4.3 Focused Beams
4.3.1 ABCD Matrix Formalism for Gaussian Beams
Focusing a Gaussian beam into a nonlinear medium requires an optical element like a lens or a curved mirror. The most powerful tool calculating the beam properties after passing these optics is the ABCD matrix formalism. It allows theoretical predictions of the ξ parameter inside a nonlinear medium, as well as the determination of stable cavity dimensions. The basis for this formalism is the paraxial ray theory[105]. In this idealized picture, a light ray is described using the slope of its propagation relative to the optical axis,dr/dz, and its transverse offset,r(z). The free space propagation of a ray preserves the slope, but changes the transverse offset tor2 =r1+Ldr/dzafter a distanceL(Figure 4.11a).
In contrast, the propagation through a thin lens with focal lengthfpreserves the transverse
(a) (b)
Figure 4.11:(a) Schematic of free ray propagation over a distanceL, changing the transverse distancerto the optical axisz. (b) Schematic of refraction of a ray at a lens, changing the slopedr/dz(adapted from[105]).
offsetr2 =r1, but changes the slope todr2/dz =−(1/f)r1+dr1/dz. The goal is to describe
with the reduced slope r0 ≡ ndr/dz, containing the local refractive index n at the ray position. This definition leads to the free space propagation matrixF and the thin lens matrix L given by In general, one important property of all ray matrices is M =AD−BC = 1. The thin lens matrixL is an approximation. A product of multiple matrices supplies a more-accurate description. For example, a biconvex lens requires an incidence matrix at a curved interface Icand free space propagation F inside the medium until the next curved interfaceIc. The resulting matrix is M =Ic·F ·Ic, with the curved interface matrixIc listed in Table 4.1.
A flat interface is described at the limitR → ∞. The incidence angle is denoted asθ. The
Chapter 4. Nonlinear Optics
Table 4.1: Ray matrices for curved interfaces with respect to incidence angle. If the ray stays in the plane of incidence, it is denoted as tangential. If it is perpendicular to the plane of incidence, it is called sagittal[105].
tangential sagittal
angles θ1 andθ2 between two media with the refraction indices n1 andn2 are described by Snell’s law. Thus, complicated optical systems including multiple lenses and mirrors are easily modeled using the mentioned matrices. After evaluating the product, one obtains a single ABCD matrix. The ABCD matrix formalism can also be used to describe the effect of an optical system on a Gaussian beam with[105]
nf
The parameters qi and qf describe the properties of the Gaussian beam before and after passing the optical system, respectively. The parameter qi can be estimated by measuring the Gaussian waist at different positions along the optical axis. Frequently, the optical system is designed to produce a specific qf parameter, corresponding to a specific ξ parameter inside a nonlinear crystal. This adjustment is necessary to get maximum conversion efficiency. A particular case is the design of an OPO cavity. The lasing mode inside the cavity is defined by the cavity dimension independent of the pump beam.
4.4 Optical Parametric Oscillator
Spontaneous down conversion inside an OPO cavity is an extremely weak process on its own, which requires a resonant cavity for parametric amplification (Section 4.2.2). In general, the trapped signal wave inside the cavity drives the nonlinear conversion. The highest frequency wave serves as a pump and passes the nonlinear medium only a single time. However, both beams are focused inside at the center of the nonlinear medium.
The resulting ξ-parameter of the pump and the signal wave control the efficiency of the nonlinear conversion (Section 4.3) In detail, maximum conversion is achieved for a slightly smaller ξp parameter relative to the ξs parameter[102]. Thus, shaping the laser beam with optics based on predictions of the ABCD matrix formalism is crucial.
4.4. Optical Parametric Oscillator 4.4.1 Bow Tie Resonator
One strength of this ABCD matrix formalism is the analysis of periodic focusing systems, for example, a set of optics that direct a beam along a closed path[105]. This system can be as simple as the two curved mirrors of an optical resonator. The periodic system of an optical cavity is characterized by the eigenvalues of the ABCD matrix describing one round trip. These are particularly helpful in analyzing the stability of the resonator. When building an OPO, one must choose an appropriate cavity design. The preferred design for maximum stability is a ring resonator in bow-tie configuration, which works without an optical diode. In a ring resonator, the signal beam passes the nonlinear medium only once by one round trip, while in a linear cavity, the beam passes the nonlinear medium twice, which might cause potential thermally induced instabilities (Section 4.4.2).
