## Bound and free waves in non-collinear second harmonic generation

Optics Express, Vol. 17, Issue 19, pp. 17000-17009 (2009)

http://dx.doi.org/10.1364/OE.17.017000

Acrobat PDF (535 KB)

### Abstract

We analyze the relationship between the bound and the free waves in the noncollinear SHG scheme, along with the vectorial conservation law for the different components arising when there are two pump beams impinging on the sample with two different incidence angles. The generated power is systematically investigated, by varying the polarization state of both fundamental beams, while absorption is included via the Herman and Hayden correction terms. The theoretical simulations, obtained for samples which are some coherence length thick show that the resulting polarization mapping is an useful tool to put in evidence the interference between bound and free waves, as well as the effect of absorption on the interference pattern.

© 2009 OSA

## 1. Introduction

1. P. D. Maker, R. W. Terhume, M. Nisenhoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. **8**(1), 21–22 (1962). [CrossRef]

_{1}induces in the nonlinear material a polarization composed by two waves tuned at 2ω

_{1}. Given the refractive index dispersion, the so-called “bound” and the “free” wave experience n(ω

_{1}) and n(2ω

_{1}), respectively, and generally travel at different velocities. The existence of these two waves, simply obtained as solutions of Maxwell’s equations [2

2. N. Bloembergen and P. S. Pershan, “Light wave at the boundary of nonlinear media,” Phys. Rev. **128**(2), 606–622 (1962). [CrossRef]

3. J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. **41**(4), 1667–1681 (1970). [CrossRef]

4. V. Figà, J. Luc, B. Kulyk, M. Baitoul, and B. Sahraoui, “Characterization and investigation of NLO properties of electrodeposited polythiophenes,” J. Eur. Opt. Soc. Rapid Publ. **4**, 09016–09021 (2009). [CrossRef]

*phase-locked*wave, in order to point out that it is located under the pump pulse and is dragged at the pump’s group velocity [5

5. V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. Bloemer, J. W. Haus, O. G. Kosareva and V. P. Kandidov, “Role of phase matching in pulsed second harmonic generation: walk off and phase locked twin pulses in negative index media,” Phys. Rev. A **76**, 033829/1–033829/12 (2007).

10. R. E. Muenchausen, R. A. Keller, and N. S. Nogar, “Surface second-harmonic and sum-frequency generation using a noncollinear excitation geometry,” J. Opt. Soc. Am. B **4**(2), 237–241 (1987). [CrossRef]

11. P. Provencher, C. Y. Côté, and M. M. Denariez-Roberge, “Surface second-harmonic susceptibility determined by noncollinear reflected second-harmonic generation,” Can. J. Phys. **71**, 66–69 (1993). [CrossRef]

12. P. Figliozzi, L. Sun, Y. Jiang, N. Matlis, B. Mattern, M. C. Downer, S. P. Withrow, C. W. White, W. L. Mochán and B. S. Mendoza, “Single-Beam and Enhanced Two-Beam Second-Harmonic Generation from Silicon Nanocrystals by Use of Spatially Inhomogeneous Femtosecond Pulses,” Phys. Rev. Lett. **94**, 047401/1–047401/4 (2005).

14. S. Cattaneo and M. Kauranen, “Determination of second-order susceptibility components of thin films by two-beam second-harmonic generation,” Opt. Lett. **28**(16), 1445–1447 (2003). [CrossRef] [PubMed]

15. S. Cattaneo, E. Vuorimaa, H. Lemmetyinen, and M. Kauranen, “Advantages of polarized two-beam second-harmonic generation in precise characterization of thin films,” J. Chem. Phys. **120**(19), 9245–9252 (2004). [CrossRef] [PubMed]

16. J. Trull, C. Cojocaru, R. Fischer, S. M. Saltiel, K. Staliunas, R. Herrero, R. Vilaseca, D. N Neshev, W. Krolikowski, and Y. S. Kivshar, “Second-harmonic parametric scattering in ferroelectric crystals with disordered nonlinear domain structures,” Opt. Express **15**(24), 15868–15877 (2007). [CrossRef] [PubMed]

17. M. C. Larciprete, F. A. Bovino, M. Giardina, A. Belardini, M. Centini, C. Sibilia, M. Bertolotti, A. Passaseo, and V. Tasco, “Mapping the nonlinear optical susceptibility by noncollinear second harmonic generation,” Opt. Lett. **34**(14), 2189–2191 (2009). [CrossRef] [PubMed]

18. D. Faccio, V. Pruneri, and P. G. Kazansky, “Noncollinear Maker’s fringe measurements of second-order nonlinear optical layers,” Opt. Lett. **25**(18), 1376–1378 (2000). [CrossRef]

*polarization map*whose pattern is characteristic of the investigated crystalline structure. This method offers the possibility to evaluate the ratio between the different non-zero elements of the nonlinear optical tensor, or the evaluation of the absolute values of the non-zero terms of the nonlinear optical tensor, without requiring sample rotation. As a result, it is extremely interesting for those conditions where the generated signal would be strongly affected by sample rotation angle, i.e. for samples which are some coherence lengths thick, when using short laser pulses, of for nano-patterned samples. With respect to the given examples, this method of polarization scan allows the characterization of the nonlinear optical tensor elements without varying the experimental conditions.

