## Calibration of the second-order nonlinear optical susceptibility of surface and bulk of glass

Optics Express, Vol. 16, Issue 12, pp. 8704-8710 (2008)

http://dx.doi.org/10.1364/OE.16.008704

Acrobat PDF (121 KB)

### Abstract

A two-beam second-harmonic generation technique is developed to calibrate the magnitude of the second-order nonlinear optical susceptibility components of surface and bulk (multipolar origin) of isotropic materials. The values obtained for fused silica calibrated against *χ _{XXX}
* of crystalline quartz are

*χ*

_{‖‖⊥}=7.9(4),

*χ*

_{⊥‖‖}+

_{γ}=3.8(4),

*χ*

_{⊥⊥⊥}+

_{γ}=59(4), and

*δ*′=7.8(4) in units of 10

^{-22}m

^{2}/V. Similar values are obtained for BK7 glass. An alternative way of calibration against

*χ*of quartz is demonstrated. The technique could also be extended to characterize the susceptibility tensor of crystals as a convenient alternative to the Maker-fringe technique.

_{XYZ}© 2008 Optical Society of America

2. G. Lupke, “Characterization of semiconductor interfaces by second-harmonic generation,” Surf. Sci. Rep. **35**, 77–161 (1999). [CrossRef]

3. P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General-considerations on optical 2nd-harmonic generation from surfaces and interfaces,” Phys. Rev. B **33**, 8254–8263 (1986). [CrossRef]

4. J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of 2nd-harmonic generation as a strictly surface probe,” Phys. Rev. B **35**, 9091–9094 (1987). [CrossRef]

5. Y. An, R. Carriles, and M. Downer, “Absolute phase and amplitude of second-order nonlinear optical susceptibility components at Si(001) interfaces,” Phys. Rev. B **75**, 241307(R) (2007). [CrossRef]

6. X. Wei, S. C. Hong, A. I. Lvovsky, H. Held, and Y. R. Shen, “Evaluation of surface vs bulk contributions in sum-frequency vibrational spectroscopy using reflection and transmission geometries,” J. Phys. Chem. B **104**, 3349–3354 (2000). [CrossRef]

7. F. J. Rodriguez, 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**, 8695–8701 (2007). [CrossRef] [PubMed]

_{2}) samples using quartz (crystalline SiO

_{2}) as reference. Both glasses are used as common substrates for thin films.

8. J. Jerphagnon and S. K. Kurtz, “Maker Fringes: A Detailed Comparison of Theory and Experiment for Isotropic and Uniaxial Crystals,” J. Appl. Phys. **41**, 1667 (1970). [CrossRef]

9. W. Herman and L. Hayden, “Maker fringes revisited - 2nd-harmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B - Optical Physics **12**, 416–427 (1995). [CrossRef]

10. I. Shoji, T. Kondo, and R. Ito, “Second-order nonlinear susceptibilities of various dielectric and semiconductor materials,” Opt. Quantum Electron. **34**, 797–833 (2002). [CrossRef]

11. R. Gehr and A. Smith, “Separated-beam nonphase-matched second-harmonic method of characterizing nonlinear optical crystals,” J. Opt. Soc. Am. B-Optical Physics **15**, 2298–2307 (1998). [CrossRef]

12. D. Armstrong, M. Pack, and A. Smith, “Instrument and method for measuring second-order nonlinear optical tensors,” Rev. Sci. Instrum. **74**, 3250–3257 (2003). [CrossRef]

13. S. Cattaneo, M. Siltanen, F. Wang, and M. Kauranen, “Suppression of nonlinear optical signals in finite interaction volumes of bulk materials,” Opt. Express **13**, 9714–9720 (2005). [CrossRef] [PubMed]

13. S. Cattaneo, M. Siltanen, F. Wang, and M. Kauranen, “Suppression of nonlinear optical signals in finite interaction volumes of bulk materials,” Opt. Express **13**, 9714–9720 (2005). [CrossRef] [PubMed]

*C*

_{∞v}symmetry) is described by a susceptibility tensor with three independent components:

*χ*

_{⊥‖‖},

*χ*

_{‖‖⊥}and

*χ*

_{⊥⊥⊥}where ⊥ denotes the surface normal and

_{‖}denotes any direction parallel to the surface [3

3. P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General-considerations on optical 2nd-harmonic generation from surfaces and interfaces,” Phys. Rev. B **33**, 8254–8263 (1986). [CrossRef]

