MM ax2 = 1 4(w2 s+w2i) (4.2) at 1 w2 p = 1 w2 s + 1 w2i (4.3) M2
M ax can then be written in form of pump and signal beam radii as following
MM ax2 = 1 4 Ã 1 w2 s − w2p w4 s ! (4.4) At a fixed pump beam waist radius, the signal radius can be adjusted so that M2
M ax
reaches a maximum. To show this, let assume w2
s =βwp2, whereβ is an arbitrary number,
then Equation 4.4 becomes
MM ax2 = 1 4w2 p µ1 β − 1 β2 ¶ (4.5) This expression has a maximum of M2
M axM = 161w2
p at β = 2, or ws =
√
2wp. Then the
pump threshold in Equation 4.1 now becomes
Pp,th=
2²on2snpcλ2pwp2
πd2
ef fL2(1−δ2)
(1−Rs) (4.6)
The crystal length can be estimated through the threshold by
L= v u u t 2²on2snpcλ2pwp2 πd2ef f(1−δ2)P p,th (1−Rs) (4.7)
The length of the crystal is found to be 2.8mm forns =np= 2.2,λp = 532nm,Rs= 90%,
δ = 2λp
λs −1 = 0.8, andPp,th= 2kW.
The confocal parameters of the pump beam with the waist radius of 50µm and signal beam with the waist radius of 70µm are bp = 64mm and bs = 120mm, respectively.
Therefore, the focusing parameters are ξp =L/bp = 0.044 andξs =L/bs = 0.023, which
are consistent with the assumption made at the beginning of the calculation, that is, the crystal length is much less than the confocal parameters of the pump and the beam.
In summary, the guide star OPO is pumped by 30W of green light with wavelength of 532nm, pulse duration of 150ps and repetition rate of 25MHz. The suitable length crystal of the MgO:PPLN crystal is 3mm. The pump and signal waist radii at the crystal are 50µm and 70µm.
4.3
Grating Period of MgO:PPLN for Guide Star OPO
The grating period of the MgO:PPLN must be carefully calculated so that the guide star OPO produces signal wavelength at 589nm using 532nm pump. The preferred temperature of the crystal is 150oC to avoid the effect of photorefractive damage, and to reduce the effect of idler absorption, which potentially can create a thermal lens in the crystal that distorts the pump and signal beams and reduces the efficiency of the guide star OPO. The fine tuning of the signal is performed by temperature tuning. Two mechanisms
42 Chapter 4: Design of Guide Star Optical Parametric Oscillator
operate during temperature tuning. Firstly, the refractive index of the crystal changes with temperature; and secondly, the grating period is a function of temperature because of thermal expansion. These two factor must be taken into account simultaneously when calculating the suitable grating period of MgO:PPLN for the guide star OPO.
The data for the refractive index of MgO doped Lithium Niobate at elevated temper- ature is limited. Shenet al. [30] measured the refractive index of 5 mol. % MgO Lithium Niobate in the spectral range from 0.54µm to 1.34µm and the temperature range from 20oC to 155oC. However, their results are only applicable in this near infrared spectral range and not suitable to the mid-IR idler wavelength of the guide star, which is 5.49µm. The idler wavelength is at the absorption edge of Lithium Niobate, as the temperature increases, the absorption edge moves toward shorter wavelength and, therefore, has an even greater effect on the refractive index in the mid-IR spectral region [26].
However, the refractive index of PPLN at elevated temperature and for the mid-IR spectral range involving IR absorption has been well studied [26]. As an illustration, the following calculation will assume PPLN as the nonlinear crystal of the guide star OPO.
