Firstly, the laser cavity was aligned with the optical cavity length of 2.92 cm.
With the real-time dynamical temporal trace, the laser cavity can easily adjusted to be in a SML operation with pulse repetition rate of 5.14 GHz, as shown in Fig. 3.3-1. We also found that the average output power is nearly independent of the crystal/mirror separation d in the range of 1-11 mm. Even so, the number of longitudinal lasing modes was found to be decreased with increasing the separation d. Therefore, the pulse width can be adjusted by changing the crystal/mirror separation to control the amount of spatial hole burning. With the high resolution (0.003 nm) optical spectrum analyzer, the optical spectrums have measured, as shown in Fig. 3.3-2. It can be clearly seen that the number of longitudinal lasing modes decrease form 14 to 4 modes as the separation d change from 1 mm to 9 mm. As the consequently, with increasing crystal/mirror spacing up to 11 mm, the mode-locked pulse width varies smoothly from 9 to 39 ps, and the spectral bandwidth varies from 77 to 15.4 GHz, as shown in Fig. 3.3-3. This result confirms the speculation that the spatial hole burning is also effective in passively mode-locked lasers [10].
Besides the pulse width can be varied by controlling the number of longitudinal lasing modes in self-mode-locked laser with SHB effect. Experimental results revealed that the stability of the mode-locked pulse train also could be significantly
65
Fig. 3.3-1. (a) Pulse trains for time span 2 ns, demonstrating the mode-locked pulses. (b) Corresponding power spectrum.
200 ps/div
Fig. 3.3-2. Optical spectrum for different crystal/mirror separations d. (a) d = 1 mm. (b) d = 2 mm. (c) d = 5 mm. (d) d = 9 mm.
Wavelength (nm)
1064.1 1064.2 1064.3 1064.4 1064.5 1064.6 1064.7
Intensity (a.u.)
1064.1 1064.2 1064.3 1064.4 1064.5 1064.6 1064.7
Intensity (a.u.)
1064.1 1064.2 1064.3 1064.4 1064.5 1064.6 1064.7
Intensity (a.u.)
1064.1 1064.2 1064.3 1064.4 1064.5 1064.6 1064.7
Intensity (a.u.)
1064.1 1064.2 1064.3 1064.4 1064.5 1064.6 1064.7
Intensity (a.u.)
1064.1 1064.2 1064.3 1064.4 1064.5 1064.6 1064.7
Intensity (a.u.)
1064.1 1064.2 1064.3 1064.4 1064.5 1064.6 1064.7
Intensity (a.u.)
1064.1 1064.2 1064.3 1064.4 1064.5 1064.6 1064.7
Intensity (a.u.)
67
Fig. 3.3-3. Mode-locked pulse width and bandwidth as function of crystal/mirror separation.
trains have some amplitude fluctuation, as shown in Fig. 3.3-4(a). The corresponding power spectrum is shown in Fig. 3.3-4(b). It can be seen that the relative frequency deviation of the power spectrum, v/v, is 3 10 4, where v is the center frequency of the power spectrum and v is the frequency deviation of full width at half maximum.
On the other hand, when the cavity is aligned for the crystal/mirror separation 11
d mm , the laser output pulses exhibit more stable with lower amplitude fluctuation and the relative frequency deviation of the power spectrum is smaller than
7.8 10 5, as seen in Fig. 3.3-4(c) and 3.3-4(d). Therefore, the more amplitude fluctuation the mode-locked lasers exhibit, the bigger the relative frequency deviation of the power spectrum is. Figure 3.3-5 depicts that the relative frequency deviation of the power spectrum versus the number of longitudinal lasing modes. As shown in Fig.
3.3-5, reducing the number of longitudinal lasing modes can diminish the amplitude fluctuation to effectively improve the SML pulse stability.
69
Fig. 3.3-4. Mode-locked pulse trains and power spectrum for two different crystal/mirror separation d. (a) Pulse trains of time span 5 μs for d 1 mm. (b) Corresponding power spectrum. (c) d 11 mm. (d) Corresponding power spectrum.
