## Interference effects at a dielectric plate applied as a high-power-laser attenuator

Optics Express, Vol. 18, Issue 4, pp. 3871-3882 (2010)

http://dx.doi.org/10.1364/OE.18.003871

Acrobat PDF (508 KB)

### Abstract

The interference effects caused by the Fresnel reflections of a Gaussian beam on the boundaries of a dielectric plate, which can be considered as a Fabry-Perot etalon, were theoretically and experimentally investigated. In addition to the incident angle and the polarization of the incident light, two additional parameters—the plate’s parallelism and the temperature—which are often neglected, were analyzed. Based on the theoretical predictions and the measured behavior of the transmittance of the dielectric plate a new, temperature-controlled variable high-power-laser attenuator is proposed. Unwanted changes in the plate’s transmittance caused by the absorption of laser pulses within the plate are also presented. These phenomena are important in many applications where dielectric plates are used for a variety of purposes.

© 2010 OSA

## 1. Introduction

1. R. O. Rice and J. D. Macomber, “Attenuation of giant laser pulses by absorbing filters,” Appl. Opt. **14**(9), 2203–2206 (1975). [CrossRef] [PubMed]

2. R. M. A. Azzam, “Tilted parallel dielectric slab as a multilevel attenuator for incident p- or s-polarized light,” Appl. Opt. **48**(2), 425–428 (2009). [CrossRef] [PubMed]

3. Y. H. Wu, Y. H. Lin, Y. Q. Lu, H. W. Ren, Y. H. Fan, J. R. Wu, and S. T. Wu, “Submillisecond response variable optical attenuator based on sheared polymer network liquid crystal,” Opt. Express **12**(25), 6382–6389 (2004). [CrossRef] [PubMed]

4. H. Lotem, A. Eyal, and A. R. Shuker, “Variable attenuator for intense unpolarized laser beams,” Opt. Lett. **16**(9), 690–692 (1991). [CrossRef] [PubMed]

5. D. Gauden, D. Mechin, C. Vaudry, P. Yvernault, and D. Pureur, “Variable optical attenuator based on thermally tuned Mach-Zehnder interferometer within a twin core fiber,” Opt. Commun. **231**(1-6), 213–216 (2004). [CrossRef]

6. K. Bennett and R. L. Byer, “Computer-controllable wedged-plate optical-variable attenuator,” Appl. Opt. **19**(14), 2408–2412 (1980). [CrossRef] [PubMed]

7. J. Staromlynska, R. A. Clay, and K. F. Dexter, “Variable optical attenuator for use in the visible spectrum,” Appl. Opt. **26**(18), 3827–3830 (1987). [CrossRef] [PubMed]

1. R. O. Rice and J. D. Macomber, “Attenuation of giant laser pulses by absorbing filters,” Appl. Opt. **14**(9), 2203–2206 (1975). [CrossRef] [PubMed]

2. R. M. A. Azzam, “Tilted parallel dielectric slab as a multilevel attenuator for incident p- or s-polarized light,” Appl. Opt. **48**(2), 425–428 (2009). [CrossRef] [PubMed]

4. H. Lotem, A. Eyal, and A. R. Shuker, “Variable attenuator for intense unpolarized laser beams,” Opt. Lett. **16**(9), 690–692 (1991). [CrossRef] [PubMed]

10. H. B. Yu, G. Y. Zhou, C. F. Siong, and L. Feiwen, “A variable optical attenuator based on optofluidic technology,” J. Micromech. Microeng. **18**(11), 115016 (2008). [CrossRef]

1. R. O. Rice and J. D. Macomber, “Attenuation of giant laser pulses by absorbing filters,” Appl. Opt. **14**(9), 2203–2206 (1975). [CrossRef] [PubMed]

6. K. Bennett and R. L. Byer, “Computer-controllable wedged-plate optical-variable attenuator,” Appl. Opt. **19**(14), 2408–2412 (1980). [CrossRef] [PubMed]

8. J. H. Lehman, D. Livigni, X. Y. Li, C. L. Cromer, and M. L. Dowell, “Reflective attenuator for high-energy laser measurements,” Appl. Opt. **47**(18), 3360–3363 (2008). [CrossRef] [PubMed]

