Analytic and experimental investigations on influence of harmonic generation on acousto-optical modulation

: In application, laser power is modulated prior to harmonic generation to a large part. Consequently modulation characteristics are influenced as a result of the non-linearity of harmonic generation. Within this paper, straight-forward approaches to calculate modulation key parameters (rise time, fall time, bandwidth and contrast ratio) of an acousto-optical modulator prior to harmonic generation stage are presented. The results will be compared to experimental data for third harmonic generation (THG) of a 1064 nm fs-laser which is power-modulated by a TeO 2 -AOM prior to THG. In the latter case, rise time and fall time are significantly reduced to approximately 66% after THG both in experimental and analytical study.


Introduction
Acousto-optical modulators (AOMs) are commonly used to modulate laser power in intraand extra-cavity setups [1][2][3][4][5]. Regarding high-power solid-state lasers, AOMs are typically used to modulate the laser's fundamental wavelength prior to extra-cavity harmonic generation (HG), though AOMs for wavelengths ranging from UV to MID-IR are available in principle [6]. The reasons are increased separation angles of 0th and 1st diffraction order after AOM and higher damage thresholds of acousto-optical crystals. In addition to that, modulation key parameters such as rise and fall time, contrast ratio and modulation bandwidth, are influenced by the non-linearity of the HG, i.e. preferable values result. Up to now, this beneficial influence was rarely taken into account when designing a laser system including HG and fast power modulation.

Analytical approach
In the following, equations for AOM power modulation of the fundamental wavelength and subsequent HG to second and third order harmonics will be given. SHG and THG are used for second and third harmonic generation (crystal) respectively. In section 2.2, the fundamental wavelength will be indexed as FUN.

Power modulation via AOM at fundamental wavelength
As preconditions to the following approach, the quotient of laser beam divergence θ 0 and acoustic beam divergence θ a has to be smaller than 0.67 (thus limiting ellipticity of the laser beam after passage of the AOM crystal to a negligible amount) and the rise time of the RF driver voltage has to be much lower than the transit time ( = time that the acoustic wave needs to travel through the beam waist diameter 2w 0 ) [7,8]. The latter precondition is fulfilled in the majority of commercially available AOMs.
When a step function is applied to the RF driver control voltage, an acoustic wave starts to transmit through the acousto-optical crystal. The front of the acoustic wave will start to travel through the optical aperture. Thus, a fraction of the laser beam, which is assumed to be of Gaussian intensity profile, is deflected due to diffraction. The amount of the diffracted laser power is dependent on the position of the front of the acoustic wave, i.e. time, and is derived as follows.
( ) The deflected, i.e. modulated average power P(t) is then given by Eq. (2), cp. knife-edge method [9], with x = Vt being the position of the front of the acoustic wave, where V equals the acoustic wave velocity, and the total laser power is P 0 = ½πI 0 w 0 2 .
To derive rise and fall times from Eq. (2), t is calculated for P(t)/P 0 = 0.1 and 0.9 respectively and Eq. (3) is obtained.
As rise and fall times are equivalent due to symmetry, the modulation bandwidth f m results from Eq. (4) (derived from [7]). Results for t rise/fall and f m are in accordance with [8].