Fig. 3.3-5. Relative frequency deviation of the power spectrum . Number of modes N
0 3 6 9 12 15
x 10-4
0 1 2 3 4
71
3.4 Theoretical Model for Estimating Maximum Longitudinal Lasing Modes in Self-Mode-Locked Lasers
Since the number of longitudinal modes play a critical role for the stability of SML operation, it is useful to develop a method for estimating the number of lasing modes. In this section, we extend the early work of Zayhowski [14] to derive an analytical formula for estimating the maximum number of longitudinal modes in term of the cavity geometry and well-know material parameters.
For a standing-wave laser consisting of N oscillating modes all with equal amplitudes and with the phases identically zero, the electric field is given by
1
where w is the beam radius. Since the cavity length is much longer than the laser l wavelength, the typical value of m is greater than 0 10 . Thus it always holds 3
expressed as
The function represents a standing-wave pattern which is fully modulated for the position near the reflecting mirrors and continuously loses contrast for the position away from the mirror. Due to the standing-wave nature, superposition of the light fields of modes results in an envelope function for the total light intensity. The spatial variation of the envelope SN(z lcav) occurs on the cavity-length scale to be much slower compared to rapid undulations of the intensities of individual modes. As N get larger, SN(z lcav) becomes narrower such that it only has weight very close to the
73
to replace the envelope function SN(z lcav). In term of gN( )z , the average intensity in Eq. (3.4.4) can be written as
standing-wave laser consisting N oscillating modes. Figure 3.4-1 depicts the calculated results for Gres( , , , )x y z N and IN( , , )x y z as a function of z, where we take N 11, 0x , and y m0 200 for the convenience of presentation.The maximum number of modes Nmax that can oscillate in a standing-wave cavity is determined from the condition that the maximum value of N in
( , , , )
Gres x y z N leads to the effective round-trip gain not less than the round-trip loss, i.e., emission lifetime, L is the round-trip loss, R is the reflectivity of the output
Fig. 3.4-1. Calculated results for Gres( , , , )x y z N (darkred line) and ( , , )
IN x y z (bule line) as function of z with N 11, x , and y 0 200
m .
0.00 0.05 0.10 0.15 0.20
<IN(0,0,z)>, Gres(0,0,z,11)
0.0 0.5 1.0 1.5 2.0
<IN(0,0,z)>
Gres(0,0,z,11)
z l
cav75
where is the absorption coefficient at the pump wavelength, w is the pump p radius, and () is the Heaviside step function. Substituting Eq. (3.4.7) and Eq.
(3.4.9) into Eq. (3.4.8) and carrying out the integration in the transverse directions, we can obtain the relationship of res( , )N d N N max , where the residual gain overlap parameter ( , )res N d is defined as
res( , ) (2 cav) g 1 lg d lg 2( ) N( ) (z d) N d l l e d cos kz g z e dz
, (3.4.10) and the effective loss-to-pump factor is defined as
ln(1 )R L
(wl2w2p) 2 hp Pabs, (3.4.11)As a result, the maximum number of modes Nmax can be determined with the criterion
with increasing d . For the experimental condition which reported above, the effective loss-to-pump factor can be found to be 6.19 10 3, where the values of the parameters are as follows: 2.5 10 cm 18 2 , 1.5 Pabs W , 100 s ,
0.85
R , L0.005 , hp 2.45 10 19 J , 0.06 wl mm , and wp 0.06 mm . Applying 6.19 10 3 into Fig. 3, the maximum number of modes Nmax can be determined as a function of d, depicted in Fig. 3.4-3. The good agreement between the theoretical estimations and the experimental data validates the usefulness of the present model.
77
Residual gain overlap parameter resN
10-4
Residual gain overlap parameter resN
10-4
Fig. 3.4-3. Theoretical estimations and experimental data for the maximum number of modes Nmax as function of d.
