratio a=b is increased, because then the problem approaches increasingly the onedimensional case. The coupling losses for a waveguide with a=b 10 and an adaptive mirror are given in Ref. [7] for ( z=by0 ) 1. These losses do not exceed 4% when l is increased by an order of magnitude and z is tuned. The losses are plotted in Fig. 4 as a function of one variable z=by0. However, in principle, the problem of loss minimisation is twodimensional when a 6 b, i.e. a local minimum corresponding to each value of z=by0 must be found by varying two radii of curvature Rx =by0 and Ry =by0 . However, such variation changes little (the relative change does not exceed 4%) the local minima (one of which is identified by a cross in Fig. 4) if a=b 5 5 and this applies to the cases when l=l0 6 1. The curve in Fig. 4 corresponding to a plane mirror demonstrates that the losses are low (not more than 1%) if z=by0 4 0:1 and this is true for any finite value of l. The other range of z where the losses are low for all wavelengths l satisfies the inequality z=bx0 ( z=by0 )(a=b)2 5 10. This range may be called the `farfield zone' of the normal Gaussian modes or the spherical optics region. In this range a spherical mirror with the radii of curvature Rx Ry z is ö in accordance with expression (4) ö adaptive and all the modes are ö in accordance with expressions (5) and (6) ö in phase, i.e. the losses are close to zero. For typical values of a, b, and l this range is however difficult to reach in practice because z and the mirror dimensions have to be large. We shall now consider a case of importance for freeelectron lasers when a plane mirror is located at a very small distance from the waveguide end and an electron beam is coupled out through a window in a vertical side wall of s 2b dimensions, where s is the window length (i.e. the distance between the mirror and the continuous part of the waveguide); 2b is the waveguide height (Fig. 1). The symmetry of the radiation relative to the x and y axes of the coordinate system in Fig. 1 is retained by assuming that there is an identical window in the opposite side wall of the waveguide. In order to lower the losses and reduce the problem to the onedimensional form, we shall assume that the horizontal size of the waveguide 2av is widened in the vicinity of the windows (i.e. 2av > 2a) by an amount needed to retain the onedimensional nature of the problem. The coupling losses are then equal to the losses in a conventional waveguide characterised by a=b 5 0:1 located at a distance z s from the mirror and, because the problem is onedimensional, the losses are a function of just one dimensionless variable s=bx0 l=a 2. Calculations show that these losses do not exceed 1% for s=bx0 4 0:1 [7]. V V Kubarev 4. Conclusions Calculations of the coupling losses for a laser cavity with a hollow rectangular or planar waveguide supporting the fundamental waveguide mode are reported. The optimal geometries of the cavities are found for various lasers, including those with a wide wavelengthtuning range, for example, freeelectron lasers. Knowing the coupling losses, we can calculate the total losses in the laser cavities since the other components of the losses (i.e. the mirror, waveguide, and other losses) can be easily determined. It is assumed that the laser emits a single mode and this is the fundamental waveguide mode. Such operation is theoretically justified and can be implemented in practice for the cavity geometries characterised by lower total losses of the fundamental mode than the losses suffered by the higher waveguide modes. for relatively short waveguides their geometry should ensure that the coupling losses of the fundamental mode are low. This is known to be valid also in the case of the most interesting, from the practical point of view, optimal cavities in which the coupling losses of the fundamental waveguide mode do not exceed a few percent. A direct experimental proof of this conclusion was provided for high values of z=b back in Ref. [9], which describes the first ever waveguide laser emitting a single fundamental waveguide mode for z=b 5:3. References 1. 2. 3. 4. 5. 6. 7. 8. 9. Kogelnik H, Li T Appl. Opt. 5 1550 (1966) Marcatili E A J, Schmeltzer R A Bell Syst. Tech. J. 43 1783 (1964) Krammer H IEEE J. Quantum Electron. QE12 505 (1976) Kubarev V V, Kurenski¯ E A Kvantovaya Elektron. (Moscow) 22 1179 (1995) [ Quantum Electron. 25 1141 (1995)] Kubarev V V, Kurenski¯ E A Kvantovaya Elektron. (Moscow) 23 311 (1996) [ Quantum Electron. 26 303 (1996)] Abrams R L IEEE J. Quantum Electron. QE8 838 (1972) Kubarev V V, Preprint No. 9767 (Novosibirsk: Institute of Nuclear Physics, 1997) Nikiforov A F, Uvarov B V Spetsial'nye Funktsii Matematichesko¯ Fiziki (Special Functions in Mathematical Physics) (Moscow: Nauka, 1984) Smith P W Appl. Phys. Lett. 19 132 (1971)