Power modulation after harmonic generation
For harmonic generation, optimal alignment (no phase-mismatch) and collinear waves are presumed. The intensity of the second harmonic beam I SHG (x,y) can be expressed by Eq. (5) [10,11], where L is the interaction length, d eff the effective non-linear coefficient (with m/V as unit), ε 0 the dielectric constant (ε 0 = 8.854 10 −12 F/m), ω = 2π c 0 /λ FUN the angular frequency and Ζ = Ζ 0 /n the impedance (Z 0 = 377 Ω). The intensity profile I FUN (x,y) is equivalent to Eq.
(1), with the difference of using I FUN0 for the peak intensity of the fundamental laser beam (instead of I 0 ) and w FUN for the 1/e 2 beam diameter of the fundamental laser beam when transmitting through the SHG crystal (instead of w 0 Variables Θ SHG (Eq. (6)) and Θ THG (Eq. (8)) summarize all non-variant parameters for SHG or THG respectively (dependent on fundamental wavelength, crystal geometry and crystal material). Subsequently, the temporal shape of the modulated average power after THG can be calculated using Eq. (9), cp. Equation (2). The upper limit of integration over x equals (w FUN /w 0 )Vt, i.e. it is assumed that the intensity profile after AOM is reimaged to HG according to geometrical optics. w FUN and w 0 are the beam radii at fundamental wavelength in the HG crystals and AOM crystal respectively.
Equation (9) is derived under the precondition that the truncated laser beam (truncated due to the working principle of AOM) will transmit to SHG and THG crystals in diffraction-free manner, which is not true in the real case. I.e. truncation will generally blur within a short range or due to additional focussing before HG. To take this fact into account, a second, simpler approach with following assumptions is eligible. The same accounts for I SHG (x,y) and I THG (x,y) with according indices. Further, it is assumed that the beam radius is unaffected by HG, i.e. w FUN = w SHG = w THG (further analysis proves minor impact of this simplification on resulting shapes of P SHG (t) and P THG (t)). As a result, Eq. (5) can be rewritten as Eqs. (10) and (11).
As before, Θ* SHG and Θ* THG summarize all non-variant variables, now including fundamental laser beam characteristics. Calculated values for Θ SHG, Θ* SHG, Θ THG and Θ* THG are expected to be higher than experimentally derived values due to neglection of walkoff, dephasing, group velocity delay and other effects. It has to be noted that there are more detailed ways to calculate SHG and THG conversion efficiencies such as [11,12]. However, for anticipated usage, the formulas stated proved to be more than adequate.
Resulting t rise/fall,SHG after SHG can be calculated using Eqs. (6) or (10) as t rise,SHG = t fall,SHG = t(P SHG /P 0,SHG = 0.9) -t(P SHG /P 0,SHG = 0.1), where P 0,SHG is the maximum power after SHG. As long as the shape of P SHG (t)/P 0,SHG does not significantly differ from P(t)/P 0 , the bandwidth f m,SHG can still be estimated to f m,SHG ≅ 0.502 2/(t rise,SHG + t fall,SHG ). Both predications apply to t rise/fall,THG and f m,THG the same way when using Eqs. (9) or (12).