The cavity design consists of two curved and two flat mirrors (Figure4.12). The
gen-Figure 4.12:(a) Schematic bow tie rig cavity design, with a periodic poled nonlinear medium of lengthLc1 and an etalon of thicknessLc2. The plane of reference is placed in the center of the nonlinear medium.
erated signal wave propagates collinear with the pump wave, except it stays inside the cavity. One round trip is completed after the reflection at the first curved, the first flat, the second flat and the second curved mirror. The two curved mirrors focus the signal wave at two points inside the cavity, at d1/2 and d2/2. The cavity angle α defines the incidence angle of the signal wave at the mirrors. It is large enough to let the laser beam pass the nonlinear crystal but as shallow as possible to keep astigmatism small. Between the two curved mirrors, at d1/2, is the nonlinear medium resting, for instance a PPLN with refraction index nc1. An etalon with the refractive index nc2 is placed at d2/2, for additional frequency selectivity of the OPO (Section 4.4.3).
The resonator matrix describes the propagation of a light ray, starting at a reference plane and ending after one round trip on the same plane. The reference plane is placed
Chapter 4. Nonlinear Optics
in the center of the nonlinear medium by the matrix
M =F(L2c1, nc1)·F(d1−d2 2,1)·Mc(R)·F(d3,1)·Mc(∞)·F(d2−L2 c2,1)· (4.37) F(Lc2, nc2)·F(d1−d2 2,1)·Mc(∞)·F(d3,1)·Mc(R)·F(d1−L2 c1,1)·F(L2c1, nc1),
(4.38) with the path wayd3 = (d1+d2)/(2 cos(2α)) between the curved and the flat mirror. The beam is characterized only at the reference plane. For analyzing the stability, the location of the reference plane is unimportant. However, for later consideration, it is convenient to know the beam waist at this position. The eingenvalues of a general ABCD matrix are[105]
λ±=m±√
m2−1, with m = A+D
2 . (4.39)
Assuming −1 ≤ m ≤ 1 leads to the complex eigenvalues λ± = cosθ±isinθ = e±iθ of magnitude one, withm ≡cosθ. If a ray vector inside the cavity is expressed as a linear combination of eigenvectors, for example, r0 =c+r++c−r−, the resulting ray vector after n round trips can be computed using[105]
rn =Mnr0 =Mn(c+r++c−r−) = c+λn+r++c−λn−r−. (4.40) Inserting the eigenvalues λ+ and λ− leads to rn = c+r+eiθ +c−r−e−iθ. Therefore, the ray oscillates about the optical axis. It does not pass the same transversal point in the reference plane after each round trip, but it stays confined. Thus, a cavity with|m|<1 is stable. A Gaussian beam is a cluster of multiple light rays. The dimensions of the cavity determine the values of the Gaussian beam parameter q that are unchanged after one round trip through the cavity. In general, the initial Gaussian beam at the reference plane with qi is described after one round trip by qf on the same plane (Equation 4.36). If the beam is self-consistent, then it returns after one round trip to its initial value qf = qi. The two stable Gaussian modes in a resonator are determined from the eigenvectors of the resonator matrix[105] One of these solutions represents a confined Gaussian beam, in case of a stable resonator, which means the waistw of the Gaussian beam has a real and positive value. In contrast, the other eigenvalue represents a nonphysical solution. Thus, the two Gaussian beam parameters of the real and geometrically stable resonator (|m|<1) are[105]
R = 2Bn
D−A and w2 = |B|λ0
π
r 1
1−m2. (4.42)
4.4. Optical Parametric Oscillator As an example, consider a cavity angle of α = 9°, a radius of curvature for the two focusing mirrors ofR = 100 mm, an yttrium aluminum garnet (YAG) etalon thickness of 3 mm and a PPLN of 50 mm length. The beam profile is only for a distinct combination of d1 and d2 circular at the center of the nonlinear medium (Figure 4.13a). Thus, setting
136 138 140 142
Figure 4.13:(a) Calculated example of possible bow tie cavity dimensions, for a radius of mirror curvature of R= 100 mm and an etalon thickness ofLc2= 3 mm (Equation (4.37)). The criterion is maximum stabilitym= 0 or a round beam profile at the center of the nonlinear medium. (b) Calculated tangential and sagittal beam waists for a fixed cavity parameterd2= 171.3 mm.