**14**(9), 2203–2206 (1975). [CrossRef] [PubMed]

6. K. Bennett and R. L. Byer, “Computer-controllable wedged-plate optical-variable attenuator,” Appl. Opt. **19**(14), 2408–2412 (1980). [CrossRef] [PubMed]

2. R. M. A. Azzam, “Tilted parallel dielectric slab as a multilevel attenuator for incident p- or s-polarized light,” Appl. Opt. **48**(2), 425–428 (2009). [CrossRef] [PubMed]

11. J. C. Cotteverte, F. Bretenaker, and A. Lefloch, “Jones matrices of a tilted plate for Gaussian beams,” Appl. Opt. **30**(3), 305–311 (1991). [CrossRef] [PubMed]

**48**(2), 425–428 (2009). [CrossRef] [PubMed]

11. J. C. Cotteverte, F. Bretenaker, and A. Lefloch, “Jones matrices of a tilted plate for Gaussian beams,” Appl. Opt. **30**(3), 305–311 (1991). [CrossRef] [PubMed]

## 2. Theory

*r*and transmission

*t*for a single dielectric boundary depend on the angle of incidence

*θ*, the polarization of the light and the ratio

_{i}*n*between the refractive indices of the incident and the transmitting media. For an uncoated plate and an arbitrary polarization they can be conveniently described using the Jones matrices:

*s*and

*p*stand for the linear polarizations perpendicular (s-polarization) and parallel (p-polarization) to the plane of incidence, respectively. The electric field transmission (

*t*) and reflectance (

_{s,p}*r*) are given by the Fresnel laws.

_{s,p}*E*and

_{s}*E*are the electric field components for both polarizations, the Jones vector

_{p}**E**for the polarized light and the reflected intensity

*I*are defined as:where

_{r}*R*(

*θ*) for an arbitrary polarization is defined as the ratio of the reflected and the incident intensities and can be written as:where

_{i}*I*stands for the incident intensity, and the angle of refraction

*θ*can be calculated using Snell’s law. The intensities for both polarizations are denoted by

_{t}*I*and

_{s}*I*, respectively. For unpolarized light, the ratios

_{p}*I*and

_{s}/I*I*are equal to 1/2.

_{p}/I*n*and thickness

*d*can be considered as a Fabry-Perot etalon (FPE) having two reflecting surfaces that have the reflectivity

*R*(

*θ*) and are separated by a distance

_{i}*d*. When a monochromatic laser light is incident upon the dielectric plate, multiple reflections take place inside the plate, leading to successive reflected and transmitted beams. If the laser beam can be described as the fundamental mode of a coherent Gaussian beam (TEM

_{00}mode) and the absorption within the plate is neglected, the power transmittance of the interference attenuator can be calculated using the Jones matrices [11

11. J. C. Cotteverte, F. Bretenaker, and A. Lefloch, “Jones matrices of a tilted plate for Gaussian beams,” Appl. Opt. **30**(3), 305–311 (1991). [CrossRef] [PubMed]

12. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. **5**(10), 1550–1567 (1966). [CrossRef] [PubMed]

**E**stands for the amplitude of the electric field at the center of the beam waist,

_{0}*k*is the wave number in the

*z*-direction,

*λ*is the wavelength in the vacuum,

*w*is the beam-waist radius,

_{0}*z*is the Rayleigh length, while

_{0}*w*(

*z*),

*ρ*(

*z*) and

*η*(

*z*) are: the beam half-width; the radius of the wave front curvature; and a phase-shift difference between the Gaussian beam and the ideal plane wave at the distance

*z*from the waist, respectively. The beam half-width, called also a spot size, corresponds to the distance from the beam’s center, where the field’s amplitude decays to 1/

*e*of its maximum.

*θ*with the surface normal. From the geometry presented in Fig. 1(a) it can be seen that the transverse displacement

_{i}*Δx*between the two adjacent, reflected beams appears in addition to the path difference between the successive transmitted beams. The electric field of the

*m*-th transmitted beam can be given by:

*w*,

*ρ*and

*η*) depend on the traveled distance

*z*, which is a function of the distance

_{m}*D*between the laser-beam source and the measured surface, the thickness of the plate

*d*, its refractive index

*n*and the angle of the incidence

*θ*. Here, the propagation through the dielectric medium with the refractive index

_{i}*n*[13

13. S. Nemoto, “Waist shift of a Gaussian-beam by a dielectric plate,” Appl. Opt. **28**(9), 1643–1647 (1989). [CrossRef] [PubMed]

*δ*and

_{0}*δ*, respectively. The vector

**A**contains the polarization as well as the reflectance and transmittance for the

*m*-th transmitted field.

