Transverse excess noise factor and transverse mode locking in a gain-guided laser

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Transverse excess noise factor and transverse mode locking in

a gain-guided laser

Ching-Hsu Chen

a,*

, Po-Tse Tai

b

, Wei-How Chiu

b

, Wen-Feng Hsieh

b,*

a

National Center for High-Performance Computing, 21 Nan-Ke 3rd Rd., Hsinshi, Tainan County 744, Taiwan

b

Department of Photonics and Institute of Electro-Optical Engineering, National Chiao Tung University, 1001 Tahsueh Rd., Hsinchu 30050, Taiwan

Received 30 May 2004; received in revised form 27 August 2004; accepted 1 October 2004

Abstract

We use the Collins integral together with a rate equation to calculate the transverse mode profiles near g1g2= 1/4 in a tightly focused end-pumped Nd:YVO4laser. The transverse mode locking is confirmed from the mode decomposition into the degenerate empty-cavity eigenmodes and from the observation of beam profile variation along the propagation distance. We obtain that the calculated transverse excess noise factor without considering the thermal effect is consistent with the previous research and it depends on the pump size. We further study the influence of the thermal lens effect on the K factor and discuss how to suppress the laser instabilities that occur near the degeneracy.

Keywords: Beam characteristics; Resonators; Diode-pumped lasers

1. Introduction

The excess noise factor was discussed ﬁrst by Petermann[1]in a gain-guided semiconductor la-ser and thus it is called Petermann K factor. The K factor is large in unstable laser cavities due to

non-orthogonality properties of the transverse modes [2,3]. In the case of a geometrically stable cavity, introducing apertures inside the cavity was shown leading to large K factors[4]. Recently, Maes and Wright [5] found that the K factor is cavity-configuration-dependent near the degener-ate cavity configurations in a geometrically stable cavity with Gaussian gain. They explained that K = 1 at the exact degeneracies is due to flat phase front coming from the phase-locked empty-cavity eigenmodes and that a large value of K beside

*

Corresponding authors. Tel.: +88 665 724 805; fax: +88 635 716 631.

E-mail addresses:[email protected](C.-H. Chen),

[email protected](W.-F. Hsieh).

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the degeneracies is due to highly curved phase front resulting from the incoherent superposition of the empty-cavity eigenmodes.

Previously, we have observed the multi-beam-waist (MBW) modes[6]near the degenerate cavity of g1g2= 1/4 in a tightly focused end-pumped Nd:YVO4laser. Our laser system that consists of a plano-concave cavity with the flat end mirror at the coated face of the Nd:YVO4crystal is the same as that of in [5]. We have understood the MBW mode at the exact degeneracy by a combina-tion picture of ray and wave optics that is similar to the notation of the asymmetric modes[7]. Fur-thermore, in the same system the cavity-configura-tion-dependent laser instabilities were found on each side of the degeneracy of g1g2= 1/4 and the thermal lens effect induced by the tightly focused pump were considered to explain the experimental data[8]. Because the thermal lens effect introduces a phase distribution into the diffraction, it will influence the transverse mode and the K factor. Therefore, it is necessary to include the thermal lens effect in the calculation of K factor. Moreover, the laser instabilities appear near the region of maximal K factor for some conditions, so the laser parameters should be chosen carefully when one wants to measure the K factor in such a laser.

In this work, we use the Collins integral to-gether with a rate equation to calculate the trans-verse mode profiles near g1g2= 1/4. We further use the genetic algorithm (GA) to decompose the calculated mode profile into the degenerate La-guerre–Gaussian (LG) modes and thus obtain their mode weightings and relative phases. Most importantly, from the experimental observation of beam profile variation along the propagation distance, we confirm phase-locking of the degener-ate transverse modes near the degeneracy. We cal-culate the K factor with and without considering the thermal effect and obtain that it is not only cavity-configuration-dependent but also pump-size-dependent. In Section 2, we describe the mode-calculation model including the thermal lens effect. In Section 3.1, we show the numerical re-sults of mode calculation together with the fitted mode expansion and then show the mode profile variation along propagation distance. The K factor as a function of the cavity length is studied in

Sec-tion 3.2 with and without considering the thermal lens eﬀect. The laser instabilities are also discussed. The conclusions are stated in Section 4.

