## Characterization of laser-induced air plasmas by third harmonic generation |

Optics Express, Vol. 19, Issue 17, pp. 16115-16125 (2011)

http://dx.doi.org/10.1364/OE.19.016115

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### Abstract

Third harmonic generation by a weak femtosecond probe pulse intersecting a pump laser-induced plasma in air is investigated and a general model is developed to describe such signal, applicable to a wide range of focusing and plasma conditions. The effect of the surrounding air on the generated signal is discussed. The third-order nonlinear susceptibility of an air plasma with electron density *N _{e}
* is determined to be

*γ*= 2 ± 1 × 10

_{p}^{−49}m

^{5}V

^{−2}and

© 2011 OSA

## 1. Introduction

1. Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. **70**, 922–924 (1997). [CrossRef]

*χ*

^{(3)}) optical properties of materials, including biological species. Since all materials have a non-vanishing

*χ*

^{(3)}, this technique can be applied to study virtually any material, as long as it is transparent at the wavelengths involved.

2. A. N. Naumov, D. A. Sidorov-Biryukov, A. B. Fedotov, and A. M. Zheltikov, “Third-harmonic generation in focused beams as a method of 3D microscopy of a laser-produced plasma,” Opt. Spectrosc. , **90**, 778–783 (2001). [CrossRef]

3. S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Filamentation-induced third-harmonic generation in air via plasma-enhanced third-order susceptibility,” Phys. Rev. A **81**, 033817 (2010). [CrossRef]

4. K. Hartinger and R. A. Bartels, “Enhancement of third harmonic generation by a laser-induced plasma,” Appl. Phys. Lett. **93**, 151102 (2008). [CrossRef]

5. S. Backus, J. Peatross, Z. Zeek, A. Rundquist, G. Taft, M. M. Murnane, and H. C. Kapteyn, “16-fs, 1-microJ ultraviolet pulses generated by third-harmonic conversion in air,” Opt. Lett. **21**, 665–667 (1996). [CrossRef] [PubMed]

*χ*

^{(3)}[3

3. S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Filamentation-induced third-harmonic generation in air via plasma-enhanced third-order susceptibility,” Phys. Rev. A **81**, 033817 (2010). [CrossRef]

*N*). Unlike in previous experiments, the intensity of the probe pulse was chosen small enough such that filamentation was avoided and the TH signal was proportional to the probe intensity raised to the third power. This setup allowed us to study the properties of the pump-induced plasma and THG without interference by probe-induced filamentation and associated nonlinearities. We measure the functional dependence of the TH signal with respect to the electron density, estimate

_{e}*χ*

^{(3)}of such a medium, and propose a method by which the electron density distribution of a laser induced plasma can be measured with temporal resolution. The conditions under which THG can be used for 3D imaging of plasmas are evaluated and the limitations discussed.

## 2. Theoretical background

*a*) vanishes due to destructive interference of the TH waves generated before and after the focus, provided the material exhibits normal dispersion, see for example [6] where

*n*

_{3ω,a}is the refractive index at the TH frequency,

*E*is the fundamental Gaussian field amplitude, Δ

_{ω}*k*= 3

_{a}*k*

_{ω,a}–

*k*

_{3ω,a}is the wave-vector mismatch between the fundamental and TH waves, and

*w*

_{0,a}corresponds to its beam waist.

7. J. F. Ward and G. H. C. New, “Optical third harmonic generation in gases by a focused laser beam,” Phys. Rev. **185**, 57–72 (1969). [CrossRef]

*E*

_{3ω}(

*z*) is plotted in the complex plane, with

*z*as the running parameter. For a particular

*z*position the resultant TH field is represented by the vector from the origin to the corresponding point on the trace. As shown in Fig. 1(a) the TH field created in the negative half-space −∞ <

*z*< 0 is canceled by the TH field created in the positive half-space 0 <

*z*< +∞. As a result the integration over the infinite medium yields no THG.