3. P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General-considerations on optical 2nd-harmonic generation from surfaces and interfaces,” Phys. Rev. B **33**, 8254–8263 (1986). [CrossRef]

14. P. Guyot-Sionnest and Y. R. Shen, “Local and nonlocal surface nonlinearities for surface optical 2nd-harmonic generation,” Phys. Rev. B **35**, 4420–4426 (1987). [CrossRef]

15. P. Guyot-Sionnest and Y. R. Shen, “Bulk contribution in surface 2nd-harmonic generation,” Phys. Rev. B **38**, 7985–7989 (1988). [CrossRef]

4. J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of 2nd-harmonic generation as a strictly surface probe,” Phys. Rev. B **35**, 9091–9094 (1987). [CrossRef]

*β*,

*γ*, and

*δ*′, which can be expressed as combinations of electric-quadrupole and magnetic-dipole tensors [16]. The SH polarization of the bulk is then

**P**

^{bulk}(

**r**)=

*β*

**e**(

**r**)[∇·

**e**(

**r**)]+

*γ*∇[

**e**(

**r**)·

**e**(

**r**)]+

*δ*′ [

**e**(

**r**)·∇]

**e**(

**r**), where

**e**(

**r**) is the fundamental field. The first term vanishes for homogeneous media and the second term behaves like the surface contribution so that the measurable quantities are the combinations

*χ*

_{⊥⊥⊥}+

*γ*and

*χ*

_{⊥‖‖}+

*γ*. Thus, the only bulk parameter that can be separated form the surface is

*δ*′. The term with

*δ*′ vanishes when the fundamental field is a single plane wave. However, it can be determined by detecting the SH signal generated jointly by two noncollinear beams at the fundamental frequency.

*s*and

*p*components of the SHG field from the glass (surface and multipolar bulk contributions) are given in transmission (lower signs) and in reflection (upper signs) by [7

7. F. J. Rodriguez, 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**, 8695–8701 (2007). [CrossRef] [PubMed]

*a*and

*b*are the amplitudes of the two fundamental beams (with frequency

*ω*), Δ

*k*

_{±}=(

*n*(cos

*θ*+cos

_{a}*θ*)±

_{b}*N*cos

*Θ*)2

*ω*/

*c*are phase mismatches and

*n*and

*N*the refractive indices of glass at the fundamental and SH frequencies, respectively. The propagation angle

*Θ*of the SH beams is given by the momentum conservation along the surface:

*N*sin

*Θ*=

*n*(sin

*θ*+sin

_{a}*θ*)/2. In all these expressions, the fields and angles are considered inside the material. To relate the external and internal quantities in a real experiment, it is necessary to use Snell’s law and the appropriate Fresnel coefficients. In our calculations of the SHG detected in transmission we also included the SHG that is generated backwards and then reflected by the front surface of the sample.

_{b}_{3}symmetry and its second-order susceptibility has four independent components:

*χ*=-

_{XXX}*χ*=-

_{XYY}*χ*=-

_{YYX}*χ*,

_{YXY}*χ*=-

_{XYZ}*χ*,

_{YXZ}*χ*=-

_{XZY}*χ*and

_{YZX}*χ*=-

_{ZXY}*χ*. In the case of SHG, the second and third are the same and the fourth is zero. The nonzero values are

_{ZYX}*χ*=0.80 pm/V and

_{XXX}*χ*=0.017 pm/V [17].

_{XYZ}*s*direction (Fig. 1) is given by the expression

*p*polarized) is the ordinary index of quartz

*N*(532 nm)=1.5469. However, the index for the fundamental wavelength depends on the polarization. For

_{o}*p*polarization it will be the ordinary index

*n*(1064 nm)=1.5341 while for

_{o}*s*polarization it will be the extraordinary index

*n*(1064 nm)=1.5428. Thus, it is necessary to consider different values of Δ

_{e}*k*

_{±}and of the angles inside the material in Eq. (2) and also different values of the Fresnel coefficients depending on the fundamental beam polarizations.