2. The calculation model

Consider a plano-concave end-pumped solid-state laser shown in Fig. 1. It consists of a laser crystal with one of its end faces high-reflection coated as a flat mirror and of a curved mirror with radius of curvature R, which are separated by dis-tance L. Let the reference plane be the place where the light beam just leaves the laser crystal toward the curved mirror. Under cylindrical symmetry, propagation of the light field toward the curved mirror and back to the flat mirror according to the Collins integral is[9]

E_mþ1ðrÞ ¼2pi Bk Z a 0 expðik2LÞEþ mðr0Þ expfðip=BkÞðAr02_{þ Dr}2_Þg J0ð2prr0=BkÞr0dr0; ð1Þ with round trip transmission matrix A B

C D

. Here Eþ_mðr0_{Þ and E}

mþ1ðrÞ are the electric fields of the mth and the (m + 1)th round trips, respectively, at the planes immediately after and before the gain medium (denoted by the superscripts + and); r0 and r are the corresponding radial coordinates, k is the wavelength of the laser, a is the aperture radius on the reference plane, and J0is a Bessel function of zero order. The aperture radius must be chosen large enough with many times of the fundamental mode radius to ensure that the diffraction loss can be neglected. In a thin-slab approximation, we can relate the electric fields Eþ_mþ1 to E_mþ1 (after and before the gain medium) in the same round trip as

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Eþ_mþ1ðrÞ ¼ qE

mþ1ðrÞ expðrDNdÞPðr=aÞ; ð2Þ where 1 q2

is the round-trip energy loss, r is the stimulated-emission cross-section, DN is the popu-lation inversion per unit volume, d is the length of the active medium, and P(r/a) is an aperture func-tion that equals 1 for r less than aperture radius a and equals 0 otherwise. Furthermore, assuming that the evolution of the population inversion fol-lows the rate equation of a four-level system, we can write the rate equation as

DNmþ1 ¼ DNmþ RpmDt cDNmDt j Emj

2 E2_s DNmDt;

ð3Þ where Rpm is the pumping rate, Dt is the round-trip time, Esis the saturation parameter of the ac-tive medium, and c is the spontaneous decay rate. It was found that a standing-wave resonator can be approximated by a ring resonator if a thin gain medium is placed close to one of the end mirrors [10]. This method is similar to the Fox–Li approach [11] and has been used to ana-lyze the decay rate of standing-wave laser cavities

[12]. For a continuous Gaussian pump proﬁle Rpm¼ Rp0expð2r2=w2pÞ with constant pump beam radius wp throughout the active medium (thin slab), the total pump rate over the entire active medium is

Z V

RpmdV ¼ Pp=hmp; ð4Þ

where Ppis the effective pump power and hmpis the photon energy of the pumping laser. Because we concerned chiefly with transverse mode profile, we did not consider the dispersion of the active medium or frequency detuning between the atomic transition and the cavity mode; thus the gain was assumed to be real. When the thermal lens effect was considered, we imposed a term exp(iDU(r)) in the diffraction integral, where DU(r) is the phase shift induced by thermal lens effect. According to Eqs. (2), (6), and (11) of [13], the phase shift can be written as DUðr0_{Þ ¼} Z d 0 kDTðr0_{; zÞ}dn dT dz; ð5Þ where DTðr0; zÞ ¼ 1 Kc Z rb r0 anPabs 2p expðazÞ 1 expð2r 2_=w2 pÞ r dr

is the temperature diﬀerence between the calcu-lated point (r0_{,z) and the boundary point (r}

b,z), wp is the pump radius, n is the fractional thermal loading, Pabs is the absorbed pump power, z is the axial coordinate, and rb, d, a, Kc, and dn/dT are the radius, thickness, absorption coefficient, thermal conductivity, and the thermal-optic coeffi-cient of the laser crystal, respectively. The thermal induced stress, the thermal deformation of the crystal, and the thermal fluctuation were neglected. In an ordinary axially pumped solid-state laser, the round-trip propagation time is many orders of magnitude shorter than the spontaneous decay time, especially in a short cavity; as a result, it would take a large number of iterations to arrive at the convergent state. To reduce computation time, we used the scaling method [12]to magnify the c value of the Nd:YVO4laser by 100 times to obtain the continuous-wave solution because the transverse mode distribution is independent of c as long as Dt 1/c. Given an initial DN and E, after the power output undergoes a procedure sim-ilar to the relaxation oscillation the field distribu-tion E converges to a cw steady soludistribu-tion. In our simulation, the aperture was chosen 600 lm that is much larger than the fundamental beam radius of 108 lm for g1g2= 1/4. To implement the Collins integral by the Romberg method, we divided the 600-lm aperture radius into 1024 segments.