the dimension d2 = 171.3 mm allows only one value d1 = 136.8 mm. The beam waist is, in this case, w = 57.5µm. For any other value, the beam is elliptical. The tangential and sagittal waist inside the reference plane is calculated separately (Figure 4.13b). The stability parameter for this cavity dimension isms =−0.1 andmt=−0.3 for the sagittal and the tangential component, respectively. This is inside the allowed range of a stable cavity of |m|<1, but apparently with a tendency to negativem values. It is convenient, for comparison, to take also a look at a cavity design closest to m = 0. Astigmatism prevents m = 0 on both beam axis, thus further calculation considering an ideal cavity with no astigmatism where α= 0°(Figure 4.13a). Keeping the dimension d2 = 171.3 mm fixed leads to a cavity parameterd1 = 140 mm. The resulting stability parameters are now ms = 0.17 andmt=−0.08 for α= 9°. The beam is no longer circular at the cavity design point, but the astigmatism increases only to (wsag−wtan)/wsag ≈4%. The maximum level of astigmatism is reached between the two flat mirrors with (wsag−wtan)/wsag ≈ 12%.
Small astigmatism also justifies the cavity design based on α = 0°, which is the design choice for the OPO in this thesis.
4.4.2 Thermal Effects
The previous consideration of the OPO stability left out absorption inside the nonlinear medium. This effect causes a rise in temperature along the propagation axis of the Gaussian laser beam. Regions of the crystal exposed to a higher field intensity have a higher temperature than those exposed to a lower intensity. This temperature gradient
Chapter 4. Nonlinear Optics
leads to a radially changing index of refraction. Instead of a free propagation inside the nonlinear medium, the beam gets focused by the thermal lens. The ABCD matrix associated with this duct is[105] The ray propagating along the z axis experiences the refraction indexn0, while a ray at an off-axis transverse position r experiences a different index of refraction, which depends on the second derivative n000 = n2. In general, it is challenging to determine the axial temperature dependence of this matrix. Thus, some authors[106] prefer instead to use a thin lens matrix (Equation (4.35)) in the middle of the crystal with the focal length[106,107]
fT = πKc αPc(dns/dT)
w2
Lc. (4.44)
Crystal heating measurements give an estimation for the absorption coefficientα≈0.08 % of a PPLN crystal at λs = 1611 nm[106,108]. The thermal conductivity of a MgO-doped (5 mol%) lithium niobate crystal isKc= 4.02 W/mK[106]and the temperature dependence of the crystal refractive index is dns/dT = 5·10−6/K[109]. Assuming a signal power of Pc = 50 W and a crystal length ofLc= 50 mm results in a focal length about fT ≈4 mm.
In the case of linear cavities, this might cause bi-stability[106]. A precaution against the thermal lens is a tightly focused beam (ξ ≈2) for higher pump depletion[106]. This method is counter-intuitive since a large waist decreases the power density and increases the focal length of the lens (Equation (4.44)). Additionally, the dimension d1 might be chosen slightly larger relative to the cold cavity stability center, to consider the thermal lens in advance[110]. However, a thermal lens is of less importance for ring cavities, due to a lower temperature increase. This statement is valid as long as the absorption of the idler beam is negligibly small. Consider a MgO-doped (5 mol%) PPLN for spectroscopy in the mid infrared between 2.5µm and 4µm, with a discrete set of poling periods of different length Λn. A typical set of seven poling periods for a mid-IR OPO goes, for example, from Λ1 = 31.5µm to Λ7 = 28.5µm in steps of 0.5µm (Figure 4.14a). Estimating the phase matching condition for planes waves with Equation (4.30) yields
1
Λn = n(λp, T)
λp − n(λs, T)
λs −n(λi, T)
λi , (4.45)
where n(λ, T) is the wavelength and temperature dependent refraction index of a MgO-doped (5 mol%) PPLN crystal[111]. The pump wavelengthsλp = 1064 nm leads to a signal wavelength of λs= (1/λp−1/λI)−1. For producing the 2.7µm to 2.9µm light needed to excite the fundamental vibrational transitions of OH, only the two longest poling periods with Λ1 = 31.5µm and Λ2 = 31.0µm satisfy the phase matching condition at reasonable
4.4. Optical Parametric Oscillator
Figure 4.14:(a) Theoretical temperature dependence of the phase matching condition of a MgO-doped (5 mol%) PPLN crystal[111]. The congruent poling periods are Λn= 32µm−n·0.5µm withn∈[1,2,··,7]. (b) Only two poling periods are suited for nonlinear conversion to target wavelengths around 2.7µm and 2.9µm driving a 1 photon and 2 photon transition in the OH, respectively. In the vicinity of the target wavelengths, there is also an absorption feature in the crystal around 2.829µm[106].