Experimental setup
The experimental setup consists of a fs-laser, an AOM modulating the laser power at fundamental wavelength, and a THG module. The laser power is measured with a Coherent LM-3 thermopile power meter for powers >30 mW and a Coherent LM2-UV silicon diode power meter for powers <30 mW. Rise time, fall time and contrast ratio are measured using a Tektronix TDS-3034C 300 MHz oscilloscope with a Thorlabs FDS010 silicon photodiode (1 ns rise time) as detector. As one oscilloscope trace of fs-pulses is not sufficient for analysis (1 µs pulse pause between two fs pulses), repetitive oscilloscope traces are collected and overlaid over each other via a long time of oscilloscope screen persistence (~10 s).
The fs-laser is emitting pulses at a repetition rate f rep = 1 MHz with a FWHM pulse duration τ = 400 fs (assuming sech 2 pulse shape), wavelength λ = 1064 nm and max. pulse energy E p,max = 2.0 µJ. The RMS pulse-to-pulse stability is approximately 1% and M 2 ≅ 1.25. The laser beam is collimated to w 0 = 0.49 mm via a Galilei-type telescope prior to the AOM.
Modulation is realized by an AOM (AA optoelectronic MT80-A1.5-1064) with TeO 2 as crystal material (V = 4200 m/s). The AOM system aperture equals 1.5 x 2 mm 2 , thus only slight clipping of the laser beam will occur. At 80 MHz carrier frequency of the acoustic wave and RF power of 2.0 W, the AOM is specified to a contrast ratio >2000:1, rise time 160 ns at 1 mm 1/e 2 beam diameter and diffraction efficiency of >85%. The separation angle of 0th and 1st order beams is 20 mrad.
The diffracted (thus modulated) 1st order laser beam is focused into the middle of the frequency doubling crystal of 5 mm thickness via a second Galilei-type telescope. The resulting focal spot radius equals approximately 250 µm (Rayleigh length ~152 mm). LiB 2 O 5 (LBO) is used as SHG crystal material. Polarization and crystal angle are aligned for optimal critical phase matching and SH conversion efficiency of Type I SHG (d eff,SHG ≅ 0.82 10 −12 m/V [13]).
After SHG the polarization of the residual fundamental beam is rotated by π/2 for subsequent Type I THG via a crystal quartz waveplate (phase retardation λ/2 for 1064 nm and λ for 532 nm). LBO of 3 mm thickness is used as THG crystal. No delay compensation between fundamental and SH beam is used. Thus, THG efficiency will be sub-optimal to some extent. As before, the crystal angle is aligned for optimal phase matching and TH conversion efficiency (d eff,THG ≅ 0.67 pm/V [13]). No additional focussing takes place before THG (due to Rayleigh length >> separation between SHG and THG crystals). After THG, residual light of SH and fundamental wavelength is subtracted by a dichroitic mirror.
The experimental setup is depicted in Fig. 1 schematically. The resulting average THG laser power in dependence to average fundamental laser power is depicted in Fig. 2. Equations (9) and (12) for t→∞ are fitted to the experimentally obtained values, where Θ SHG and Θ THG or Θ* SHG and Θ* THG respectively are used as independent variables. Additionally, a polynomial fit of fourth order (P THG (P) = a P 4 + b P 3 + c P 2 + d P + e) is shown in Fig. 2 for comparison. In the latter case, the independent variables d and e are set to zero, to represent P THG (0) = 0 and dP THG (0)/dP FUN = 0. Polynomial fits of lower order (2nd, 3rd) delivered less accurate agreement between regression and experimental values. Polynomial fits of higher order (5th, 6th) did not increase agreement significantly. This is understandable when solving Eqs. (10) and (12) for low values of P(t). P THG then becomes P THG ~P(t) 3 -Θ* SHG 2 P(t) 4 . The polynomial fit equals Eq. (14). Θ SHG , Θ THG , Θ* SHG and Θ* THG are calculated using the stated parameters and Eqs. (6), (8), (11), and (13). Due to missing delay compensation, the experimentally obtained values for Θ THG and Θ* THG are significantly lower than the calculated values as of neglection of group velocity delay. Results of calculated and experimentally derived values are summarized in Table 1.

Experimental results
In the following, experimental results on modulation key parameters both prior to and after THG are presented. Analytical results for Eqs. (9), (12) and (14) will be compared, where the values for Θ SHG , Θ THG , Θ* SHG and Θ* THG correspond to the values of regressions in Table 1. In all diagrams, experimental results are shown as point plots and calculated results as line plots.

Temporal AOM modulation shape prior to and after THG
The temporal development of modulated laser power is measured. Analytical results for the temporal development before THG are derived according to Eq. (3). Experimental and analytical results are depicted in Fig. 3 for both max. average fundamental power before THG P FUN = 1.51 W (Fig. 3 left) and P FUN = 0.36 W (Fig. 3 right).

Dependence of modulation parameters on THG efficiency
Rise and fall times are measured. Before THG, the rise time equals 156 ns and the fall time 148 ns (independent on P FUN ). The average value of 152 ns is in good accordance with the specification of 160 ns/mm ⋅ 0.98 mm = 157 ns. Rise and fall times after THG are measured for varying average fundamental power P FUN , thus varying THG efficiency. Analytical results are provided by solving both P THG (t 90% ) = 0.9P THG,max and P THG (t 10% ) = 0.1P THG,max for t rise/fall = t 90% -t 10% with numerical methods. The bandwidth f m is calculated via Eq. (4) and equals 3.3 MHz before THG. The results are depicted in Fig. 4, where the upper limit of the rise/fall time ordinate and lower limit of the bandwidth ordinate are set to the value prior to THG. The contrast ratio before THG is measured and equals 3125:1, which exceeds the specification of 2000:1. At max. average power of the fundamental laser beam P FUN = 1.51 W, analytical values of the contrast ratio after THG exceed 10 9 :1 for Eq. (9) and (12) and 10 7 :1 for Eq. (14). Experimentally, the contrast ratio after THG is higher than the signal-to-noiseratio of the diode signal of 10 4 :1.