3. Results and discussion 3.1. Transverse mode locking

The parameters used for mode calculation are the same as in[6]. The important control parameters in this work are the cavity length L and the pump ra-dius wp. The other parameters for calculating the thermal lens eﬀect are n = 0.23, a = 1930 m1, Kc= 5.23 W m1K1, and dn/dT = 8.5· 106 K1. We show the normalized intensity proﬁle and

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the phase profile on the reference plane with solid circles inFigs. 2(a) and (b)at the degeneracy (L = 6 cm) for wp= 30 lm and the effective pump power of 100 mW without considering the thermal lens effect. In order to show the good fitting of mode decomposition using GA, we plot the fitted results inFigs. 2(a)–(d) with open circles and use the loga-rithm scale inFig. 2(a). The mode decomposition is done with 13 fitting parameters including six ampli-tude weightings and seven relative phases. For the aperture radius of 600 lm on the reference plane, we expand the calculated mode profile into the 1/ 3-degenerate LGpm modes with p = 0, 3, 6, . . . ,18 and m = 0, where p is the radial mode index and m

is the angular index. The normalized electric ﬁeld of LGp0mode can be expressed as

Ep0ðr; zÞ ¼ Ap0ðr; zÞ exp r2 wðzÞ2 ! exp i kr 2 2RðzÞ exp i kz ð2p þ 1Þtan1 z zR þ dp ; where Ap0ðr; zÞ ¼ E0 wc wðzÞL 0 p 2r2 wðzÞ2 !

is the modal function, E0is the normalization con-stant, zRis the Rayleigh length, w(z) is the beam

0 30 60 0 1 1 3 0 100 200 300 400 500 600 10- 6 10- 5 10- 4 10- 3 10- 2 10- 1 100 Normalized intensity Radial coordinater (µm) 0 100 200 300 400 500 600 -4 -3 -2 -1 0 1 2 3 4 Phase (radian) Radial coordinater (µm) (a) (b) 0 30 60 0 1 1 3 0 100 200 300 400 500 600 10- 6 10- 5 10- 4 10- 3 10- 2 10- 1 100 Normalized intensity Radial coordinater (µm) -5 -4 -3 -2 -1 0 1 2 3 Phase (radian) Radial coordinater (_µm) 0 100 200 300 400 500 600 (c) (d)

Fig. 2. The intensity profile in logarithm scale (a) and the phase profile (b) at the exact degeneracy (L = 6.0 cm) obtained from the mode calculation (solid circles) and from the fitted result of mode decomposition (empty circles). Inset in (a) are the intensity profile in linear scale (solid curve) and the saturated gain distribution (dashed curve). (c) and (d) are, respectively, the intensity and phase profiles for L = 6.01 cm.

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radius, R(z) is the radius of curvature of the phase front, r and z are, respectively, the radial and axial coordinates, and L0

pis the Laguerre polynomial for mode index p. We assume all the excited LGp0 modes have the same wavenumber and then the intensity profile |g0E00+ g3E30+ + g18E18,0|2 with seven amplitude weightings g(g0 be fixed unity) and seven relative phases dp is fitted to the mode-calculation profile. We see that the resultant fitted profiles match with the mode-calculation profiles extremely well in Figs. 2(a)–(d). From

Fig. 2(a) the central lobe of the intensity profile is near-Gaussian with the waist radius of 30 lm (approximately equals to the pump radius, see the solid curve in the inset with linear scale), which shows that the laser is strongly gain-guided. Note that the radius of the fundamental mode, wo, is 108 lm. The seriously saturated gain distribu-tion is shown with the dashed curve in the inset ofFig. 2(a). The gain distribution is obtained from the term exp(rDNd) in Eq.(2), where DN is r-de-pendent. Fig. 2(b) shows that the phase profile is flat within r = 200 lm but discontinuously jumps pphase at some positions of r, e.g., the first phase jump at r = 200 lm corresponds to the position of the second intensity zero of the LG3,0 mode. The relative phases of the degenerate LG modes for L = 6.0 cm show the degenerate LG modes are not only phase-locked but also nearly in-phase on the reference plane. The unusual result of flat wavefront on the flat end mirror, obtained in our mode calculation including gain, is the same as in[5]and this was discussed therein.