*D*is introduced with different linear and/or nonlinear optical parameters, see Fig. 2, the TH field

*E*

_{3}

*will in general no longer vanish. The corresponding integral for the TH field can be split into three components - before the slab, the slab and after the slab. After adding the three contributions, taking into consideration that at the interface of two nonlinear media the phase mismatch between the fundamental and harmonic waves has to be continuous, the TH field takes on the form*

_{ω}*p*and

*a*refer to the slab material and medium

*a*, respectively. The coefficients

*t*

^{ω,3ω}are the transmission factors for the fundamental and harmonic waves. We also allowed for an arbitrary position

*z*of the beam waist by introducing

_{w}*Z*=

*z*–

*z*. In Eq. (2) we assume a homogeneous distribution of both Δ

_{w}*k*and

_{p}*a*and in the slab.

*p*) region in air (material

*a*) centered around the focus of the probe. According to the Drude model the refractive index of the plasma at frequency

*ω*can be written as [8]: where

*n*

_{ω,a}is the refractive index of air at frequency

*ω*,

*N*is the electron density, and

_{e}*N*=

_{c}*ɛ*

_{0}

*m*

_{e}ω^{2}/

*e*

^{2}is the critical electron density where laser and plasma frequency are equal,

*e*and

*m*are the electron charge and mass, respectively. For

_{e}*λ*= 800 nm, the critical electron density is

*N*≈ 1.8 × 10

_{c}^{27}m

^{−3}.

*k*= 3

_{a}*ω*(

*n*

_{ω,a}−

*n*

_{3ω,a})/

*c*is the wave-vector mismatch in air and

*q*= −4

*ω*/(3

*cN*).

_{c}7. J. F. Ward and G. H. C. New, “Optical third harmonic generation in gases by a focused laser beam,” Phys. Rev. **185**, 57–72 (1969). [CrossRef]

*ansatz*: where

*γ*is a constant to be determined, representing the effective second order hyperpolarizability of a single electron in the plasma. This

_{p}*ansatz*is consistent with the notion that the third-order susceptibility of the plasma

10. A. B. Fedotov, S. M. Gladkov, N. I. Koroteev, and A. M. Zheltikov, “Highly efficient frequency tripling of laser radiation in a low-temperature laser-produced gaseous plasma,” J. Opt. Soc. Am. B **8**, 363–366 (1991). [CrossRef]

*w*

_{0}≳ 1

*μ*m), which is a consequence of

*E*

_{3}

*from Eq. (2) being the solution to the paraxial wave equation, and radially symmetric electron density distributions. Equation (2) can be simplified assuming*

_{ω}*D*≪

*z*

_{0}to take on the form

*n*

_{3ω,p}≈

*n*

_{3ω,a}=

*n*

_{3ω}in the prefactors,

*z*

_{0,p}≈

*z*

_{0,a}=

*z*

_{0}, and assumed

*t*

^{ω,3ω}≈ 1 and

*z*= 0. The first term in Eq. (6), with

_{w}*N*≪

_{e}*N*) as is typical for weakly ionized plasmas, and sufficiently small plasma thickness

_{c}*D*, the phase mismatch |Δ

*k*| ≪ 1. This allows us to expand the trigonometric functions and the exponential prefactor in Eq. (6), using Eq. (4), to first order in

_{p}D*N*. We obtain Since the TH signal

_{e}*S*

_{3ω}∝ |

*E*

_{3ω}|

^{2}we expect that

*D*= 100

*μ*m) for weak and strong focusing conditions of the probe beam, respectively. The solid blue lines in Figures 3(a) and 3(b) show the exact results using Eq. (2), while the solid red lines neglect the contribution from air to the generated TH signal. For weak focusing conditions, Fig. 3(a), the contribution from air cannot be neglected. The TH generated in air and in the plasma are of the same order of magnitude but opposite signs. For tight focusing conditions, Fig. 3(b), the contribution from air is important for the low electron densities. Only for high electron densities this contribution becomes negligible, since the effect of the “quasi-free” electrons on the signal becomes dominant, cf. Eq. (5). In general, the air contribution can be approximately neglected if

*γ*) and

_{p}N_{e}D*N*> 10

_{e}^{24}m

^{−3}and

*D*= 100

*μ*m, |Δ

*k*| ≳ 1 and the approximations leading to Eq. (7) are not valid. The oscillations of the TH signal are a consequence of the phase mismatch approaching and exceeding

_{p}D*π*.