*r*, while the

_{s,p}*δ*in Eq. (3) stands for the phase change. For the purpose of our calculations, only the internal reflections need to be taken into consideration. The phase change

_{r}*δ*is defined as [14

_{r}14. E. Hecht, *Optics* (2nd Edition, Addison Wesley, 1987), pp. 100. [PubMed]

*θ*stands for the Brewster angle. The expressions for the angles above the critical angle are omitted from Eq. (4), since the angle of refraction

_{B}*θ*never exceeds the critical angle. According to Eq. (4), the term 2

_{t}*δ*in Eq. (3) is either 0 or 2π and therefore does not change the electric field of the

_{r}*m*-th transmitted beam.

**E**

*and the corresponding intensity*

_{t}*I*at any point can be expressed as the sum of the successive output beams:

_{t}*w*,

*ρ*and

*η*) are approximately constant and the transmitted power

*P*can be calculated from the transmitted intensity as:

_{t}*R*(

*θ*

_{i}), defined by Eq. (1), is a function of the incident angle and depends on the polarization of the incident light. Using the relations defined by Eq. (2), the term in the curly brackets in Eq. (7) reduces to

15. H. Abu-Safia, R. Al-Tahtamouni, I. Abu-Aljarayesh, and N. A. Yusuf, “Transmission of a Gaussian-beam through a Fabry-Perot interferometer,” Appl. Opt. **33**(18), 3805–3811 (1994). [CrossRef] [PubMed]

*δ*between each succeeding reflection is given in Eq. (3).

*m = l*or

*Δx*= 0, i.e., for normal incidence. In all other cases it is smaller than 1 and therefore reduces the amplitude of the fringes corresponding to the plane wave. This phenomenon can be explained by the overlapping between the reflected beams. Since the Gaussian beam is limited, the overlapping between neighboring beams (i.e.,

*m ≠ l*) at large angles is reduced. From the term in the curly brackets in Eq. (8) it can be concluded that the limited beam effects depend on the ratio between the transverse displacement

*Δx*and the beam-waist radius

*w*, and does not depend on the distance between the beam source and the plate’s position.

_{0}## 3. Experimental setup

*n*= 1.52 and a calculated thermal diffusivity

*κ*= 0.96 W m

^{−1}K

^{−1}stands for the thermal conductivity,

*ρ*= 2510 kg m

^{−3}is the density, and

*c*800 J kg

_{p}=^{−1}K

^{−1}is the specific heat capacity. The flatness of the plate was λ/10 at 633 nm. We tested three plates with equal thickness and flatness, but different parallelism. The parallelism was measured by a homodyne quadrature laser interferometer [16

16. P. Gregorčič, T. Požar, and J. Možina, “Quadrature phase-shift error analysis using a homodyne laser interferometer,” Opt. Express **17**(18), 16322–16331 (2009). [CrossRef] [PubMed]

*λ*= 633 nm) using the beam-waist radius

*w*= 400 μm. The spot size on the plate was

_{0}*w =*448 μm and the radius of curvature was

*ρ*= 1.98 m. The cylindrical head of the polarized He-Ne laser was rotated so that the linear polarization of the probe beam was perpendicular to the plane of incidence (s-polarization). The absorption of a He-Ne light within the thin plate can be neglected, since the absorption coefficient

*μ*was estimated to be less than 0.7 m

^{−1}.

*λ*= 266 nm) as a heating source (the excitation laser) due to its high absorption coefficient in borosilicate glass. The absorption coefficient was evaluated by UV-vis spectrophotometer (Hewlett Packard 8453) to be higher than 7 × 10

^{4}m

^{−1}. The duration of the excitation laser pulse was 5 ns, the repetition rate was 20 Hz, the beam radius was 1.5 mm, and the pulse energy was 13 mJ. A fraction of the excitation-laser energy (0.5 mJ per pulse) was reflected from the front surface of the plate, while the rest (12.5 mJ per pulse) was absorbed within the plate.