When the cavity length is slightly tuned away from the degeneracy to L = 6.01 cm, the central lobe of the intensity proﬁle shows a slightly dis-torted Gaussian in the inset of Fig. 2(c) with the solid curve in linear scale. Also shown with the dashed curve in the inset is the saturated gain dis-tribution. We can see in Fig. 2(d) that the phase pattern is already highly curved for r < 100 lm and no longer has p-jumps at some positions of r. Note that the phase is continuous at r = 223 lm because the phase jump is 2p. Besides, at L = 6.01 cm the degenerate LGp0 modes are no longer in-phase on the reference plane but have monotonically increasing relative phases with the increase of p. Even so, these LGp0modes are still

phase-locked to form a stationary mode. Such a stationary mode exhibits proﬁle variation along the propagation distance due to the variation of Gouy phases of the LGp0modes and it is in fact an optical bottle beam that has been presented in

[14]. It is worthy to note that nearly the same behavior for the case of L = 5.99 cm except that the phase pattern is inverted within r = 100 lm and the relative phases of the LGp0modes decrease monotonically with increase of p.

At L = 6.05 cm, the intensity proﬁle is much distorted from Gaussian and the phase pattern is highly curved for r < 150 lm. We will show later that this cavity length corresponds to where the maximal K factor is. The mode weightings and the relative phases of the LGp0modes for various cavity lengths are summarized inFigs. 3(a) and (b)

except for the absence of L = 6.04 cm because the laser instability occurs there. The self-pulsing and chaotic time evolution have been observed there and also on the other side of the degeneracy when the pump radius is smaller than 40 lm[8]. We can see inFig. 3that the mode weightings for the case of L = 6.05 cm have signiﬁcant decrease for p = 3, 6, 9 as compared with the case of L = 6.03 cm and that the relative phases no longer monotonically increase but alternate for p > 6.

Although the far-field intensity pattern looks like a Gaussian profile when the cavity is tuned far away from the degeneracy to L = 6.10 cm, the mode profile still varies along the propagation. We can see fromFig. 4(a)that the calculated mode profile exhibits a dark center from z = 6.8 to 8.0 cm. We therefore used this z-dependent profile to verify the phase-locking of degenerate transverse modes as L is tuned from 5.90 to 6.10 cm. The pro-file will appear a dark center approximately from 19 to 23 cm after a convergent lens with focal length of 7 cm when the convergent lens is put be-hind the output coupler at a distance of 12 cm.

Fig. 4(b) shows the photograph taken at a distance of 20.5 cm after the convergent lens from our Nd:YVO4 laser as L is set on the right edge of the phase-locking region. We found experimen-tally that the phase-locking region has been shifted 500 lm toward the short cavity side by the ther-mal lens eﬀect for wp= 30 lm and pump power of 150 mW.

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3.2. Transverse excess noise factor

We calculate the K factor according to the rela-tion given by [4,5], K¼ ½R₀1j EðrÞj2rdr2=jR₀1 E2ðrÞr drj2, that is valid for a plano-concave cavity with the electric field E(r) at the flat end mirror[5]. The upper limit of the integral in the expression of K may be replaced by the aperture radius as long as the aperture radius is large enough. Our calcu-lation shows that the K factor is indeed independ-ent of the aperture radius for a convergindepend-ent solution E(r) except for L = 6.04 cm where the laser insta-bility appears. Thus, the variation of K factor shown below does not result from diffraction via the aperture[4]but from the inherent mode prop-erties[5]near the degenerate cavity.