*z*

_{0}for constant probe energy. If focused tightly, the probe will generate TH in a small volume, representing a small fraction of the plasma, with the limiting case where the TH signal becomes negligible for a bulk sample. For large

*z*

_{0}the small probe intensity prevents efficient THG. Optimal focusing conditions are found for 2

*z*

_{0}≈

*D*, for the case where |Δ

*k*| ≪ 1.

_{p}D*z*,

*N*(

_{e}*z*) =

*N*

_{e}*f*(

*z*). Such an electron density distribution produces a gradient in the plasma non-linearity

*k*(

_{p}*z*), cf. Eq. (4). The total TH field can be calculated using a relation similar to Eq. (2). The integral corresponding to the plasma region is split into smaller regions of thickness Δ

*z*within which

*N*(

_{e}*z*) can be regarded constant. As before, when adding these contributions, the accumulated phase in one region has to be correctly added to the phase in the following region. The TH field takes on the form

*z*=

*D*/(

*M*– 1), with

*m*= 1,2,...,

*M*. Under these conditions it is still possible to show analytically that

*k*in Eq. (8) to first order in

_{p}*N*.

_{e}## 3. Experimental setup

## 4. Results and discussion

*D*≈ 140

*μ*m was produced by 1.5 mJ pulses using a lens L1 with

*f*= 1 m. The probe was focused with an

*f*= 30-cm lens (L2) producing a beam waist of

*w*

_{0}≈ 17

*μ*m at

*z*= 0. For small probe energies (≲ 5

_{w}*μ*J) the TH signal shows the expected cubic dependence. For larger energies the slope is less than three, which was also observed in [13

13. S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Efficient third-harmonic generation through tailored IR femtosecond laser pulse filamentation in air,” Opt. Express **17**, 3190–3195 (2009). [CrossRef] [PubMed]

_{2}and O

_{2}as a function of laser intensity [11

11. J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B **71**, 877–879 (2000). [CrossRef]

12. A. Talebpour, J. Yang, and S. L. Chin, “Semi-empirical model for the rate of tunnel ionization of N2 and O2 molecule in an intense Ti:sapphire laser pulse,” Opt. Commun. **163**, 29–32 (1999). [CrossRef]

*N*≈ 4 × 10

_{e}^{24}m

^{−3}at probe energies of about 20

*μ*J.

*μ*J under the focusing conditions of Fig. 5 and no plasma present, the TH signal was below our detection limit. From experimentally obtained ionization rates of N

_{2}and O

_{2}as a function of laser intensity [11

11. J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B **71**, 877–879 (2000). [CrossRef]

12. A. Talebpour, J. Yang, and S. L. Chin, “Semi-empirical model for the rate of tunnel ionization of N2 and O2 molecule in an intense Ti:sapphire laser pulse,” Opt. Commun. **163**, 29–32 (1999). [CrossRef]

*N*≈ 3 × 10

_{e}^{17}m

^{−3}for such a probe pulse.