Jundt [26] assumed that there was an IR absorption contribution to the refractive index and the Sellmeier equation was in the following form:
n2e =a1+b1f+ a2+b2f λ2−(a 3+b3f)2 +a4+b4f λ2−a2 5 − a6λ2 (4.8)
where ai and bj are given in Table 4.1; temperature parameter f is the square of the
absolute temperature in degree Kelvin, with an added offset to make it vanish at the reference temperature To = 24.5o. For temperature T expressed in degrees Celsius, f is
given by f = (T−To)(T+To+ 2×273.16) = (T−24.5oC)(T + 570.82) (4.9) Parameter Value a1 5.35583 a2 0.100473 a3 0.20692 a4 100 a5 11.34927 a6 1.5334×10−2 b1 4.629×10−7 b2 3.832×10−8 b3 -0.89×10−8 b4 2.657×10−5
Table 4.1: Sellmeier coefficients for congruently grown Lithium Niobate for Equation 4.8 Equations 4.8 and 4.9 allow the refractive index of Lithium Niobate to be calculated at any temperature from 20oC to 200oC and at any wavelength from 0.4µm to 5µm. For instance, at 120oC, the refractive indices of Lithium Niobate at 532nm 589nm and 5497nm are 2.24026, 2.21978, 1.97338, respectively. The grating period of the PPLN for phase matching these wavelengths is calculated using Equation 2.53 or
§4.3 Grating Period of MgO:PPLN for Guide Star OPO 43 Λ = np 1 λp − ns λs − ni λi = 12.000µm (4.10)
The grating period is first calculated at the elevated temperature, then using the thermal expansion Equation 4.11 the room temperature grating period can be accurately deduced. The thermal expansion coefficients α and β describe the crystal length l at temperature T normalised to the length of 25oC,l
25oC:
l=l25oC[1 +α(T −25oC) +β(T−25oC)2] (4.11) where α = 1.54×10−5K−1 and β = 5.3×10−9K−1. As the crystal expands the grating
period also increases in the propagation direction. For example, the grating period of the PPLN of the guide star OPO working at T=120o is 11.985µm at room temperature. The accuracy of this calculation has been verified experimentally by Huang et al. [1]. They produced Sodium lines at 589nm using PPLN optical parametric generation pumped 532nm.
The same calculation could be repeated if the Sellmeier equations for MgO Lithium Niobate was well known enough. Shenet al. [30] measured the refractive index of 5 mol. % MgO Lithium Niobate in the spectral range from 0.54µm to 1.34µm and the temperature range from 20oC to 155oC. Their results are given in the following form
n2e=A+ B
λ2−C −Dλ
2 (4.12)
where the parameters A, B, C and D are temperature dependent as shown in Table 4.2. Coefficient Temperature dependent toC
A = +4.7676×10−9t3+ 4.2653×10−9t2+ 8.3758×10−5t+ 4.5436
B =−5.4883×10−9t3+ 7.0901×10−7t2+ 4.6147×10−5t+ 0.0926
C = +1.0230×10−8t3−1.3172×10−6t2−3.3074×10−5t+ 0.0500 D =−9.0039×10−10t3−1.5281×10−7t2+ 5.0802×10−5t−0.02382
Table 4.2: Temperature-dependent Sellmeier coefficients for congruently grown Lithium Niobate with 5 mol.% MgO
Using this data, the refractive indices of MgO:PPLN at 120o are 2.23231, 2.21287 for 532nm and 589nm, respectively. The refractive index at 5497nm is not calculable because this approximation only works in the range from 0.5µm to 1.3µ.
However, some approximation can be made to estimate the grating period of MgO:PPLN. These refractive indices at 532nm and 589nm are only 0.3% less than those of PPLN. Furthermore, when Equation 4.10 is used, the effect of ni is much smaller than
the effect of np and ns on the grating period as the wavelength of the idler is 10 times
longer than the pump and the signal. Hence, it is safe to assume that the refractive index of MgO:PPLN at 5497nm isni = 1.96746, that is 0.5% less than as for PPLN. The grating
period is then equal to 12.317µm at 120oC, or 12.30µm at 25oC.
The grating period of MgO:PPLN that can be used to create 589nm by the 532nm pump should be close to the value of 12.30µm. The exact value requires accurate data of refractive index at those wavelengths and temperatures. However, many grating periods in the proximity of 12.30µm can be made on the same crystal; for example, the grating periods of 12.10µm, 12.20µm, 12.30µm, and 12.40µm can be fabricated. This series of grating in conjunction with the temperature tuning should allow the OPO to be tuned to
44 Chapter 4: Design of Guide Star Optical Parametric Oscillator
generate a signal at 589nm.