By decreasing the aperture radius on the refer-ence plane from 600 down to 500 lm, we found that the K factor is unchanged for the stable out-put but the laser instabilities (e.g., L = 6.04 cm) are suppressed.Fig. 5(a) shows the K factor as a function of L for wp= 40 lm with solid triangles and wp= 30 lm with solid circles when the aper-ture radius is 500 lm without considering the ther-mal lens eﬀect. We obtain that the K factor is pump-size-dependent besides cavity-conﬁgura-tion-dependent. However, further decreasing the pump size smaller than 30 lm does not increase the K factor. It seems that there is a proper pump size that leads to the largest K-factor. For compar-ing the value of K with the results of [5]around

0 3 6 9 12 15 18 0.0 0.2 0.4 0.6 0.8 1.0 Mode weight

Radial mode index p L6.00 L6.01 L6.02 L6.03 L6.05 L6.06 L6.07 0 3 6 9 12 15 18 0.0 0.5 1.0 Phase/ π

Radial mode index p

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Fig. 3. The mode weightings (a) and the relative phases (b) of the LGp0modes as L is tuned away from the degeneracy.

Fig. 4. (a) The numerical beam proﬁle variation along the propagation distance z for L = 6.10 cm. The intensity proﬁle with a dark center can be seen at z = 6.88.0 cm that is transformed to a distance of 19–23 cm after the convergent lens. (b) The photograph experimentally taken at 20.5 cm after the convergent lens.

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g1g2= 1/2, we calculated two cases of wp= 58 lm (wp/wo= 1/2, solid triangles) and wp= 40 lm (solid circles) inFig. 5(b). The case of wp= 58 lm is

con-sistent with Fig. 2 of[5]. The value of K factor is comparable for wp= 30 lm in Fig. 5(a) and for wp= 40 lm inFig. 5(b) that means the degeneracy of g1g2= 1/4 is also a good choice for measuring the cavity-conﬁguration-dependent K factor.

As aforementioned, a proper small pump size results in large K but it may lead to the instabilities and strengthen the thermal lens effect. Because the thermal lens effect introduces a phase distribution into the electric field, in turn it will influence the K factor. When the thermal lens effect is included, the result plotted as open circles inFig. 5(c) shows that the K-curve of wp= 30 lm is shifted and dis-torted from the feature of without the thermal lens effect. Also, the K-factor is decreased by the ther-mal lens effect. When the therther-mal lens effect is con-sidered, the laser instabilities occur at L 5.91 cm that is shifted from L 5.96 cm without consider-ing the thermal lens effect. Further decreasconsider-ing the aperture radius to 450 lm is able to fully suppress the laser instabilities. In experiment, decreasing the aperture may be achieved by focusing the pumping beam near the rim of the crystal instead of setting a real hard aperture against the crystal inside the cavity. It is worthy to note that the instabilities are also found near the degeneracy of g1g2= 1/2, for which the instability occurs at L = 3.91–3.94 cm and L = 4.064.09 cm for wp= 40 lm when the aperture radius is larger than 500 lm. There-fore, the laser parameters must be chosen carefully for measuring the cavity-configuration-dependent K factor near the degeneracies.

4. Conclusion

We have confirmed the phase-locking of degen-erate transverse modes near the degeneracy of g1g2= 1/4 in a tightly focused end-pumped Nd: YVO4laser that can be verified by observation of beam profile variation along the propagation dis-tance within a large cavity-length detuning from the degeneracy. We also decomposed the station-ary lasing mode into the degenerate Laguerre– Gaussian modes with their relative locked phase. Furthermore, we obtained that the K factor is cavity-configuration-dependent and pump-size-dependent. When the thermal lens effect is taken

5. 90 5. 95 6.00 6. 05 6.10 1 10 K factor Cavity length L (cm) 3.9 4.0 4.1 1 10 K factor Cavity length L (cm) 5.9 6.0 6.1 1 10 K factor Cavity length L (cm) (a) (b) (c)

Fig. 5. (a) The K factor as a function of L for wp= 30 lm (solid

circles) and 40 lm (solid triangles) near g1g2= 1/4 without

considering the thermal lens eﬀect. (b) The cavity-dependent K factor for wp= 40 lm (solid circles) and 58 lm (solid triangles)

near g1g2= 1/2 without considering the thermal lens eﬀect. Note

that wo= 116 lm for L = 4 cm near g1g2= 1/2. (c) The

cavity-dependent K factor for wp= 30 lm and the eﬀective pump

power of 100 mW near g1g2= 1/4 with (open circles) and

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into account the cavity-conﬁguration-dependent K factor is shifted and decreased. Although the laser instabilities appear near the region of maximal K factor they can be suppressed.

Acknowledgment

The research was partially supported by the Na-tional Science Council of the Republic of China under Grant NSC93-2112-M-009-035.

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