*E*=

_{i}*f*

_{ijk}*a*

_{j}*b*, where

_{k}*i*,

*j*and

*k*are

*s*or

*p*and we assume summation over repeated indices. The coefficients

*f*for glass and quartz can be obtained from Eqs. (1) and (2), respectively. To calibrate the glass parameters against

_{ijk}*χ*of quartz we choose

_{XXX}*p*-polarized fundamental beams and detect

*p*-polarized SHG. In this way, we are able to measure the signals from both glass and quartz without the need to change the polarizations. It also makes the analysis simpler because only the ordinary refractive index of quartz is used. If the intensity of the fundamental beams is kept constant, the ratio of the SHG intensity between the glass sample and the quartz reference when all the fields are

*p*polarized is

*I*/

^{glass}_{SHG}*I*=|

^{quartz}_{SHG}*f*|

^{glass}_{ppp}^{2}/|

*f*|

^{quartz}_{ppp}^{2}. The ratio of SHG intensity is measured experimentally by changing the glass sample to the quartz reference and using neutral density filters to keep the photomultiplier tube used for the detection in the range of linear response. The value of

*f*is calculated from Eq. (2) and the tabulated susceptibility of quartz. With all this we obtain a calibrated value of

^{quartz}_{ppp}*f*.

^{glass}_{ppp}7. F. J. Rodriguez, 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**, 8695–8701 (2007). [CrossRef] [PubMed]

*p*, with a rotating quarter-wave plate. The SHG generated by the two beams is detected after passing through a polarizer oriented at different angles. We use four measurements of SHG vs. quarter-wave plate angle for various combinations of polarizations (

*s*,

*p*or 45°) of the linearly polarized fundamental beam and the SH beam. The fitting of three such measurements already yields the relative values of

*f*,

^{glass}_{ppp}*f*,

^{glass}_{pss}*f*and

^{glass}_{ssp}*f*. The fourth measurement can be used as a check of the experimental setup or to find the best simultaneous fit of the four measurements for more accurate values. From the calibrated values of all the

^{glass}_{sps}*f*we obtain the susceptibility tensor components using Eq. (1).

^{glass}_{ijk}*θ*=30° (linearly polarized beam) and

_{a}*θ*=61° (quarter-wave plate beam). By fitting the four polarization measurements simultaneously with the theoretical expressions (Fig. 2) we obtained the relative values of all the

_{b}*f*and their absolute values from the calibrated

^{glass}_{ijk}*f*. The tensor components derived from them are shown in Table I.

^{glass}_{ppp}*δ*′ parameter of glass against

*χ*of quartz. To make the measurement in glass depend uniquely on the bulk

_{XYZ}*δ*′ parameter, we set one beam (

*a*) to have

*p*polarization and to be at normal incidence with respect to the surface (

*θ*=0) and the other beam (

_{a}*b*) to have

*s*polarization. Eq. (1) then becomes

*s*direction is only sensitive to

*χ*and is given, for both X- and Y-cut quartz by the expression

_{XYZ}*b*. The expression for the phase mismatch is also slightly affected: Δ

*k*

_{±}=(

*n*cos

_{o}*θ*+

_{a}*n*cos

_{e}*θ*±

_{b}*N*cos

_{o}*Θ*)2

*ω*/

*c*as is also the angle of the SHG beam

*N*sin

_{o}*Θ*=(

*n*sin

_{o}*θ*+

_{a}*n*sin

_{e}*θ*)/2. The measurement in quartz is very sensitive to the alignment of the (

_{b}*s*) polarization of beam

*b*and the Z axis of quartz. Any small angle between the two directions will introduce a contribution from the much larger

*χ*component of quartz and result in a higher SHG signal.

_{XXX}*θ*=-31° and X-cut quartz as the reference we obtained the

_{b}*δ*′ value shown in Table I for fused silica and BK7 glass. We calculated the rest of the tensor components using the relative values obtained in the polarization analysis described above for the fused silica and the one made previously [7

**15**, 8695–8701 (2007). [CrossRef] [PubMed]

*b*and the quartz crystal were aligned with a precision better than 15′ by minimizing the SH signal to avoid any contribution from

*χ*. However, we still detected about 30% of the SH intensity having

_{XXX}*s*polarization while it should have been zero as seen from Eq. (4). This is presently a drawback of this method and can lead to errors in the experimentally determined values. The origin of the problem is unclear, but it may be associated with small polarization rotation in this geometry because of the optical activity of quartz or with a multipolar response of quartz.

