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

Optics Express, Vol. 20, Issue 17, pp. 19168-19175 (2012)

http://dx.doi.org/10.1364/OE.20.019168

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### Abstract

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.

© 2012 OSA

## 1. Introduction

1. N. Savage, “Acousto-optic devices,” Nat. Photonics **4**(10), 728–729 (2010). [CrossRef]

5. J. H. Garcia-López, V. Aboites, A. V. Kir’yanov, M. J. Damzen, and A. Minassian, “High repetition rate Q-switching of high power Nd:YVO_{4} slab laser,” Opt. Commun. **218**(1-3), 155–160 (2003). [CrossRef]

6. R. Mazelsky and D. K. Fox, “An introduction to acousto-optic materials,” Mater. Sci. Forum **61**, 1–6 (1990). [CrossRef]

## 2. Analytical approach

### 2.1 Power modulation via AOM at fundamental wavelength

_{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

7. D. Maydan, “Acoustooptical pulse modulators,” IEEE J. Quantum Electron. **6**(1), 15–24 (1970). [CrossRef]

_{0}is the peak intensity at x = 0 and y = 0 (r = 0) and w

_{0}the 1/e

^{2}beam radius [9

9. J. M. Khosrofian and B. A. Garetz, “Measurement of a gaussian laser beam diameter through the direct inversion of knife-edge data,” Appl. Opt. **22**(21), 3406–3410 (1983). [CrossRef] [PubMed]

9. J. M. Khosrofian and B. A. Garetz, “Measurement of a gaussian laser beam diameter through the direct inversion of knife-edge data,” Appl. Opt. **22**(21), 3406–3410 (1983). [CrossRef] [PubMed]

_{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

7. D. Maydan, “Acoustooptical pulse modulators,” IEEE J. Quantum Electron. **6**(1), 15–24 (1970). [CrossRef]

_{rise/fall}and f

_{m}are in accordance with [8].

### 2.2 Power modulation after harmonic generation

_{SHG}(x,y) can be expressed by Eq. (5) [10, 11

11. D. Eimerl, “High average power harmonic generation,” IEEE J. Quantum Electron. **23**(5), 575–592 (1987). [CrossRef]

_{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}). As a pulsed laser is used in the experiments, it is chosen that

_{FUN}is the average fundamental laser power, f

_{rep}the repetition rate of the laser and τ FWHM the pulse duration (sech

^{2}shape assumed). This proved to be most adequate as approximation, though it has to be noted that other pre-factors for

_{FUN}- I

_{SHG}due to previous SHG conversion. For low SHG conversion efficiency, the residual fundamental pump intensity can be estimated as I

_{FUN}- I

_{SHG}≈I

_{FUN}. 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.

_{FUN}is used (instead of Eq. (1)) and set as I

_{FUN}(x,y) = I

_{FUN,0}/2 = P

_{FUN}/(A

_{FUN}f

_{rep}τ) for

_{FUN}is therefore

_{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). The same accounts for Eqs. (7), (12), and (13) result. As a remark, Eq. (12) expresses the temporal shape of the modulated laser power analytically, where Eq. (9) has to be solved numerically generally. 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

11. D. Eimerl, “High average power harmonic generation,” IEEE J. Quantum Electron. **23**(5), 575–592 (1987). [CrossRef]

12. J. A. Armstrong, N. N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. **127**(6), 1918–1939 (1962). [CrossRef]

_{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).

## 3. Experimental setup

_{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.

_{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.

_{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

13. S. P. Velsko, M. Webb, L. Davis, and C. Huang, “Phase-matches harmonic generation in lithium triborate (LBO),” IEEE J. Quantum Electron. **27**(9), 2182–2192 (1991). [CrossRef]

_{eff,THG}≅ 0.67 pm/V [13

13. S. P. Velsko, M. Webb, L. Davis, and C. Huang, “Phase-matches harmonic generation in lithium triborate (LBO),” IEEE J. Quantum Electron. **27**(9), 2182–2192 (1991). [CrossRef]

_{SHG}and Θ

_{THG}or Θ*

_{SHG}and Θ*

_{THG}respectively are used as independent variables.

_{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 .

## 3. Experimental results

_{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.

### 3.1 Temporal AOM modulation shape prior to and after THG

_{FUN}= 1.51 W (Fig. 3 left) and P

_{FUN}= 0.36 W (Fig. 3 right).

### 3.2 Dependence of modulation parameters on THG efficiency

_{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.

_{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-noise-ratio of the diode signal of 10

^{4}:1.

## 4. Discussion

_{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.