## 2. Wavevector conservation in noncollinear second harmonic generation

*ω1*=

*ω2*=

*ω,*having two different incidence angles, with respect to surface normal,

*α*and

_{1}*α*, and different polarization state,

_{2}20. N. Bloembergen, “Conservation laws in nonlinear optics,” J. Opt. Soc. Am. B **70**(12), 1429–1436 (1980). [CrossRef]

21. F. J. Rodríguez, F. X. Wang, B. K. Canfield, S. Cattaneo, and M. Kauranen, “Multipolar tensor analysis of second-order nonlinear optical response of surface and bulk of glass,” Opt. Express **15**(14), 8695–8701 (2007). [CrossRef] [PubMed]

_{1}and 2ω

_{2}, respectively. The two bound waves, having wavevectors

*α’*and

_{1,B}*α’*, are equal to

_{2,B}*α’*and

_{1}*α’*, respectively, and depend on the polarization state of the corresponding pump beam but not on the chosen polarization state of the SH. On the other side, the two free waves experience higher refractive indices, due to material dispersion, thus their wavevectors,

_{2}*α’*and

_{1,F}*α’*, are dependent only on the polarization state of the SH beam.

_{2,F}*α*= 26° and

_{1}*α*= 44°, i.e. the two pump beams have an aperture angle of 18° with respect to each other. The fundamental beam is tuned at 830 nm and the extraordinary and ordinary refractive index are assumed to be

_{2}*α’*and

_{1,B}*α’*as a function of polarization state of the corresponding pump beam.

_{2,B}*α’*= 9.16° for a

_{1,F}*α’*= 9.17° for an

_{1,F}*α’*= 14.61° for a

_{2,F}*α’*= 14.63° for an

_{2,F}*α’*from the conservation of the Poynting vector:While

_{3,F}*α’*is obtained from the vectorial sum of the two pumps’ wavectors, in the three waves interaction:The obtained propagation angle of the noncollinear bound wave,

_{3,B}*α’*, is represented in Fig. 3 , and results as a combination of the propagation angles of the two pump beams.

_{3,B}## 3. Noncollinear second harmonic generation

*P*

_{ω1 + ω2}, as a function of incidence angles, including the effect of absorption, trough the extinction coefficient at the fundamental,

*A*and

_{1}*A*are the fundamental beams transverse areas onto sample surface, retrieved from the pump beam area (

_{2}*A*) as

*L*is sample thickness. It’s worth noting that Fresnel coefficients are in general complex, but for small extinction coefficients they can be assumed real. The power of the incident fundamental beams is taken into account in the term

*φ*,

_{1}*φ*and

_{2}*φ*as well.

_{3}23. T. K. Lim, M. Y. Jeong, C. Song, and D. C. Kim, “Absorption effect in the calculation of a second-order nonlinear coefficient from the data of a maker fringe experiment,” Appl. Opt. **37**(13), 2723–2728 (1998). [CrossRef]

*d*

_{eff}(

*α*) in Eq. (3) represents the effective susceptibility tensor, being dependent on the second order nonlinear optical tensor, the polarization state of both pumps and generated beams and, of course, on the fundamental beams incidence angles,

*α*and

_{1}*α*.

_{2}## 4. Examples for some different crystal structures

*α*= 26° and

_{1}*α*= 44°, with respect to surface normal, in both examples. The generated beam is projected along the bound SH angle, i.e. approximately along the bisector of the two pump beams aperture angle.

_{2}*d*,

_{15}= d_{24}*d*and

_{31}= d_{32}*d*, the final expressions for

_{33}*d*

_{eff}(

*α*) as a function of polarization angle of the two pumps,

*φ*and

_{1}*φ*is easily written:

_{2}_{ω}= 830, and the linear refractive indices already used for the calculations shown in Fig. 2 and Fig. 3 [22]. The SH power for

*φ*and

_{1}*φ*.

