## Non-collinear generation of third harmonic of IR ultrashort laser pulses by PTR glass volume Bragg gratings

Optics Express, Vol. 17, Issue 5, pp. 3564-3573 (2009)

http://dx.doi.org/10.1364/OE.17.003564

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### Abstract

Three conditions for non-collinear third harmonic generation by a PTR glass volume Bragg grating are demonstrated using infrared ultrashort pulse illumination. Each condition corresponds to a different angle of grating orientation and a separate generation mechanism. We identify the mechanisms as corresponding to sum-frequency generation, Bragg diffraction of 3ω, and a non-resonant Bragg condition involving three ω photons interacting with a nonlinear grating vector. Theoretical modeling is performed using wave vector additions and the results are compared to experimental measurements.

© 2009 Optical Society of America

## 1. Introduction

### 1.1 PTR Glass

^{-3}(1000 ppm) and are associated with low losses. This photosensitivity is sufficient for recording high efficiency Bragg gratings in PTR glass samples having thicknesses of a few millimeters [1

1. O. M. Efimov, L.B. Glebov, and V. I. Smirnov, “High-frequency Bragg gratings in a photothermorefractive glass,” Opt. Lett. **25**, 1693–1695 (2000). [CrossRef]

^{2}of CW irradiation by a 1085 nm Yb-doped fiber laser focused to a spot diameter of 300 μm [2

2. L. B. Glebov, V. I. Smirnov, C. M. Stickley, and I. V. Ciapurin, “New approach to robust optics for HEL systems,” Proc. SPIE **4724**, 101–109 (2002). [CrossRef]

^{2}, but is locally decreased to 7 J/cm

^{2}when Pt inclusions are present [3]. The high laser damage threshold of PTR glass volume Bragg gratings make them ideal for use in high power laser systems.

### 1.2 THG by a volume Bragg grating in PTR glass

5. S. Juodkazis, E. Gaizauskas, V. Jarutis, J. Reif, S. Matsuo, and H. Misawa, “Optical third harmonic generation during femtosecond pulse diffraction in a Bragg grating,” J. Phys. D: Appl. Phys. **39**, 50 (2006). [CrossRef]

## 2. Experimental observations

*θ*of the TBG was set to Bragg angle for 780 nm and calculated according to Bragg’s law

*n*(

*λ*) is the refractive index of PTR glass as a function of wavelength and ∧ is the spatial period of the grating. Figure 2 shows that after propagation through the TBG, two THG beams, 3ω

^{(i)}and 3ω

^{(ii)}, appeared between the diffracted, ω

_{D}, and transmitted, ω

_{T}, beams. We call this configuration two-beam THG and distinguish the beams by labeling THG closest to the transmitted beam as 3ω

^{(i)}and THG closest to the diffracted beam as 3ω

^{(ii)}.

^{12}W/cm

^{2}. A TBG in PTR glass with 4 μm spatial period, 0.97 mm thickness, and amplitude of refractive index modulation of 607 ppm was placed near the focal plane. The angles of the TBG at which non-collinear THG was generated were measured and given in Table 1. At the wavelengths 1300 nm and 1588 nm, it was again observed that for the TBG oriented at Bragg angle for fundamental, two THG beams appeared between the transmitted and diffracted beams. We will designate the Bragg angle for fundamental as

*θ*

_{1}. However, in addition to THG at

*θ*

_{1}two other angles also resulted in non-collinear generation of third harmonic. These two interactions are illustrated in Figs. 3(a)&(b) along with the assumed wave vector conditions responsible for their generation. At angle

*θ*

_{2}, THG is attributed to Bragg diffraction for incident light at wavelength

*λ*/3. This interaction is likely due to the generation of third harmonic at the front interface of the glass grating and then subsequent diffraction of this surface generated third harmonic. This phenomenon could not be seen with fundamental pulses at 780 nm because of absorption of 266 nm light in the bulk of PTR glass after generation by the front surface.

*θ*

_{2}as surface diffracted THG. The appearance of THG at angle

*θ*

_{3}represents a non-Bragg resonance condition where three fundamental photons interact with a grating vector to generate the third harmonic. We label the THG process at angle

*θ*

_{3}as generation and diffraction by a nonlinear grating. In the next section we impose phase-matching conditions on the three assumed wave vector interactions and derive theoretical values for the angles at which THG is expected. A comparison of these theoretical values is then done with the experimentally measured values of Table 1.

## 3. Phase-matching conditions

**k**

_{T}(

*λ*,

*θ*) and the diffracted wave vector

**k**

_{D}(

*λ*,

*θ*) are given by

**k**(

*λ*,

*θ*) and grating vector

**K**are

*λ*is expressed in microns and the values of A, B, C, D, E and F are given in Table 2.