Separation d (mm)
0 2 4 6 8 10
Number of longitudinal modes N
2 4 6 8 10 12 14 16 18
Experimental data Theoretical estimaition
79
3.5 Conclusion
We have experimentally confirmed that a reliable spontaneous mode-locked (SML) can occur in short-cavity Nd:YVO4 lasers without employing an extra nonlinearity. We further found that the stability of SML pulses could be significantly improved by reducing the number of longitudinal lasing modes to diminish the phase fluctuation. Considering the SHB effect, we have driven an analytical formula to establish the relationship between the number of longitudinal lasing modes and the crystal/mirror separation. The theoretical estimations for the number of longitudinal lasing modes were shown to be in good agreement with experimental observations.
[1] H. C. Liang, R. C. C. Chen, Y. J. Huang, K. W. Su, and Y. F. Chen, “Compact efficient multi-GHz Kerr-lens mode-locked diode-pumped Nd:YVO4 laser,” Opt.
Express 16, 21149 (2008).
[2] H. C. Liang, H. L. Chang, W. C. Huang, K. W. Su, Y. F. Chen, and Y. T. Chen,
“Self-mode-locked Nd:GdVO4 laser with multi-GHz oscillations: manifestation of third-order nonlinearity,” Appl. Phys. B 97, 451 (2009).
[3] H. C. Liang, Y. J. Huang, W. C. Huang, K. W. Su, and Y. F. Chen, “High-power, diode-end-pumped, multigigahertz self-mode-locked Nd:YVO4 laser at 1342 nm,” Opt. Lett. 35, 4 (2010).
[4] J. A. Armstrong, “Measurement of picosecond laser pulse widths,” Appl. Phys.
Lett 10, 16 (1967).
[5] J. A. Giordmaine, P. M. Rentzepis, S. L. Shapiro, and K. W. Wecht, “Two photon excitation of fluorescence by picosecond light pulses,” Appl. Phys. Lett. 11, 216 (1967).
[6] H. P. weber, “Generation and measurement of ultrashort light pulses,” J. Appl.
Phys. 39, 6041 (1968).
[7] R. Dändliker, H. P. Weber, and A. A. Grütter, “Influence of systematic phase
81
(1969).
[8] A. A. Grütter, R. Dändliker, H. P. Weber, “The output intensity of a non-ideally mode-locked laser,” J. Appl. Math. Phys. 20, 574 (1969).
[9] R. Paschotta, J. Aus der Au, G. J. Spühler, S. Erhard, A. Giesen, and U. Keller,
“Passive mode locking pg this-disk laser: effect of spatial hole burning,” Appl.
Phys. B 72, 267 (2001).
[10] C. J. Flood, D. R. Walker, and H. M. van Driel, “Effect of spatial hole burning in a mode-locked diode end-pumped Nd:YAG laser,” Opt. Lett. 20, 58 (1995).
[11] H. S. Kim, S. K. Kim, and B. Y. Kim, “Longitudinal mode control in few-mode erbium-doped fiber lasers,” Opt. Lett. 21, 1144 (1996).
[12] L. Jiang and L. V. Asryan, “How many longitudinal modes can oscillate in a quantum-dot laser: an analytical estimate,” IEEE Photon. Technol. Lett. 20, 1661 (2008).
[13] G. J. Kintz and T. Baer, “Single frequency operation in solid-state laser materials with short absorption depths,” IEEE Quantum Electron. 26, 1457 (1990).
[14] J. J. Zayhowski, “Limits imposed by spatial hole burning on the single-mode operation of standing-wave laser cavities,” Opt. Lett. 15, 431 (1990).