_{2}*k*= 0. The oscillation of SH signal as a function of different pump beams polarization state is due to the high thickness, with respect to the coherence length of the process. At the given incidence angles

_{2ω}*α*and

_{1}*α*, in fact, the coherence length, defined in noncollinear SHG as

_{2}*φ*= 90° and

_{1}*φ*= 90°, and its minimum value at

_{2}*φ*= 0° and

_{1}*φ*= 0° (

_{2}24. R. Héliou, J. L. Brebner, and S. Roorda, “Role of implantation temperature on residual damage in ion-implanted 6H–SiC,” Semicond. Sci. Technol. **16**(10), 836–843 (2001). [CrossRef]

*k*= 0.003, corresponding to a linear transmittance at λ

_{2ω}_{2ω}= 415nm of approximately 10%.

*k*= 0.005, corresponding to a linear transmittance at 2ω of approximately 2%.

_{2ω}*-*polarized, i.e. when

*φ*and

_{1}*φ*are both 0° or 180°, while relative maxima (saddle point) occur when both pumps are

_{2}*-*polarized, i.e. when polarization angles of both pumps are set to ± 90°. Conversely, when the two pump beams have crossed polarization, i.e when

*φ*= 0° and

_{1}*φ*= 90° and viceversa, the nonlinear optical tensor do not allow SH signal which is

_{2}*-*polarization, the maxima generally occur when the two pump beams have crossed polarization, as shown in Fig. 6b, i.e. when the first pump is

*φ*

_{1}= ± 90° and

*φ*

_{2}is equal to either 0° or 180°. Relative maxima occur in the reverse situation, when the first pump is

*φ*

_{1}= 0° or ± 180°, and the second pump

*φ*

_{2}= 90°. Finally, when the two pumps are equally polarized, either

_{ω}= 1500 nm so that the generated beam falls within the absorption band of GaAs (E

_{g}~1.42 eV). The parameters provided by refractive index dispersion are

*n*= 3.37 at the fundamental beam frequency and

_{ω}*n*= 3.70 at the SH frequency, without birefringence, along with a high absorption coefficient such that at 2ω

_{2ω}*k*= 0.1 [25]. Due to its high-symmetry nonlinear

_{2ω}**-**optical susceptibility tensor, the only non-zero elements are

*d*. The expressions for

_{14}= d_{25}= d_{36}*d*as a function of polarization angle of the two pumps,

_{eff}(α)*φ*and

_{1}*φ*is thus simplified:

_{2}*d*. The pattern of the noncollinear SH signal, in fact, can be modified depending on the different crystalline structure that is considered.

_{eff}(α)## 5. Conclusions

## Acknowledgments.

## References and links

1. | P. D. Maker, R. W. Terhume, M. Nisenhoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. |

2. | N. Bloembergen and P. S. Pershan, “Light wave at the boundary of nonlinear media,” Phys. Rev. |

3. | J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. |

4. | V. Figà, J. Luc, B. Kulyk, M. Baitoul, and B. Sahraoui, “Characterization and investigation of NLO properties of electrodeposited polythiophenes,” J. Eur. Opt. Soc. Rapid Publ. |

5. | V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. Bloemer, J. W. Haus, O. G. Kosareva and V. P. Kandidov, “Role of phase matching in pulsed second harmonic generation: walk off and phase locked twin pulses in negative index media,” Phys. Rev. A |

6. | W. N. Herman and L. M. Hayden, “Maker fringes revisited: second harmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B |

7. | J. Rams and J. M. Cabrera, “Second harmonic generation in the strong absorption regime,” J. Mod. Opt. |

8. | M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer and M. Scalora, “Inhibition of linear absorption in opaque materials using phase-locked harmonic generation,” Phys. Rev. Lett. |

9. | E. Fazio, F. Pettazzi, M. Centini, M. Chauvet, A. Belardini, M. Alonzo, C. Sibilia, M. Bertolotti, and M. Scalora, “Complete spatial and temporal locking in phase-mismatched second-harmonic generation,” Opt. Express |

10. | R. E. Muenchausen, R. A. Keller, and N. S. Nogar, “Surface second-harmonic and sum-frequency generation using a noncollinear excitation geometry,” J. Opt. Soc. Am. B |

11. | P. Provencher, C. Y. Côté, and M. M. Denariez-Roberge, “Surface second-harmonic susceptibility determined by noncollinear reflected second-harmonic generation,” Can. J. Phys. |

12. | P. Figliozzi, L. Sun, Y. Jiang, N. Matlis, B. Mattern, M. C. Downer, S. P. Withrow, C. W. White, W. L. Mochán and B. S. Mendoza, “Single-Beam and Enhanced Two-Beam Second-Harmonic Generation from Silicon Nanocrystals by Use of Spatially Inhomogeneous Femtosecond Pulses,” Phys. Rev. Lett. |