Discussion
Experimentally, a significant reduction of rise and fall time of the AOM after THG down to 60% … 71% of rise and fall time prior to THG is proved for THG efficiencies P FUN /P THG > 1% (P THG > 5 mW), cp. Figure 4. This results in a raise of bandwidth to 141% … 152% compared to the value prior to THG. For low THG efficiencies of < 1% (P THG < 5 mW), rise and fall times slightly increase apparently. Neither model represents this. The effect is accredited to leakage of the SHG and fundamental laser beam after the dichroitic mirror. Sensitivity of the photodiode used in experiments is 7 times higher for 532 nm compared to 355 nm and 1064 nm. Solving Eq. (10), the rise time at 532 nm is approximately 125 ns. The rise time at the fundamental wavelength is 152 ns. A leakage of the SHG laser beam of ~0.2 mW, fundamental laser beam of ~1 mW or a combination of both therefore explains the deviation.
The analytical solution via Eq. (9) -the integration of the THG intensity profile under assumption of diffraction-free propagation of laser beams -does not represent the measured time delay between temporal modulated average power development prior to and after THG, which is measured to be in a range of 20 ns .. 40 ns, cp. Figure 2. This can directly be appointed to neglection of diffraction. Then again, Eq. (9) delivers values for rise and fall times which deviate from the experimental ones less than +/− 8% for THG efficiencies > 1.5% (P THG > 10 mW).
The simplified model via Eq. (12), which delivers an analytical solution for temporal modulated power development, neglects truncation of the beam and therefore represents time delay well, cp. Figure 2. The values for rise and fall time are stable over a wide interval and deviate from the experimental ones by + 5% … + 15% for THG efficiencies > 1% (P THG > 5 mW).
The direct polynomial solution, Eq. (14), shows similar behavior as Eq. (12), but it has to be noted that it does not represent the physical properties of the HG process and therefore will not be useful to estimate values far away from the regression interval (e.g. THG efficiency > 3.5%).
The contrast ratio after THG is exceeding the measurement range towards excellent values of >10 4 :1.

Summary and outlook
The experimental results show a significant reduction of rise and fall time of laser power modulated via AOM due to harmonic generation. The quotient of rise and fall time prior to THG (directly after AOM) to after THG equals ~0.66, thus the bandwidth is increased by a factor of ~1.5. This quotient is directly correlated to the characteristics of harmonic generation, i.e. its non-linearity. Also due to non-linearity, the temporal development of the modulated laser power is delayed by approximately 30 ns.
Two analytical methods to estimate modulation parameters after THG are introduced. On the one hand Eq. (9) is well suited to calculate resulting rise and fall times to an accuracy of +/− 8% but does not represent the temporal delay of the modulation. On the other hand Eq. (12) represents the delay very well, and results for rise and fall time after HG deviate from the experimental values by + 5% … + 15%. Therefore both methods are suited to predict the behavior after HG, where Eq. (12) is simpler to use, since no numerical methods have to be applied to gain the temporal development of the modulated power after HG.
With small adaptions, the presented methods are applicable to other harmonic generations, such as FHG or direct THG (not via sum frequency, but direct tripling of the fundamental wavelength). The same accounts to other modulation devices, such as electro-optic (e.g. pockels cell) or interferometric modulators (e.g. Mach-Zehnder modulator). Mainly Eq. (1) and Eq. (2) would have to be changed in the latter case.