Chapter 4
Measurements of Refractive Indexes and Thermal Optical
Coefficient: Application of
Self-Mode-Lock ed Lasers
83
4.1 Refractive Indexes and Thermal Optical Coefficient
The refractive index of the material medium is an important optical parameter since it exhibits the optical properties of the material. Its values are usually required to interpret various types of spectroscopic data. The common method for measuring the refractive index is using the ellipsometer. In generally, this method only can measure the thin film materials but not the bulk materials. Besides the refractive indexes of crystals, thermal optical coefficient is another important parameter of crystals, since it directly influences the pumped-power-induced thermal lensing that expressed by [1]
2 1
In solid-state lasers, this pumped induced thermal lens is of primary importance because of its significant influence on laser stability, oscillation mode size, maximum achievable average power, efficiency, and output beam quality [2-4]. Therefore, a knowledge of the thermal optical coefficient in the direction parallel and perpendicular to the c-axis of the anisotropic crystals like Nd:YVO4 and Nd:GdVO4
are necessary. Recently, several researchers have measured the thermal optical coefficient of a Nd:GdVO4 crystal by measuring the focal length of thermal lens as a function of the pumped power [5,6]. More recently, the stable self-mode-locked Nd:YVO4 lasers are successful developing at the lasing wave length of 1064 nm and
for measuring the refractive index of crystals. Here, we have demonstrated a novel method to measure the ordinary and extraordinary refractive indexes of Nd:YVO4
with different doped concentration by measuring the repetition rate shifts of the mode-locked Nd:YVO4 laser as a crystal is placed inside the laser cavity. We also have measured several crystals such as Nd:GdVO4, Nd:YGdVO4, and KTP in the same laser configuration. On the other hand, the thermal optical coefficients were experimentally obtained by heating the crystal with an oven. In this method, with the measured variation of the pulse repetition rate as a function of the temperature, we can estimate the thermal optical coefficient of Nd:YVO4 crystal. The experimental results reveal that both of the measurement are in agreement with that reported in the literature [9-15].
85
4.2 Experimental Setup
Figure 4.2-1 depicts the configuration the experimental setup. Firstly, the stable self-mode-locked lasers are achieved for operating at 1064 nm and 1342 nm. The mode-locked cavity is a simple concave-plano resonator. The gain medium is a-cut 0.2 at. % Nd:YVO4 crystal with a length of 10 mm. Both end surface of the Nd:YVO4
crystal were wedge 2∘to suppress the Fabry-Perot etalon effect. The laser crystal was wrapped with indium foil and mounted in a water-cooled copper holder. The water temperature was maintained around 20 °C to ensure stable laser output. The input mirror was a 504 mm radius-of-curvature concave mirror with antireflection coating at 808 nm on the entrance face and high transmittance coating at 808 nm on the second surface. Two kinds of output coupler are used for different operation at 1064 nm and 1342 nm. The pump source was a 3-W 808-nm fiber-coupled laser diode with a core diameter of 100 m and a numerical aperture of 0.16. Focusing lens with 25 mm focal length and 85% coupling efficiency was used to re-image the pump beam into the laser crystal. The average pump size was approximately 70 μm. The samples are placed behind of the laser gain medium near the output coupler.
The optical cavity length was set to be approximately 5.4 cm with the corresponding free spectral range (FSR) of 2.75 GHz. The mode-locked pulses were detected by a high-speed InGaAs photodetector (Electro-optics Technology Inc.
ET-3500 with rise time 35 ps), whose output signal was connected to a digital
analyzed by an RF spectrum analyzer (Advantest, R3265A) with bandwidth of 8.0 GHz.
87
Fig. 4.2-1. The configuration of the refractive indexes measuring system.
Nd:YVO4 Pumping
beam
Laser diode Focusing
lens
Cavity mirror
Sample
Stage Stage
Output coupler
4.3 Experimental Results and Analysis
First of all, the stable self-mode-locked are established for operating at 1064 nm and 1342 nm. The pulse repetition rates are measured at the frequency of 2.748 GHz.