*τ*between pump and probe pulses and determined the time-dependent electron density in the plasma

*N*(

_{e}*τ*) by means of a diffraction experiment [14

14. Z. Sun, J. Chen, and W. Rudolph, “Determination of the transient electron temperature in a femtosecond-laser-induced air plasma,” Phys. Rev. E **83**, 046408 (2011). [CrossRef]

*N*(

_{e}*τ*). We used an

*f*= 30-cm focusing lens (L2) for the probe producing a beam waist of ≈ 17

*μ*m at

*z*= 0. The laser amplifier repetition rate was reduced to 250 Hz to avoid thermal effects (see below). Figure 6 shows the so obtained TH signal from the probe as a function of the electron density in the plasma. The solid line illustrates the good agreement between the experiment and the predictions from Eq. (8). The simulations also took into account the change of the electron density spatial profile [14

_{w}14. Z. Sun, J. Chen, and W. Rudolph, “Determination of the transient electron temperature in a femtosecond-laser-induced air plasma,” Phys. Rev. E **83**, 046408 (2011). [CrossRef]

*N*(

_{e}*z*) changes from Gaussian at large

*N*(0) (small delays) to a more flat-top profile for low

_{e}*N*(0) (large delays) because of the bimolecular nature of the electron-ion recombination.

_{e}*γ*= 2 ± 1 × 10

_{p}^{−49}m

^{5}V

^{−2}, where the main sources of error came from the uncertainties on the beam waist and

*M*

^{2}value of the probe, and on

*N*≈ 1.5 × 10

_{e}^{23}m

^{−3}[14

14. Z. Sun, J. Chen, and W. Rudolph, “Determination of the transient electron temperature in a femtosecond-laser-induced air plasma,” Phys. Rev. E **83**, 046408 (2011). [CrossRef]

7. J. F. Ward and G. H. C. New, “Optical third harmonic generation in gases by a focused laser beam,” Phys. Rev. **185**, 57–72 (1969). [CrossRef]

*N*

_{air}is the total number of air molecules ≈ 2 × 10

^{25}m

^{−3}). In reference [3

3. S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Filamentation-induced third-harmonic generation in air via plasma-enhanced third-order susceptibility,” Phys. Rev. A **81**, 033817 (2010). [CrossRef]

^{−24}m

^{2}/V

^{2}was reported for

*N*≈ 2 × 10

_{e}^{25}m

^{−3}, in agreement with what we obtain using our value for

*γ*and Eq. (5).

_{p}*y*direction) across the probe beam, which was realized by moving the focusing lens of the pump beam (L1) along the

*y*direction using a translation stage. This resulted in the filament moving by the same amount Δ

*y*with respect to the probe beam, which is only true in the vicinity of the focal plane of lens L1, in which case aberrations and a change in the time delay between pump and probe beams can be neglected. The results of the scan are shown in Fig. 7. We used an

*f*= 30-cm focusing lens (L2) for the probe producing a beam waist of ≈ 17

*μ*m at

*z*= 0. The solid line in Fig. 7 was obtained from Eq. (8), and shows a distribution with a width of ≈ 100

_{w}*μ*m. For each

*y*position we assumed a constant

*N*over the probe beam with respect to

_{e}*y*and integrated over

*z*. Since

*N*(

_{e}*y*) has a width of ≈ 140

*μ*m.

*μ*m. The laser amplifier repetition rate was set to 1 kHz, since the thermal effects were negligible under these experimental conditions. Figure 8 shows the TH signal while the plasma was scanned laterally (

*y*direction) across the probe beam for several pump pulse energies (left column), and simulation results (right column).

*w*

_{0}≈ 50

*μ*m, a value larger than the predicted waist generated by a 10-cm-focal-length lens. For the high intensities used, however, the generation of a plasma and subsequent defocusing prevents the pump from reaching such small waists. The value assumed for the simulations was chosen based on the best agreement found between the simulations and the experimental results. It should be noted that the pump intensities involved lead to tunneling ionization and are higher than intensities for which rates have been determined experimentally [12

12. A. Talebpour, J. Yang, and S. L. Chin, “Semi-empirical model for the rate of tunnel ionization of N2 and O2 molecule in an intense Ti:sapphire laser pulse,” Opt. Commun. **163**, 29–32 (1999). [CrossRef]

*N*

_{e}*f*(

*y, z*) were then used in Eq. (8) to obtain the TH signal, which is shown as a solid line. We find good overall qualitative agreement between the model predictions and the experimental data. For the higher pump energies

*N*(

_{e}*y, z*) broadens and its peak value increases. The model predicts local minima of the TH produced in the plasma center even though this is where

*N*(

_{e}*y, z*) peaks, which is a consequence of the phase mismatch exceeding

*π*, cf. Fig. 3(a). Therefore, structure found in the TH signal of high-density plasmas does not necessarily point to similar structures in the electron density itself.