**15**, 8695–8701 (2007). [CrossRef] [PubMed]

5. Y. An, R. Carriles, and M. Downer, “Absolute phase and amplitude of second-order nonlinear optical susceptibility components at Si(001) interfaces,” Phys. Rev. B **75**, 241307(R) (2007). [CrossRef]

*χ*

_{⊥‖‖}+

*γ*and 3 orders smaller for

*χ*

_{‖‖⊥}and

*χ*

_{⊥⊥⊥}+

*γ*. The value of the bulk parameter ζ that appears in cubic crystals is also 3 orders of magnitude larger than the

*δ*′ of fused silica. The large difference in the magnitudes can be understood by the resonant conditions used in the crystalline silicon measurements and the non-resonant conditions in our fused silica measurements. In addition, Wei,

*et al.*, [6

6. X. Wei, S. C. Hong, A. I. Lvovsky, H. Held, and Y. R. Shen, “Evaluation of surface vs bulk contributions in sum-frequency vibrational spectroscopy using reflection and transmission geometries,” J. Phys. Chem. B **104**, 3349–3354 (2000). [CrossRef]

^{-22}m

^{2}/V, very close to the ones reported.

## References and Links

1. | T. F. Heinz, “Second-order nonlinear optical effects at surfaces and interfaces,” in |

2. | G. Lupke, “Characterization of semiconductor interfaces by second-harmonic generation,” Surf. Sci. Rep. |

3. | P. Guyot-Sionnest, W. Chen, and Y. R. Shen, “General-considerations on optical 2nd-harmonic generation from surfaces and interfaces,” Phys. Rev. B |

4. | J. E. Sipe, V. Mizrahi, and G. I. Stegeman, “Fundamental difficulty in the use of 2nd-harmonic generation as a strictly surface probe,” Phys. Rev. B |

5. | Y. An, R. Carriles, and M. Downer, “Absolute phase and amplitude of second-order nonlinear optical susceptibility components at Si(001) interfaces,” Phys. Rev. B |

6. | X. Wei, S. C. Hong, A. I. Lvovsky, H. Held, and Y. R. Shen, “Evaluation of surface vs bulk contributions in sum-frequency vibrational spectroscopy using reflection and transmission geometries,” J. Phys. Chem. B |

7. | F. J. Rodriguez, 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 |

8. | J. Jerphagnon and S. K. Kurtz, “Maker Fringes: A Detailed Comparison of Theory and Experiment for Isotropic and Uniaxial Crystals,” J. Appl. Phys. |

9. | W. Herman and L. Hayden, “Maker fringes revisited - 2nd-harmonic generation from birefringent or absorbing materials,” J. Opt. Soc. Am. B - Optical Physics |

10. | I. Shoji, T. Kondo, and R. Ito, “Second-order nonlinear susceptibilities of various dielectric and semiconductor materials,” Opt. Quantum Electron. |

11. | R. Gehr and A. Smith, “Separated-beam nonphase-matched second-harmonic method of characterizing nonlinear optical crystals,” J. Opt. Soc. Am. B-Optical Physics |

12. | D. Armstrong, M. Pack, and A. Smith, “Instrument and method for measuring second-order nonlinear optical tensors,” Rev. Sci. Instrum. |

13. | S. Cattaneo, M. Siltanen, F. Wang, and M. Kauranen, “Suppression of nonlinear optical signals in finite interaction volumes of bulk materials,” Opt. Express |

14. | P. Guyot-Sionnest and Y. R. Shen, “Local and nonlocal surface nonlinearities for surface optical 2nd-harmonic generation,” Phys. Rev. B |

15. | P. Guyot-Sionnest and Y. R. Shen, “Bulk contribution in surface 2nd-harmonic generation,” Phys. Rev. B |

16. | M. Kauranen and S. Cattaneo, “Polarization techniques for surface nonlinear optics,” in |

17. | R. W. Boyd, |

18. | V. Rodriguez, “Quantitative determination of linear and second-harmonic generation optical effective responses of achiral or chiral materials in planar structures: Theory and materials,” J. Chem. Phys. |

**OCIS Codes**

(160.2750) Materials : Glass and other amorphous materials

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.4350) Nonlinear optics : Nonlinear optics at surfaces