The relative frequency deviation of the power spectrum, v/v, is smaller than 6 10 5, where v is the center frequency of the power spectrum and v is the frequency deviation of full width at half maximum. After the sample inserts into the optical resonator, the pulse repetition rates are observed a finite shit corresponding the change of optical path length and can be expressed as:
'
2 'opt 2 opt
c c
f f f
L L
, (4.3.1) )
where Lopt is the optical path length with a sample inside. By measuring the shift of pulse repetition rate, the change of optical path length Lopt L'opt Lopt can be estimate. However, the difference of the optical path length is associated with the length and refractive index of crystal. Therefore, the refractive index will be obtained with a given crystal length. However the refractive index which estimated by a mode-locked pulse is called group index n [16], a convenient way for determining g the refractive index n according to p
89
Here, we have measured three kinds of Nd3+-doped crystals with different host materials including Nd:YVO4, Nd:GdVO4, and Nd:YGdVO4 and the self-mode- locked lasers are not only operating at 1064 nm but also operating at 1342 nm. Table 4.3.1 shows the experimental results for the measurement of ordinary refractive indexes and extraordinary refractive indexes at 1064 nm and 1342 nm. The refractive indexes measuring in this method are found to be agreement with the values that reported in the literatures [9-15]. Figure 4.3-1 demonstrates the experimental results for measuring the ordinary refractive index (n ) and extraordinary refractive index o (n ) of Nd:YVOe 4 with different doped concentration in the range of 0.2 - 0.8 at. %.
There are three samples for each doped concentration. The ordinary refractive indexes are found in the range of 1.9997 - 2.0002 and the extraordinary refractive indexes are found in the range of 2.2232 - 2.2240 as a function of doped concentration.
It can be seen that the crystals exhibit higher refractive index with the higher doped concentration and the variation are observed in the order of 104..
In the second part, we have measured the thermal optical coefficient of 0.1 at. % Nd:YVO4 crystal with a length of 12 mm. In the experimental system, an oven is used for heating the crystal inside the optical cavity. By heating the crystal form 30 °C to 200 °C, the pulse repetition rate of the mode-locked laser was found to decrease gradually. The frequency shifts are resulted from the increase of the optical path
by:
( 1)
opt cav c
L L (4.3.3) n l
where Lcav is the cavity length and n is the refractive index of the crystal. As the crystal is heated, the optical length path has changed and can be written as
' ( 1) (1 )
where dn dT is the thermal optical coefficient, is the thermal expansion coefficient, and T is the increase of temperature. Thus, the difference of the optical path length is obtained by
1
2However, the last term in Eq. (4.3.4) is quite small usually and can be neglect. With the frequency shifts which were obtained experimentally, the difference of optical path length would be carrying out. Therefore, with the parameterslc 12 mm ,
1.996
no , 2.223ne , and 4.43 10 6 K, the thermal optical coefficient can be found to be 7.6 10 6 K and 4.2 10 6 K for the direction perpendicular and parallel to the c-axis of crystal respectively. The experimental results are in good
91
agreement with that reported in the literature (the value of thermal optical coefficient are 8.5 10 6 K and 3 10 6 K, respectively, for a-axis and c-axis of Nd:YVO4
crystal) [12].
Table. 4.3.1. The experimentally measuring refractive indexes for different material at operating wavelength of 1064 nm and 1342 nm.
Materials 1064 nm 1342 nm
1.7958 1.8872 [15] 1.7685 1.8536 [15]
ny nz ny nz
93
Fig. 4.3-1. The experimental results for measuring refractive indexes of Nd:YVO4 with different doped concentrations. (a) For ordinary refractive indexes. (b) For extraordinary refractive indexes.
(b) (a)
Fig. 4.3-2. Frequency shift versus the temperature of oven.
Temperature (K)
300 320 340 360 380 400 420 440 460 480
Frequency shift f (KHz)
-1000 -800 -600 -400 -200 0 200
a-axis crystal c-axis of crystal
95
4.4 Conclusion
In summary, we have demonstrated a novel method to estimate the refractive indexes of several crystals. The ideal is based on the shift of pulse repetition rate as a crystal is placed inside the optical cavity. The ordinary and extraordinary refractive indexes of Nd:YVO4 crystals with different doped concentration are experimentally found that the refractive indexes increasing as the doped concentration increasing in the range from 0.2 at.% to 0.8 at.%. Besides, the Nd:YVO4 crystal, we also measuring the Nd:GdVO4 and Nd:GdYVO4 crystals. The experimental results are consist with the value which have reported. By heating the crystal, the thermal optical coefficient of Nd:YVO4 crystal are experimentally observed. The thermal optical coefficients are 7.6 10 6 K and 4.2 10 6 K at different axis of Nd:YVO4 crystal and are in good agreement with the results in other reports.