*θ*

_{2}and

*θ*

_{3}we write

**K**

_{NL}because the three photon interaction is a

*χ*

_{3}process that interacts with the nonlinear refractive index

*n*

_{2}and not the linear refractive index for which the grating vector

**K**is defined for. Nevertheless we evaluate both

**K**and

**K**

_{NL}using Eq. (7). This implicitly assumes grating modulation of the nonlinear index follows the modulation in the linear refractive index. The angle

*θ*in each of the above wave vector equations is solved for by imposing the phase-matching condition given by Eq. (9). The grating we label the angles that satisfy Eqs. (10) & (11) as

*θ*

_{2}and

*θ*

_{3}respectively. The resulting solutions are for angles inside a medium of refractive index

*n*and are converted to angles in air by Snell’s law

*θ*

_{2}and

*θ*

_{3}agree with the experimentally measured values. Also, the theory is able to account for the large change in angle

*θ*

_{3}as the wavelength changed from 1300 nm to 1588 nm. Hence we have justified the assumed wave vector equations given by Eqs. (10) & (11). The theoretical angle for

*θ*

_{1}is derived from Eq. (1), but the argument for a SFG interaction that causes two-beam THG will be made in the next section that discusses the angular selectivity of each of the THG beams.

## 4. Angular selectivity of two-beam THG

*θ*

_{1}. Let us see if we can support an SFG interaction by measuring the angular selectivity of the two THG beams. Angular detuning from Bragg condition affects the relative intensities of transmitted and diffracted beams. Therefore, if THG is a result of interaction between transmitted and diffracted photons, the intensities of the THG beams will be affected differently. A Ti:sapphire regenerative amplifier laser system operating at 780 nm, ~120 fsec, 1 kHz, and pulse energies up to 1 mJ was used with a TBG with the following parameters: ∧ = 4 μm, L = 0.85 mm,

*n*

_{1}= 467 ppm. A computer controlled rotation stage controlled the angle of the TBG while an amplified GaP photodetector measured the intensity of THG. Due to the bandwidth sensitivity of the detector, no light was detected from transmitted or scattered fundamental radiation, and only radiation from THG was detected. Figure 4 shows the experimentally obtained angular selectivity profiles for the two THG beams. It is evident that the 3ω

^{(i)}and 3ω

^{(ii)}beams show different angular dependencies.

*κ*is a constant,

*I*

_{ωT}is the intensity of the transmitted beam and

*I*

_{ωD}is the intensity of the diffracted beam. Assuming that the spectral selectivity of the TBG is larger than the bandwidth of the laser it is possible to neglect the integration between the spectral profile of the beam and the diffraction efficiency of the TBG. When the grating selectivity is greater than the laser spectral bandwidth the intensity of the diffracted and transmitted beams can be written as

*I*

_{0}is the incident intensity and

*η*(

*θ*) is the diffraction efficiency of the TBG as a function of incident angle. The behavior of volume Bragg gratings is well modeled using Kogelnik’s coupled wave theory [6]. The diffraction efficiency for a TBG at resonant frequency is

7. I. V. Ciapurin, L. B. Glebov, and V. I. Smirnov, “Modeling of Gaussian beam diffraction on volume Bragg gratings in PTR glass,” Proc. SPIE **5742**, 183–194 (2005). [CrossRef]

^{(i)}and 3ω

^{(ii)}that account for the main intensity fluctuations seen in the experimental measurements. Lobe maxima and minima are in agreement for both experiment and theory. The model however does not predict the asymmetry seen in the experimental measurements. This asymmetry can be a consequence of the asymmetry of the fundamental pulse spectrum, shown in Fig. 6, which was used for performing these THG experiments.

## 5. Nonlinear refractive index grating

*θ*

_{3}and generates THG requires three fundamental photons to interact with a grating vector. This is a

*χ*

_{3}process and no interaction can occur with the linear grating vector of the TBG recorded inside PTR glass. Therefore the interaction occurs between the incident wave vectors and a grating vector arising from the nonlinear contribution of

*χ*

_{3}. This is possible if we assume that modulation in

*χ*

_{3}occurs concurrently with modulation in the linear refractive index. The nonlinear susceptibility can then be written as

**K**

_{NL}. Using a Green’s formulation [8

8. J. X. Cheng and X. S. Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B **19**, 1604–1610 (2007). [CrossRef]

*θ*

_{3}with three incident fundamental photons we can write the electric fields as

**K**

_{NL}with

**K**. In this case Eq. (25) is equivalent to Eq. (11). Thus the THG condition given by Eq. (11) is justified by assuming a nonlinear grating arising from modulation in

*χ*

_{3}in PTR glass. The Green’s formulation was applied to explain the nonlinear THG but is not limited to this case. It is also possible to formulate the SFG interaction and derive Eqs. (2) & (3). We write the electric fields in the case of SFG involving two transmitted photons and one diffracted photon as

*unphase*-matched THG leads to low conversion efficiency. The efficiency of THG for the SFG interaction was estimated using the responsivity of GaP photodetectors to be on the order of 10

^{-4}. A larger conversion efficiency would be expected for volume gratings in phosphate glasses rather than alkali-silicate glasses such as PTR glass because of higher third-order susceptibility

*χ*

_{3}

^{(0)}in those glasses.