*φ*of 25 degrees with respect to the plate’s normal. Both beams were crossed concentrically on the plate. A band-pass filter (BPF) was placed in front of the photodiode to eliminate the light from the excitation laser.

## 4. Results and discussion

*θ*. Therefore, at that angle, defined as tan(

_{B}*θ*) =

_{B}*n*, the transmittance oscillations disappear.

**30**(3), 305–311 (1991). [CrossRef] [PubMed]

*w*is large in comparison with the plate’s thickness

_{0}*d*.

*R*(

*θ*) is the power reflectivity defined by Eq. (1). The results of Eq. (10) are shown in Fig. 2 with the solid, black curve.

_{i}*n*= 1.52) as a function of the incident angle is presented in Figs. 2(c) and 2(d). Figure 2(c) shows the transmittance for the plate with a parallelism of 15 μrad. A comparison of these results with the corresponding theoretical results in Fig. 2(a) reveals a good agreement between the experiment and the theory. However, in contrast to the theoretical predictions, the experimental results show a larger decrease in the interference maxima at large angles. This happens because the dielectric plate is not perfectly parallel; instead, it forms a small wedge. The wedge changes the plate’s thickness, so particular points of the probe-beam profile experience a different transmittance. To prove this we measured the transmittance of plates with equal thickness and flatness, but different parallelisms. The gray and the red curves in Fig. 2(d) show the measured transmittance for the plates with parallelisms of 45 μrad and 200 μrad, respectively. It is clear that the interference maxima decrease when the wedge increases from 15 μrad (the gray curve in Fig. 2 (c)) to 45 μrad (the gray curve in Fig. 2(d)). In the case of the plate with a parallelism of 200 μrad, i.e., the 200-nm change in plate’s thickness at 1 mm, which approximately corresponds to a λ/4-change of the plate’s thickness within the probe-beam diameter, the measured transmittance conforms to the transmittance of the incoherent light. This phenomenon can be seen by a comparison between the black and the red curves in Fig. 2 (d). The similarity with the results of the incoherent light is a consequence of measuring the transmittance of the whole beam, which gives an average transmittance of the distorted beam profile. However, the intensity at any particular point of the transmitted beam will show the interference oscillations.

*δ*, in a linear approximation depends on the plate’s temperature change as:

*ΔT*denotes the temperature change;

*n*and

_{0}*d*are initial refractive index and the plate’s thickness, respectively;

_{0}*α*is the linear temperature coefficient; and

*α*and

*α*.

_{d,n}*θ*= 0° and

_{i}*θ*= 70°, respectively. It is clear that the temperature change of

_{i}*ΔT*~75 K alters the transmittance at normal incidence from its trough to its peak value. Therefore, at normal incidence, one fringe occurs during the

*ΔT*~150 K.

*ΔT*for particular angles

*θ*= 0° and

_{i}*θ*= 70° is shown as the circles in Figs. 3(b) and 3(c), respectively. The black curves show a theoretical fit according to Eq. (9), where the temperature-dependent phase-shift difference

_{i}*δ*(

*ΔT*), defined by Eq. (11), was involved. The fitting parameters,

*d*and

_{0}*α*, obtained with the least-squares method, are equal to 140 μm ± 0.080 μm, and 10

_{d,n}^{−5}K

^{−1}± 10

^{−6}K

^{−1}, respectively. For normal incidence (Fig. 3(b)) the transmittance trough value equals ~0.84, while the transmittance peak value reaches ~1. The transmittance changes periodically with a period of

*ΔT*~150 K. This period increases with the angle of incidence and is equal to 180 K at

*θ*= 70° (Fig. 3(c)). Here, the measured transmittance is in the interval between 0.29 and 0.9. Therefore, the dielectric plate applied as an attenuator at this incident angle has a 4 times greater dynamic range than at normal incidence. This is a consequence of the higher reflectivity of the plate’s surface according to Eq. (1). The difference in the peak values between the fitted (the black curve) and the measured transmittance appears at large angles because the limited beam and the wedge effects are not taken into account in Eq. (9).