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[2] Y. F. Chen, T. M. Huang, C. F. Kao, C. L. Wang, and S. C. Wang, “Optimization in scaling fiber-coupled laser-diode end-pumped lasers to higher power:
influence of thermal effect,” IEEE J. Quantum Electron. 33, 1424 (1997).
[3] Y. F. Chen, C. F. kao, T. M. Huang, C. L. Wang, and S. C. Wang, “Influence of thermal effect on output power optimization in fiber-coupled laser-diode end-pumped lasers,” IEEE J. Sel. Top. Quantum Electron. 3, 29 (1997).
[4] W. A. Clarkson, “Thermal effects and their mitigation in end-pumped solid-state lasers,” J. Phys. D 34, 2381 (2001).
[5] H. Zhang, J. Liu, J. Wang, C. Wang, L. Zhu, Z. Shao, X. Meng, X. Hu, M. Jiang, and Y. T. Chow, “Characterization of the laser crystal Nd:GdVO4,” J. Opt. Soc.
Am. B 19, 18 (2002).
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Sharma, and T. P. S. Nathan, “Experimental determination of the thermo-optic coefficient (dn dT) and the effective stimulated emission cress-section ( )e
97
Phys. B 77, 81 (2003).
[7] H. C. Liang, Ross C. C. Chen, Y. J. Huang, K. W. Su, and Y. F. Chen, “Compact efficient multi-GHz Kerr-lens mode-locked diode-pumped Nd:YVO4 laser,” Opt.
Express 16, 21149 (2008).
[8] H. C. Liang, Y. J. Huang, W. C. Huang, K. W. Su, and Y. F. Chen, “High-power, diode-end-pumped, multigigahertz self-mode-locked Nd:YVO4 laser at 1342 nm,” Opt. Lett. 35, 4 (2010).
[9] B. H. T. Chai, G. Loutts, J. Lefaucheur, X. X. Zhang, P. Hong, and M. Bass,
“Comparison of laser performance of Nd-doped YVO4, GdVO4, Ca5(PO4)3F, Sr5(PO4)3F and Ca5(VO4)3F,” in Advanced Solid-State Lasers, T. Y. Fan and B. H.
T. Chai, eds., Vol. 20 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D. C., 1994), 41.
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Zhao, and G. Zheng, “Thermal conductivity and refractive indices of Nd:GdVO4
crystals,” Cryst. Res. Technol. 38, 793 (2003).
[12] J. C. Bermudez G., V. J. Pinto-Robledo, A. V. Kir’yanov, and M. J. Damzen,
“The thermo-lensing effect in a grazing incidence, diode-side-pumped Nd:YVO4
laser,” Opt. Commun. 210, 75 (2002).
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Appl. Opt. 27, 3314 (1988).
[16] H. G. Danielmeyer and H. P. Weber, “Direct measurement of the group velocity of light,” Phys. Rev. A 3, 1708 (1971).
99
Chapter 5
Self-Mode-Locked Lasers for High-Order
Transverse Modes
5.1 Paraxial Approximation Maxwell’s Equation : Wave Functions of Spherical Laser Cavity
It is well-know that the wave propagation in a source-free medium follows the Maxwell’s equation which can be expressed as [1,2]
0
Thus, the electric field can represented as
2 2
101
this is the well-know Helmholtz equation, where k is the wave vector. For a wave which propagates primarily along the z direction, ( , , )E x y z can be written as
this is the well-know Helmholtz equation, where k is the wave vector. For a wave which propagates primarily along the z direction, ( , , )E x y z can be written as