Chapter 2 Passively Q-switched Eye-safe Laser with Optical Parametric Oscillator
2.3 The influence of birefringence on pulse behavior in OPO
2.3.1 Theoretical analysis and discussion in birefringence effect
Figure 2.3-2 depicts the typical thermally induced birefringence effect in a Nd:YAG rod with [111] crystal orientation [15]. The principle axes of the induced birefringence are radially and tangentially directed at each point in the rod cross section. In the propagation of electric field along the rod orientation, a linearly polarized field would suffer a phase difference between radial and tangential direction and turn into elliptically polarized. When a linearly polarized field is along y-axis of the rod cross section, it will be gradually depolarized after propagation and resolved into two components along x- and y-axis. This depolarization indicates an energy coupling and transformation between two axes. If the resonator contains a polarizing element to maintain the polarization of fields on y-axis, the field depolarized into x-axis would be filtered out and becomes a depolarization loss. Without any polarizing element in the resonator, on the contrary, the energy will flow back after multiple round trips.
The proportion of energy transformation, Γ, between two orthogonal axes after a round trip could be estimated from the phase difference δ given by [15]
( )r 2π n r( ) 2l dlcr
where λ denotes the optical wavelength, Δn is the difference of refractive index changes between tangential (Δnψ) and radial (Δnr) direction, lcr is the length of Nd:YAG rod, r and φ are the radius and azimuth angle of any interesting point in the cylindrical coordinate system of the rod cross section as depicted in Fig. 2.3-2, and the upper limit r0 of integral about r is confined to the beam radius of wave in the crystal. The difference of refractive changes is mainly related to the deposited heat in the crystal. For Nd:YAG, it could be expressed as [15]
Δn(r)=Δnφ −Δnr =
(
3.2×10−4)
⋅Q(l)r2, (2.1-6) where Q is the heat deposited in the crystal per unit volume in the dimension of watts per cubic meter and r is in the dimension of meters. For an end-pumped scheme in our experiment,Chapter 2 Passively Q-switched Eye-safe Laser with Optical Parametric Oscillator
Q is an exponential-decay function along the axis of rod because of the exponential absorption of pump radiation.
To investigate the influence of polarization transformation, we start from the modified rate equations of Q-switched intracavity OPO modeled by Y.F. Chen et al. [16]. Since the oscillator only resonates fundamental and signal fields, the evolution equation of the idler field was eliminated. In the following analyses, y-axis is set to be the ordinary-wave direction which is the phase-matching direction of fundamental and signal wave. By concerning the depolarization term, the rate equations for the four-level Q-switched laser are re-modified as below:
where N and φ are the population inversion density and photon density; the subscript p, s, and x, y stand for the fundamental wave, signal wave on x- and y-axis, respectively; tr is the roundtrip transit time of light in the resonator; σ is the stimulated emission cross section of gain medium and lcr is the length of gain medium; lca is the effective cavity length; Δφs,y is the signal intensity of vacuum noise along y-axis; Γ is the transformation ratio of photon density between two orthogonal axes in a round trip resulted from depolarization; Rs is the signal output reflectivity and L is the roundtrip dissipative optical loss; σopo is the OPO effective cross section and can be derived from small parametric gain coefficient and is given by
2 2
Fig. 2.3-2. The schematic of depolarization due to thermally induced birefringence effect.
nφ nr
r x '
' y
z x
y
( ) [1 1 1] E r
r
Chapter 2 Passively Q-switched Eye-safe Laser with Optical Parametric Oscillator
where the subscript i stands for idler wave, ω is the angular frequency, n is the refractive indices, c is the speed of light, deff is the effective nonlinear coefficient, Ap and As are the mode areas of fundamental and signal wave, respectively, and ε0 is the vacuum permittivity.
The second terms in the right hand sides of Eq. (5) and (6) represent the losses induced from optical parametric process. There are two assumptions in the modified equations. First, the saturable absorber, Cr4+:YAG, is nearly transparent due to sufficiently high intensity in intracavity OPO scheme and the transmission is set to be unity when the lasing threshold is achieved. Secondly, in the Eq. (7), since the photon density of fundamental wave in the resonator is much higher than the signal wave, we neglect the influence of transformation of signal photons and only consider the signal photons in phase-matching direction (y-axis).
From numerical results, these two assumptions have been demonstrated acceptable.
Figure 2.3-3 depicts the calculated results of the temporal pulse behavior of signal wave at 1573 nm for the dependence of output reflectivity according to Eq. (2.1-7)-(2.1-11). The cavity parameters are used with the following value: ωs = 1.198×1015 sec-1, ωi = 5.733×1015 sec-1, deff = 3.1 pm/V, σ = 2.8×10-19 cm2, lcr = 2 cm, lnl = 2 cm, lca = 9 cm, np = 1.748, ns = 1.737, ni = 1.771, Rp = 0.998, Lp = 0.01, Ls = 0.03, Γ = 0.03. , Ap = 0.5π×(1.6 mm)2, As = 0.5π×(1.6 mm)2. It can be seen that under the existence of depolarization induced energy transformation, Γ, the value of OPO output reflectivity will greatly influence the satellite pulse behavior. This can be explained as below. The parasitic pulse results from the energy transformation. With higher output reflectivity, higher component of photons on x-axis is stored in the resonator. Therefore, more conspicuous pulse compared with first peak could be obtained. For Γ = 0.03, the numerical results show good agreement with the experimental results in temporal pulses. This indicates that the theoretical model is successful in predicting the dynamic behavior of signal pulse.
Figure 2.3-4 shows the theoretical and experimental result of the signal output energy versus output reflectivity for different value of Γ from 0.02 to 0.05. The theoretical curve of signal energy Es was obtained according to Eq. (2.1-12) given as:
,
0
ln 1 ( )
s s ca
s s y
r s
A hv l
E t dt
t ⎛R ⎞ ∞ϕ
= ⋅ ⎜ ⎟
⎝ ⎠
∫
(2.1-12) where hvs is the photon energy of signal wave. It is noted that with higher value of Γ, the higher output energy could be obtained. With the increasing of output reflectivity from 9% to 50%, the experimental output pulse energy varies from 11 mJ to 9 mJ. On the other hand, with lower signal output reflectivity of 9% to reduce the influence of birefringence effect, the effective pulse width is narrower and an output peak power up to 2.5 MW was obtained.Chapter 2 Passively Q-switched Eye-safe Laser with Optical Parametric Oscillator
Fig. 2.3-3. The temporal pulse of simulation result for the value of Γ = 0 and 0.03. The result shows the influence of depolarization and the dependence of signal output reflectivity.
-20 0 20 40
Fig. 2.3-4. The output energy versus different output signal reflectivity. The solid lines are numerical results of theoretical analysis for different value of Γ. The empty circles with error bar are experimental results.
0.0 0.2 0.4 0.6 0.8
4 6 8 10 12 14
Pulse Energy of Signal (mJ)
Reflectivity of output coupler
Γ = 0.03 Γ = 0.02 Γ = 0.04 Γ = 0.05
Chapter 2 Passively Q-switched Eye-safe Laser with Optical Parametric Oscillator