_{i}*ΔT*~4 K. This value is comparable with the temperature rise calculated by the specific heat equation, where the heat equals the absorbed pulse energy (12.5 mJ).

*t*= 0.4 s in Fig. 4(b), the single pulse changes the transmittance by 1%. On the other hand, the transmittance does not change at the troughs and peaks, where the FPE’s response is zero.

*t =*10 s, i.e., when the first 200 pulses were absorbed into the plate. The steady-state conditions depend on the excitation-laser power, absorption coefficient and the thermal properties of the plate.

17. T. Požar, P. Gregorčič, and J. Možina, “Optical measurements of the laser-inducedultrasonic waves on moving objects,” Opt. Express **17**(25), 22906–22911 (2009). [CrossRef]

18. P. Gregorčič, R. Petkovšek, J. Možina, and G. Močnik, “Measurements of cavitation bubble dynamics based on a beam-deflection probe,” Appl. Phys., A Mater. Sci. Process. **93**(4), 901–905 (2008). [CrossRef]

## 5. Conclusion

## References and links

1. | R. O. Rice and J. D. Macomber, “Attenuation of giant laser pulses by absorbing filters,” Appl. Opt. |

2. | R. M. A. Azzam, “Tilted parallel dielectric slab as a multilevel attenuator for incident p- or s-polarized light,” Appl. Opt. |

3. | Y. H. Wu, Y. H. Lin, Y. Q. Lu, H. W. Ren, Y. H. Fan, J. R. Wu, and S. T. Wu, “Submillisecond response variable optical attenuator based on sheared polymer network liquid crystal,” Opt. Express |

4. | H. Lotem, A. Eyal, and A. R. Shuker, “Variable attenuator for intense unpolarized laser beams,” Opt. Lett. |

5. | D. Gauden, D. Mechin, C. Vaudry, P. Yvernault, and D. Pureur, “Variable optical attenuator based on thermally tuned Mach-Zehnder interferometer within a twin core fiber,” Opt. Commun. |

6. | K. Bennett and R. L. Byer, “Computer-controllable wedged-plate optical-variable attenuator,” Appl. Opt. |

7. | J. Staromlynska, R. A. Clay, and K. F. Dexter, “Variable optical attenuator for use in the visible spectrum,” Appl. Opt. |

8. | J. H. Lehman, D. Livigni, X. Y. Li, C. L. Cromer, and M. L. Dowell, “Reflective attenuator for high-energy laser measurements,” Appl. Opt. |

9. | M. J. Mughal and N. A. Riza, “Compact acoustooptic high-speed variable attenuator for high-power applications,” IEEE Photon. Technol. Lett. |

10. | H. B. Yu, G. Y. Zhou, C. F. Siong, and L. Feiwen, “A variable optical attenuator based on optofluidic technology,” J. Micromech. Microeng. |

11. | J. C. Cotteverte, F. Bretenaker, and A. Lefloch, “Jones matrices of a tilted plate for Gaussian beams,” Appl. Opt. |

12. | H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. |

13. | S. Nemoto, “Waist shift of a Gaussian-beam by a dielectric plate,” Appl. Opt. |

14. | E. Hecht, |

15. | H. Abu-Safia, R. Al-Tahtamouni, I. Abu-Aljarayesh, and N. A. Yusuf, “Transmission of a Gaussian-beam through a Fabry-Perot interferometer,” Appl. Opt. |

16. | P. Gregorčič, T. Požar, and J. Možina, “Quadrature phase-shift error analysis using a homodyne laser interferometer,” Opt. Express |

17. | T. Požar, P. Gregorčič, and J. Možina, “Optical measurements of the laser-inducedultrasonic waves on moving objects,” Opt. Express |

18. | P. Gregorčič, R. Petkovšek, J. Možina, and G. Močnik, “Measurements of cavitation bubble dynamics based on a beam-deflection probe,” Appl. Phys., A Mater. Sci. Process. |

**OCIS Codes**

(140.0140) Lasers and laser optics : Lasers and laser optics

(140.6810) Lasers and laser optics : Thermal effects

(260.3160) Physical optics : Interference

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: December 9, 2009

Revised Manuscript: January 13, 2010

Manuscript Accepted: January 28, 2010

Published: February 11, 2010

**Citation**

Peter Gregorčič, Aleš Babnik, and Janez Možina, "Interference effects at a dielectric plate
applied as a high-power-laser attenuator," Opt. Express **18**, 3871-3882 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-4-3871