_{THG}> 10 mW).

_{THG}> 5 mW).

^{4}:1.

## 5. Summary and outlook

## Acknowledgments

## References and links

1. | N. Savage, “Acousto-optic devices,” Nat. Photonics |

2. | A. W. Warner and D. A. Pinnow, “Miniature acousto-optic modulators for optical communications,” IEEE J. Quantum Electron. |

3. | A. J. DeMaria, R. Gagosz, and G. Barnard, “Ultrasonic-refraction shutter for optical master oscillators,” Appl. Phys. (Berl.) |

4. | D. Maydan, “Fast modulator for extraction of internal laser power,” Appl. Phys. (Berl.) |

5. | J. H. Garcia-López, V. Aboites, A. V. Kir’yanov, M. J. Damzen, and A. Minassian, “High repetition rate Q-switching of high power Nd:YVO |

6. | R. Mazelsky and D. K. Fox, “An introduction to acousto-optic materials,” Mater. Sci. Forum |

7. | D. Maydan, “Acoustooptical pulse modulators,” IEEE J. Quantum Electron. |

8. | I. C. Chang, “Acoustooptic devices and applications” in |

9. | J. M. Khosrofian and B. A. Garetz, “Measurement of a gaussian laser beam diameter through the direct inversion of knife-edge data,” Appl. Opt. |

10. | B. E. A. Saleh and M. C. Teich, |

11. | D. Eimerl, “High average power harmonic generation,” IEEE J. Quantum Electron. |

12. | J. A. Armstrong, N. N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. |

13. | S. P. Velsko, M. Webb, L. Davis, and C. Huang, “Phase-matches harmonic generation in lithium triborate (LBO),” IEEE J. Quantum Electron. |

**OCIS Codes**

(140.7090) Lasers and laser optics : Ultrafast lasers

(190.2620) Nonlinear optics : Harmonic generation and mixing

(220.4830) Optical design and fabrication : Systems design

(230.1040) Optical devices : Acousto-optical devices

(230.4110) Optical devices : Modulators

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: May 23, 2012

Manuscript Accepted: July 19, 2012

Published: August 6, 2012

**Citation**

Peter Bechtold, Oliver Hentschel, and Michael Schmidt, "Analytic and experimental investigations on influence of harmonic generation on acousto-optical modulation," Opt. Express **20**, 19168-19175 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-17-19168

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### References

- N. Savage, “Acousto-optic devices,” Nat. Photonics4(10), 728–729 (2010). [CrossRef]
- A. W. Warner and D. A. Pinnow, “Miniature acousto-optic modulators for optical communications,” IEEE J. Quantum Electron.9(12), 1155–1157 (1973). [CrossRef]
- A. J. DeMaria, R. Gagosz, and G. Barnard, “Ultrasonic-refraction shutter for optical master oscillators,” Appl. Phys. (Berl.)34, 453–456 (1963).
- D. Maydan, “Fast modulator for extraction of internal laser power,” Appl. Phys. (Berl.)41, 1552–1559 (1970).
- J. H. Garcia-López, V. Aboites, A. V. Kir’yanov, M. J. Damzen, and A. Minassian, “High repetition rate Q-switching of high power Nd:YVO4 slab laser,” Opt. Commun.218(1-3), 155–160 (2003). [CrossRef]
- R. Mazelsky and D. K. Fox, “An introduction to acousto-optic materials,” Mater. Sci. Forum61, 1–6 (1990). [CrossRef]
- D. Maydan, “Acoustooptical pulse modulators,” IEEE J. Quantum Electron.6(1), 15–24 (1970). [CrossRef]
- I. C. Chang, “Acoustooptic devices and applications” in Handbook of Optics V, M. Bass, ed. (McGraw-Hill Professional, 2010)
- J. M. Khosrofian and B. A. Garetz, “Measurement of a gaussian laser beam diameter through the direct inversion of knife-edge data,” Appl. Opt.22(21), 3406–3410 (1983). [CrossRef] [PubMed]
- B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley Series in Pure & Applied Optics, 2007), Chap. 21.
- D. Eimerl, “High average power harmonic generation,” IEEE J. Quantum Electron.23(5), 575–592 (1987). [CrossRef]
- J. A. Armstrong, N. N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev.127(6), 1918–1939 (1962). [CrossRef]
- S. P. Velsko, M. Webb, L. Davis, and C. Huang, “Phase-matches harmonic generation in lithium triborate (LBO),” IEEE J. Quantum Electron.27(9), 2182–2192 (1991). [CrossRef]

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