13. | S. Cattaneo and M. Kauranen, “Polarization-based identification of bulk contributions in surface nonlinear optics,” Phys. Rev. B |

14. | S. Cattaneo and M. Kauranen, “Determination of second-order susceptibility components of thin films by two-beam second-harmonic generation,” Opt. Lett. |

15. | S. Cattaneo, E. Vuorimaa, H. Lemmetyinen, and M. Kauranen, “Advantages of polarized two-beam second-harmonic generation in precise characterization of thin films,” J. Chem. Phys. |

16. | J. Trull, C. Cojocaru, R. Fischer, S. M. Saltiel, K. Staliunas, R. Herrero, R. Vilaseca, D. N Neshev, W. Krolikowski, and Y. S. Kivshar, “Second-harmonic parametric scattering in ferroelectric crystals with disordered nonlinear domain structures,” Opt. Express |

17. | M. C. Larciprete, F. A. Bovino, M. Giardina, A. Belardini, M. Centini, C. Sibilia, M. Bertolotti, A. Passaseo, and V. Tasco, “Mapping the nonlinear optical susceptibility by noncollinear second harmonic generation,” Opt. Lett. |

18. | D. Faccio, V. Pruneri, and P. G. Kazansky, “Noncollinear Maker’s fringe measurements of second-order nonlinear optical layers,” Opt. Lett. |

19. | V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, |

20. | N. Bloembergen, “Conservation laws in nonlinear optics,” J. Opt. Soc. Am. B |

21. | F. J. Rodríguez, F. X. Wang, B. K. Canfield, S. Cattaneo, and M. Kauranen, “Multipolar tensor analysis of second-order nonlinear optical response of surface and bulk of glass,” Opt. Express |

22. | W. J. Choyke, and E. D. Palik, “Silicon Carbide (SiC)”, in |

23. | T. K. Lim, M. Y. Jeong, C. Song, and D. C. Kim, “Absorption effect in the calculation of a second-order nonlinear coefficient from the data of a maker fringe experiment,” Appl. Opt. |

24. | R. Héliou, J. L. Brebner, and S. Roorda, “Role of implantation temperature on residual damage in ion-implanted 6H–SiC,” Semicond. Sci. Technol. |

25. | E. D. Palik, “Gallium Arsenide (GaAs)”, in |

**OCIS Codes**

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.5970) Nonlinear optics : Semiconductor nonlinear optics including MQW

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: June 25, 2009

Revised Manuscript: August 4, 2009

Manuscript Accepted: August 4, 2009

Published: September 9, 2009

**Citation**

M. C. Larciprete, F. A. Bovino, A. Belardini, C. Sibilia, and M. Bertolotti, "Bound and free waves in non-collinear second harmonic generation," Opt. Express **17**, 17000-17009 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-19-17000