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### References

- R. O. Rice and J. D. Macomber, “Attenuation of giant laser pulses by absorbing filters,” Appl. Opt. 14(9), 2203–2206 (1975). [CrossRef] [PubMed]
- R. M. A. Azzam, “Tilted parallel dielectric slab as a multilevel attenuator for incident p- or s-polarized light,” Appl. Opt. 48(2), 425–428 (2009). [CrossRef] [PubMed]
- Y. H. Wu, Y. H. Lin, Y. Q. Lu, H. W. Ren, Y. H. Fan, J. R. Wu, and S. T. Wu, “Submillisecond response variable optical attenuator based on sheared polymer network liquid crystal,” Opt. Express 12(25), 6382–6389 (2004). [CrossRef] [PubMed]
- H. Lotem, A. Eyal, and A. R. Shuker, “Variable attenuator for intense unpolarized laser beams,” Opt. Lett. 16(9), 690–692 (1991). [CrossRef] [PubMed]
- D. Gauden, D. Mechin, C. Vaudry, P. Yvernault, and D. Pureur, “Variable optical attenuator based on thermally tuned Mach-Zehnder interferometer within a twin core fiber,” Opt. Commun. 231(1-6), 213–216 (2004). [CrossRef]
- K. Bennett and R. L. Byer, “Computer-controllable wedged-plate optical-variable attenuator,” Appl. Opt. 19(14), 2408–2412 (1980). [CrossRef] [PubMed]
- J. Staromlynska, R. A. Clay, and K. F. Dexter, “Variable optical attenuator for use in the visible spectrum,” Appl. Opt. 26(18), 3827–3830 (1987). [CrossRef] [PubMed]
- J. H. Lehman, D. Livigni, X. Y. Li, C. L. Cromer, and M. L. Dowell, “Reflective attenuator for high-energy laser measurements,” Appl. Opt. 47(18), 3360–3363 (2008). [CrossRef] [PubMed]
- M. J. Mughal and N. A. Riza, “Compact acoustooptic high-speed variable attenuator for high-power applications,” IEEE Photon. Technol. Lett. 14(4), 510–512 (2002). [CrossRef]
- H. B. Yu, G. Y. Zhou, C. F. Siong, and L. Feiwen, “A variable optical attenuator based on optofluidic technology,” J. Micromech. Microeng. 18(11), 115016 (2008). [CrossRef]
- J. C. Cotteverte, F. Bretenaker, and A. Lefloch, “Jones matrices of a tilted plate for Gaussian beams,” Appl. Opt. 30(3), 305–311 (1991). [CrossRef] [PubMed]
- H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5(10), 1550–1567 (1966). [CrossRef] [PubMed]
- S. Nemoto, “Waist shift of a Gaussian-beam by a dielectric plate,” Appl. Opt. 28(9), 1643–1647 (1989). [CrossRef] [PubMed]
- E. Hecht, Optics (2nd Edition, Addison Wesley, 1987), pp. 100. [PubMed]
- H. Abu-Safia, R. Al-Tahtamouni, I. Abu-Aljarayesh, and N. A. Yusuf, “Transmission of a Gaussian-beam through a Fabry-Perot interferometer,” Appl. Opt. 33(18), 3805–3811 (1994). [CrossRef] [PubMed]
- P. Gregorčič, T. Požar, and J. Možina, “Quadrature phase-shift error analysis using a homodyne laser interferometer,” Opt. Express 17(18), 16322–16331 (2009). [CrossRef] [PubMed]
- T. Požar, P. Gregorčič, and J. Možina, “Optical measurements of the laser-inducedultrasonic waves on moving objects,” Opt. Express 17(25), 22906–22911 (2009). [CrossRef]
- P. Gregorčič, R. Petkovšek, J. Možina, and G. Močnik, “Measurements of cavitation bubble dynamics based on a beam-deflection probe,” Appl. Phys., A Mater. Sci. Process. 93(4), 901–905 (2008). [CrossRef]

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