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: March 27, 2008

Revised Manuscript: May 22, 2008

Manuscript Accepted: May 26, 2008

Published: May 29, 2008

**Citation**

Francisco J. Rodriguez, Fu Xiang Wang, and Martti Kauranen, "Calibration of the second-order nonlinear optical susceptibility of surface and bulk of glass," Opt. Express **16**, 8704-8710 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-12-8704

Sort: Year | Journal | Reset

### References

- T. F. Heinz, "Second-order nonlinear optical effects at surfaces and interfaces," in Nonlinear Surface Electromagnetic Phenomena, H.-E. Ponath, and G. I. Stegeman, eds., (Elsevier, Amsterdam, 1991).
- G. Lupke, "Characterization of semiconductor interfaces by second-harmonic generation," Surf. Sci. Rep. 35, 77-161 (1999). [CrossRef]
- P. Guyot-Sionnest, W. Chen, and Y. R. Shen, "General-considerations on optical 2nd-harmonic generation from surfaces and interfaces," Phys. Rev. B 33, 8254-8263 (1986). [CrossRef]
- J. E. Sipe, V. Mizrahi, and G. I. Stegeman, "Fundamental difficulty in the use of 2nd-harmonic generation as a strictly surface probe," Phys. Rev. B 35, 9091-9094 (1987). [CrossRef]
- Y. An, R. Carriles, and M. Downer, "Absolute phase and amplitude of second-order nonlinear optical susceptibility components at Si(001) interfaces," Phys. Rev. B 75, 241307(R) (2007). [CrossRef]
- X. Wei, S. C. Hong, A. I. Lvovsky, H. Held, and Y. R. Shen, "Evaluation of surface vs bulk contributions in sum-frequency vibrational spectroscopy using reflection and transmission geometries," J. Phys. Chem. B 104, 3349-3354 (2000). [CrossRef]
- F. J. Rodriguez, 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, 8695-8701 (2007). [CrossRef] [PubMed]
- J. Jerphagnon and S. K. Kurtz, "Maker Fringes: A Detailed Comparison of Theory and Experiment for Isotropic and Uniaxial Crystals," J. Appl. Phys. 41, 1667 (1970). [CrossRef]
- W. Herman and L. Hayden, "Maker fringes revisited - 2nd-harmonic generation from birefringent or absorbing materials," J. Opt. Soc. Am. B 12, 416-427 (1995). [CrossRef]
- I. Shoji, T. Kondo, and R. Ito, "Second-order nonlinear susceptibilities of various dielectric and semiconductor materials," Opt. Quantum Electron. 34, 797-833 (2002). [CrossRef]
- R. Gehr, and A. Smith, "Separated-beam nonphase-matched second-harmonic method of characterizing nonlinear optical crystals," J. Opt. Soc. Am.Bs 15, 2298-2307 (1998). [CrossRef]
- D. Armstrong, M. Pack, and A. Smith, "Instrument and method for measuring second-order nonlinear optical tensors," Rev. Sci. Instrum. 74, 3250-3257 (2003). [CrossRef]
- S. Cattaneo, M. Siltanen, F. Wang, and M. Kauranen, "Suppression of nonlinear optical signals in finite interaction volumes of bulk materials," Opt. Express 13, 9714-9720 (2005). [CrossRef] [PubMed]
- P. Guyot-Sionnest and Y. R. Shen, "Local and nonlocal surface nonlinearities for surface optical 2nd-harmonic generation," Phys. Rev. B 35, 4420-4426 (1987). [CrossRef]
- P. Guyot-Sionnest and Y. R. Shen, "Bulk contribution in surface 2nd-harmonic generation," Phys. Rev. B 38, 7985-7989 (1988). [CrossRef]
- M. Kauranen and S. Cattaneo, "Polarization techniques for surface nonlinear optics," in Progress in Optics, E. Wolf, ed., (Elsevier, Amsterdam, 2008).
- R. W. Boyd, Nonlinear Optics (Academic Press, San Diego, CA, 2003).
- V. Rodriguez, "Quantitative determination of linear and second-harmonic generation optical effective responses of achiral or chiral materials in planar structures: Theory and materials," J. Chem. Phys. 128, 064707 (2008). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.