temperatures (Figure 4.14b). Concerning the idler absorption inside the PPLN crystal, two wavelength regions are worth mentioning, the phonon absorption above 4µm and an absorption peak around 2.829µm[106]. At the peak of the absorption feature, a 5 cm long crystal absorbs more than 80 %[106]. Without the MgO-doping of the crystal the absorption peak would overlap with the wavelength needed for two-photon vibrational spectroscopy on OH[112](Figure 4.14b). Fortunately, the MgO-doping shifts the absorption line in between the two wavelengths of interest for OH spectroscopy.
4.4.2.1 Spectral Instabilities
The temperature distribution inside the crystal plays an important role in selecting the phase matching bandwidth. Additionally, a temperature rise caused by absorption leads to spectral instabilities. Above a critical pump level, the OPO signal line width might experience a broadening[113]. Increasing the pump power further, the OPO starts to emit multiple modes over a frequency range larger than the bandwidth of the gain profile.
One reason is spontaneous Raman scattering, caused by phonons inside the crystal[114,115]. At high powers, this leads to a stimulated Raman scattering of the signal wave, adding satellite peaks to the spectrum[113]. The frequency shift of Raman lines relative to the signal frequency is constant and independent of the poling period. However, some modes change monotonically with the poling period of the PPLN crystal. These modes are associated with cascade optical parametric oscillations[116]. The ideal OPO operation describes the conversion of a pump wave λp into an idler λ(1)i and a signal wave λ(1)s
(Section 4.0.3). The wave vectors of the pump and the idler wave are both in the forward
Chapter 4. Nonlinear Optics
direction (Figure 4.15a). At high powers, however, the signal acts as pump wave itself,
(a) (b)
Figure 4.15:(a) Schema of the quasi wave vector mismatch for the ideal OPO operation. (b) Conversion to an additional wave withk(2)s caused by backward oscillation of an idler wave withk(2)i (adapted from[116]).
which is converted into a second signal wave with λ(2)s (Figure 4.15b). The frequency difference between the initial and the additional signal waves depends on the phase matching condition. In contrast to the previous wave vector orientations, the idler can also propagate in the direction opposite to the signal and pump, denoted as a parametric backward oscillation (Figure 4.15b). As an example, forλ(1)s = 1.7µm and a period length of Λ2 = 31µm, the secondary wavelengths of signal and idler are aroundλ(2)s = 2.8µm and λ(2)i = 4.3µm. Those wavelengths are in general unwanted. A simple solution to this issue along with the Raman scattering is the reduction of the intra-cavity signal power. It is convenient to couple the signal power out with one partly transmissive mirror. Optimum OPO operation of a bow tie cavity has been observed at a signal wave output of around 3 %[108]. The suggested signal output of a linear cavity is 4 % or larger, to ensure reliable single mode operation[106]. The limit of the circulating power inside the cavity additionally increases the idler power, as well enhances the beam quality factor M2, corresponding to an ideal Gaussian beam[108].