## 6. Conclusion

## Acknowledgments

## References and links

1. | O. M. Efimov, L.B. Glebov, and V. I. Smirnov, “High-frequency Bragg gratings in a photothermorefractive glass,” Opt. Lett. |

2. | L. B. Glebov, V. I. Smirnov, C. M. Stickley, and I. V. Ciapurin, “New approach to robust optics for HEL systems,” Proc. SPIE |

3. | L. B. Glebov, L. N. Glebova, V. I. Smirnov, M. Dubinskii, L. D. Merkle, S. Papernov, and A.W. Schmid, “Laser Damage Resistance of Photo-Thermo-Refractive Glass Bragg Gratings,” Proceedings of Solid State and Diode Lasers Technical Review. Albuquerque, Poster-4 (2004). |

4. | V. I. Smirnov, S. Juodkazis, V. Dubikovsky, J. Hales, B. Ya. Zel’dovich, H. Misawa, and L. B. Glebov, “Resonant third harmonic generation by femtosecond laser pulses on Bragg grating in photosensitive silicate glass,” Conference on Lasers and Electro-Optics CLEO-2002 paper CTuG7 (2002). |

5. | S. Juodkazis, E. Gaizauskas, V. Jarutis, J. Reif, S. Matsuo, and H. Misawa, “Optical third harmonic generation during femtosecond pulse diffraction in a Bragg grating,” J. Phys. D: Appl. Phys. |

6. | H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell. Syst. Tech. J. |

7. | I. V. Ciapurin, L. B. Glebov, and V. I. Smirnov, “Modeling of Gaussian beam diffraction on volume Bragg gratings in PTR glass,” Proc. SPIE |

8. | J. X. Cheng and X. S. Xie, “Green’s function formulation for third-harmonic generation microscopy,” J. Opt. Soc. Am. B |

**OCIS Codes**

(050.7330) Diffraction and gratings : Volume gratings

(190.4720) Nonlinear optics : Optical nonlinearities of condensed matter

(320.2250) Ultrafast optics : Femtosecond phenomena

(160.5335) Materials : Photosensitive materials

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: December 16, 2008

Revised Manuscript: February 17, 2009

Manuscript Accepted: February 18, 2009

Published: February 23, 2009

**Citation**

Leo A. Siiman, Julien Lumeau, Lionel Canioni, and Leonid B. Glebov, "Non-collinear generation of third harmonic of IR ultrashort laser pulses by PTR glass volume Bragg gratings," Opt. Express **17**, 3564-3573 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-5-3564

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### References

- O. M. Efimov, L.B. Glebov, and V. I. Smirnov, "High-frequency Bragg gratings in a photothermorefractive glass," Opt. Lett. 25, 1693-1695 (2000). [CrossRef]
- L. B. Glebov, V. I. Smirnov, C. M. Stickley, and I. V. Ciapurin, "New approach to robust optics for HEL systems," Proc. SPIE 4724, 101-109 (2002). [CrossRef]
- L. B. Glebov, L. N. Glebova, V. I. Smirnov, M. Dubinskii, L. D. Merkle, S. Papernov, and A.W. Schmid, "Laser Damage Resistance of Photo-Thermo-Refractive Glass Bragg Gratings," Proceedings of Solid State and Diode Lasers Technical Review. Albuquerque, Poster-4 (2004).
- V. I. Smirnov, S. Juodkazis, V. Dubikovsky, J. Hales, B. Ya. Zel’dovich, H. Misawa, and L. B. Glebov, "Resonant third harmonic generation by femtosecond laser pulses on Bragg grating in photosensitive silicate glass," Conference on Lasers and Electro-Optics CLEO-2002 paper CTuG7 (2002).
- S. Juodkazis, E. Gaizauskas, V. Jarutis, J. Reif, S. Matsuo and H. Misawa, "Optical third harmonic generation during femtosecond pulse diffraction in a Bragg grating," J. Phys. D: Appl. Phys. 39,50 (2006). [CrossRef]
- H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell. Syst. Tech. J. 48, 2909 (1969).
- I. V. Ciapurin, L. B. Glebov, and V. I. Smirnov, "Modeling of Gaussian beam diffraction on volume Bragg gratings in PTR glass," Proc. SPIE 5742, 183-194 (2005). [CrossRef]
- J. X. Cheng and X. S. Xie, "Green’s function formulation for third-harmonic generation microscopy," J. Opt. Soc. Am. B 19, 1604-1610 (2007). [CrossRef]

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