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### References

- P. D. Maker, R. W. Terhume, M. Nisenhoff, and C. M. Savage, “Effects of dispersion and focusing on the production of optical harmonics,” Phys. Rev. Lett. 8(1), 21–22 (1962). [CrossRef]
- N. Bloembergen and P. S. Pershan, “Light wave at the boundary of nonlinear media,” Phys. Rev. 128(2), 606–622 (1962). [CrossRef]
- J. Jerphagnon and S. K. Kurtz, “Maker fringes: a detailed comparison of theory and experiment for isotropic and uniaxial crystals,” J. Appl. Phys. 41(4), 1667–1681 (1970). [CrossRef]
- V. Figà, J. Luc, B. Kulyk, M. Baitoul, and B. Sahraoui, “Characterization and investigation of NLO properties of electrodeposited polythiophenes,” J. Eur. Opt. Soc. Rapid Publ. 4, 09016–09021 (2009). [CrossRef]
- V. Roppo, M. Centini, C. Sibilia, M. Bertolotti, D. de Ceglia, M. Scalora, N. Akozbek, M. Bloemer, J. W. Haus, O. G. Kosareva and V. P. Kandidov, “Role of phase matching in pulsed second harmonic generation: walk off and phase locked twin pulses in negative index media,” Phys. Rev. A 76, 033829/1–033829/12 (2007).
- W. N. Herman and L. M. Hayden, “Maker fringes revisited: second harmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B 12(3), 416–427 (1995). [CrossRef]
- J. Rams and J. M. Cabrera, “Second harmonic generation in the strong absorption regime,” J. Mod. Opt. 47, 1659–1669 (2000).
- M. Centini, V. Roppo, E. Fazio, F. Pettazzi, C. Sibilia, J. W. Haus, J. V. Foreman, N. Akozbek, M. J. Bloemer and M. Scalora, “Inhibition of linear absorption in opaque materials using phase-locked harmonic generation,” Phys. Rev. Lett. 101, 113905/1–113905/4 (2008).
- E. Fazio, F. Pettazzi, M. Centini, M. Chauvet, A. Belardini, M. Alonzo, C. Sibilia, M. Bertolotti, and M. Scalora, “Complete spatial and temporal locking in phase-mismatched second-harmonic generation,” Opt. Express 17(5), 3141–3147 (2009). [CrossRef] [PubMed]
- R. E. Muenchausen, R. A. Keller, and N. S. Nogar, “Surface second-harmonic and sum-frequency generation using a noncollinear excitation geometry,” J. Opt. Soc. Am. B 4(2), 237–241 (1987). [CrossRef]
- P. Provencher, C. Y. Côté, and M. M. Denariez-Roberge, “Surface second-harmonic susceptibility determined by noncollinear reflected second-harmonic generation,” Can. J. Phys. 71, 66–69 (1993). [CrossRef]
- P. Figliozzi, L. Sun, Y. Jiang, N. Matlis, B. Mattern, M. C. Downer, S. P. Withrow, C. W. White, W. L. Mochán and B. S. Mendoza, “Single-Beam and Enhanced Two-Beam Second-Harmonic Generation from Silicon Nanocrystals by Use of Spatially Inhomogeneous Femtosecond Pulses,” Phys. Rev. Lett. 94, 047401/1–047401/4 (2005).
- S. Cattaneo and M. Kauranen, “Polarization-based identification of bulk contributions in surface nonlinear optics,” Phys. Rev. B 72, 033412/1–033412/4 (2005).
- S. Cattaneo and M. Kauranen, “Determination of second-order susceptibility components of thin films by two-beam second-harmonic generation,” Opt. Lett. 28(16), 1445–1447 (2003). [CrossRef] [PubMed]
- S. Cattaneo, E. Vuorimaa, H. Lemmetyinen, and M. Kauranen, “Advantages of polarized two-beam second-harmonic generation in precise characterization of thin films,” J. Chem. Phys. 120(19), 9245–9252 (2004). [CrossRef] [PubMed]
- J. Trull, C. Cojocaru, R. Fischer, S. M. Saltiel, K. Staliunas, R. Herrero, R. Vilaseca, D. N Neshev, W. Krolikowski, and Y. S. Kivshar, “Second-harmonic parametric scattering in ferroelectric crystals with disordered nonlinear domain structures,” Opt. Express 15(24), 15868–15877 (2007). [CrossRef] [PubMed]
- M. C. Larciprete, F. A. Bovino, M. Giardina, A. Belardini, M. Centini, C. Sibilia, M. Bertolotti, A. Passaseo, and V. Tasco, “Mapping the nonlinear optical susceptibility by noncollinear second harmonic generation,” Opt. Lett. 34(14), 2189–2191 (2009). [CrossRef] [PubMed]
- D. Faccio, V. Pruneri, and P. G. Kazansky, “Noncollinear Maker’s fringe measurements of second-order nonlinear optical layers,” Opt. Lett. 25(18), 1376–1378 (2000). [CrossRef]
- V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer, Berlin, 1997).
- N. Bloembergen, “Conservation laws in nonlinear optics,” J. Opt. Soc. Am. B 70(12), 1429–1436 (1980). [CrossRef]
- F. J. Rodríguez, F. X. Wang, B. K. Canfield, S. Cattaneo, and M. Kauranen, “Multipolar tensor analysis of second-order nonlinear optical response of surface and bulk of glass,” Opt. Express 15(14), 8695–8701 (2007). [CrossRef] [PubMed]
- W. J. Choyke, and E. D. Palik, “Silicon Carbide (SiC)”, in Handbook of Optical Constants of Solids, E. D. Palik, ed., (Academic Orlando, Fla., 1985).
- T. K. Lim, M. Y. Jeong, C. Song, and D. C. Kim, “Absorption effect in the calculation of a second-order nonlinear coefficient from the data of a maker fringe experiment,” Appl. Opt. 37(13), 2723–2728 (1998). [CrossRef]
- R. Héliou, J. L. Brebner, and S. Roorda, “Role of implantation temperature on residual damage in ion-implanted 6H–SiC,” Semicond. Sci. Technol. 16(10), 836–843 (2001). [CrossRef]
- E. D. Palik, “Gallium Arsenide (GaAs)”, in Handbook of Optical Constants of Solids, E.D. Palik, ed. (Academic Orlando, Fla., 1985).

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