*z*one would need focusing conditions of the probe for which the confocal parameter 2

*z*

_{0}≲ Δ

*z*. At the same time, to ensure

*k*Δ

_{p}*z*≪ 1 is required. On the other hand, if 2

*z*

_{0}≪ Δ

*z*the TH field would vanish due to destructive interference of the TH waves generated before and after the focus, adding another source of ambiguity in interpreting the structure found on the TH signal. As an example, boundaries of relatively uniform regions of thickness

*D*≫ 2

*z*

_{0}and individual structures with dimensions ≪ 2

*z*

_{0}separated by

*D*could give rise to similar TH signals. In certain cases this ambiguity could be addressed by performing series of scans with different focusing conditions and scan directions. The detection limit of the experiment will dictate how strong the probe beam can be focused while keeping the intensity small enough to ensure a cubic power law for the TH conversion and to avoid significant contributions to the TH signal in the absence of the pump. As an example, for an air plasma with an electron density

*N*= 10

_{e}^{25}m

^{−3}, the upper limit on the dimension

*D*which can be probed without ambiguities using a single lateral scan is

*D*≈ 1/Δ

*k*≈ 16

_{p}*μ*m.

## 5. Summary

*γ*= 2 ± 1 × 10

_{p}^{−49}m

^{5}V

^{−2}was determined for the third-order susceptibility of the plasma. Under certain conditions (cylindrically symmetric plasma, phase mismatch ≪ 1, and a confocal parameter of the probe approximately equal to the size of the plasma), lateral scans of the probe beam through the plasma can be used to determine 2D plasma density profiles with sub picosecond time resolution. In the general case, phase mismatch critically affects the scan profiles and must be taken into account to derive electron density profiles. 3D mapping of plasma densities using THG seems possible when additional information is available, such as the peak densities and approximate dimension of the structures. The latter could be obtained via modeling or confocal fluorescence imaging combined with up-conversion if temporal resolution is needed.

## Acknowledgments

## References and links

1. | Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. |

2. | A. N. Naumov, D. A. Sidorov-Biryukov, A. B. Fedotov, and A. M. Zheltikov, “Third-harmonic generation in focused beams as a method of 3D microscopy of a laser-produced plasma,” Opt. Spectrosc. , |

3. | S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Filamentation-induced third-harmonic generation in air via plasma-enhanced third-order susceptibility,” Phys. Rev. A |

4. | K. Hartinger and R. A. Bartels, “Enhancement of third harmonic generation by a laser-induced plasma,” Appl. Phys. Lett. |

5. | S. Backus, J. Peatross, Z. Zeek, A. Rundquist, G. Taft, M. M. Murnane, and H. C. Kapteyn, “16-fs, 1-microJ ultraviolet pulses generated by third-harmonic conversion in air,” Opt. Lett. |

6. | R. W. Boyd, |

7. | J. F. Ward and G. H. C. New, “Optical third harmonic generation in gases by a focused laser beam,” Phys. Rev. |

8. | A. A. Fridman and L. A. Kennedy, |

9. | P. E. Ciddor, “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt. |

10. | A. B. Fedotov, S. M. Gladkov, N. I. Koroteev, and A. M. Zheltikov, “Highly efficient frequency tripling of laser radiation in a low-temperature laser-produced gaseous plasma,” J. Opt. Soc. Am. B |