4.4.3 Frequency Selection
Consider the OPO cavity is stable at single-mode emission. Now it is convenient for spectroscopic measurements, to tune the lasing frequency. A coarse way to change the emission frequency is increasing or lowering the temperature of the PPLN crystal. One approximation of the phase matching gain profile is
I = sinc2
2πLc
Λ(λp, λs, T)− 2πLc
Λ2
. (4.46)
The following calculations assume the bow tie cavity design of Section 4.4.1, a PPLN crystal of length Lc = 5 cm and a period length Λ2 = 31µm. The temperature and wavelength dependent period length Λ(λp, T) is calculated with Equation (4.45). Thus, the resulting phase matching gain profile has a full width at half maximum (FWHM) around 100 GHz (Figure 4.16a). A stable single-mode OPO lases at the cavity mode closest
4.4. Optical Parametric Oscillator
Figure 4.16: (a) Schema of the frequency selectivity inside an OPO cavity. The phase matching profile of the PPLN crystal allows a coarse tuning of the lasing wavelength. (b) A more precise frequency selectivity is supplied by a thin etalon. The material and the thickness of the etalon determines the final gain profile.
to the highest gain. The cavity modes are separated by the free spectral range
FSR = c
ndtotal = 428.46 MHz, (4.47)
with the refraction index in air n ≈ 1 and the total optical round trip path length dtotal = 699.7 mm.
The phase matching condition of the PPLN supplies nearly no frequency selectivity (Figure 4.16b). Without frequency selective elements inside the cavity, the lasing mode is defined by parasitic etalon effects or absorption features caused by impurities of the nonlinear medium. These spurious frequency selective losses modify the gain profile and create local maxima and minima[106]. The lasing mode can stay at the absorption related maximum of the gain medium and resist temperature fluctuations of the PPLN crystal of as much as 100 mK[106]. Thus, thermal-locking cancels noise fluctuations induced by temperature changes, but also prevents tuning of the OPO signal frequency. However, frequency tuning can be achieved by adding an etalon into the OPO cavity[117]. The low etalon thicknessdallows a significant spacing between the transmitted interference fringes.
The second important parameter of a Fabry-Perot interferometer like the etalon or OPO cavity is the coefficient of finesse F = 4R/(1−R)2, which is a measure of the spectral width of the modes inside the interferometer and increases with the reflectivityR. A higher reflectivity of the cavity mirrors leads to a narrower mode profile. The reflectivity at the etalon surfaces is dependent on the refraction indexn of the material and the polarization state of the light wave. In the case of p-polarized light, the reflection coefficient is given by the Fresnel equation
Chapter 4. Nonlinear Optics
Snell’s law sinθo =nsinθi describes the relation between the angle outside the medium θo and the angle inside the medium θi. Since the reflectivity of the etalon surfaces is much lower than the reflectivity of the cavity mirrors, the finesse of the etalon is several orders of magnitude smaller than that of the cavity. It is convenient to calculate the transmission profile of an etalon to illustrate the influence of different etalon materials and thicknesses.
The transmission of an etalon is described in approximation by the Airy function[118]
I = 1
1 +F sin2(φ/2) with φ= 4π
λ ndcos(θi). (4.49) However, to study the etalon in more detail, it is necessary to calculate the transmission for a Gaussian electromagnetic field (Equation (4.32)). First, the incident beam gets refracted at the etalon surface. The propagation direction inside the etalon changes according to Snell’s law. Some fraction of the beam experiences a back and forth reflection inside the etalon, which introduces the walk-off X = 2dsin(θt) relative to the initial beam. The magnitude of the walk-off depends critically on the etalon thickness d and the internal reflection angleθi of the etalon. Multiplem ∈Nof this steps lead to the total transmitted electric field[118]
Et(x, y, z) = X∞ m=0
(1−R)RmEm(xm, y, zm), (4.50) with xm =x−mX and zm =z+ 2mdcos(θi). As an example, the transmissions curves have been calculated for a 800µm thick silicon etalon and a 3 mm thick YAG etalon at a zero incidence angle (Figure 4.16a). The higher refractive index nSi ≈3.4[119] of silicon relative to the refractive index nYAG ≈ 1.8[120] of YAG leads to a higher finesse of the silicon etalon. Thus, transmission fringes are more pronounced (Figure 4.16a). However, the smaller FSRYAG=c/(2nd) = 27 GHz of the YAG etalon relative to the silicon etalon FSRSi≈54 GHz partly compensates for it. The resulting mode selection inside the cavity improves with both etalons (Figure 4.16b). The silicon etalon is here superior to the YAG etalon.
Assuming the lasing mode is successfully selected with the etalon, the next step is to find a way to tune the frequency of the etalon’s transmission maximum. One possibility
Assuming the lasing mode is successfully selected with the etalon, the next step is to find a way to tune the frequency of the etalon’s transmission maximum. One possibility