11. | J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B |

12. | A. Talebpour, J. Yang, and S. L. Chin, “Semi-empirical model for the rate of tunnel ionization of N2 and O2 molecule in an intense Ti:sapphire laser pulse,” Opt. Commun. |

13. | S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Efficient third-harmonic generation through tailored IR femtosecond laser pulse filamentation in air,” Opt. Express |

14. | Z. Sun, J. Chen, and W. Rudolph, “Determination of the transient electron temperature in a femtosecond-laser-induced air plasma,” Phys. Rev. E |

15. | L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys. JETP |

**OCIS Codes**

(320.2250) Ultrafast optics : Femtosecond phenomena

(180.4315) Microscopy : Nonlinear microscopy

(280.5395) Remote sensing and sensors : Plasma diagnostics

**ToC Category:**

Microscopy

**History**

Original Manuscript: May 26, 2011

Revised Manuscript: July 19, 2011

Manuscript Accepted: July 24, 2011

Published: August 8, 2011

**Citation**

Cristina Rodríguez, Zhanliang Sun, Zhenwei Wang, and Wolfgang Rudolph, "Characterization of laser-induced air plasmas by third harmonic generation," Opt. Express **19**, 16115-16125 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-16115

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### References

- Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997). [CrossRef]
- A. N. Naumov, D. A. Sidorov-Biryukov, A. B. Fedotov, and A. M. Zheltikov, “Third-harmonic generation in focused beams as a method of 3D microscopy of a laser-produced plasma,” Opt. Spectrosc. , 90, 778–783 (2001). [CrossRef]
- S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Filamentation-induced third-harmonic generation in air via plasma-enhanced third-order susceptibility,” Phys. Rev. A 81, 033817 (2010). [CrossRef]
- K. Hartinger and R. A. Bartels, “Enhancement of third harmonic generation by a laser-induced plasma,” Appl. Phys. Lett. 93, 151102 (2008). [CrossRef]
- S. Backus, J. Peatross, Z. Zeek, A. Rundquist, G. Taft, M. M. Murnane, and H. C. Kapteyn, “16-fs, 1-microJ ultraviolet pulses generated by third-harmonic conversion in air,” Opt. Lett. 21, 665–667 (1996). [CrossRef] [PubMed]
- R. W. Boyd, Nonlinear Optics (Academic Press, 2008).
- J. F. Ward and G. H. C. New, “Optical third harmonic generation in gases by a focused laser beam,” Phys. Rev. 185, 57–72 (1969). [CrossRef]
- A. A. Fridman and L. A. Kennedy, Plasma Physics and Engineering (Taylor & Francis, 2004).
- P. E. Ciddor, “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt. 35, 1566–1573 (1996). [CrossRef] [PubMed]
- A. B. Fedotov, S. M. Gladkov, N. I. Koroteev, and A. M. Zheltikov, “Highly efficient frequency tripling of laser radiation in a low-temperature laser-produced gaseous plasma,” J. Opt. Soc. Am. B 8, 363–366 (1991). [CrossRef]
- J. Kasparian, R. Sauerbrey, and S. L. Chin, “The critical laser intensity of self-guided light filaments in air,” Appl. Phys. B 71, 877–879 (2000). [CrossRef]
- A. Talebpour, J. Yang, and S. L. Chin, “Semi-empirical model for the rate of tunnel ionization of N2 and O2 molecule in an intense Ti:sapphire laser pulse,” Opt. Commun. 163, 29–32 (1999). [CrossRef]
- S. Suntsov, D. Abdollahpour, D. G. Papazoglou, and S. Tzortzakis, “Efficient third-harmonic generation through tailored IR femtosecond laser pulse filamentation in air,” Opt. Express 17, 3190–3195 (2009). [CrossRef] [PubMed]
- Z. Sun, J. Chen, and W. Rudolph, “Determination of the transient electron temperature in a femtosecond-laser-induced air plasma,” Phys. Rev. E 83, 046408 (2011). [CrossRef]
- L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys. JETP 20, 1307–1314 (1965).

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