## Optical functionalities of dielectric material deposits obtained from a Lambertian evaporation source

Optics Express, Vol. 15, Issue 9, pp. 5451-5459 (2007)

http://dx.doi.org/10.1364/OE.15.005451

Acrobat PDF (518 KB)

### Abstract

The thickness profile of deposits obtained from Lambertian evaporation sources is highlighted concerning its transmission optical functionality, in the case of dielectric materials. Fresnel diffraction is used to characterize the lateral resolution and intensity on the optical axis of an input gaussian laser beam. Functionality similar to logarithmic axicons, with uniform lateral resolution and also uniform on-axis intensity, is theoretically derived. It is also shown for this particular optical structure that the intensity slope along the optical axis can be changed from positive to negative values by only changing the input beam width.

© 2007 Optical Society of America

## 1. Introduction

3. J. M. González-Leal, R. Prieto-Alcón, J. A. Ángel, D. A. Minkov, and E. Márquez, “Influence of the substrate absorption on the optical and geometrical characterization of thin dielectric films,” Appl. Opt. **41**, 7300 (2002). [CrossRef] [PubMed]

4. A. T. Friberg, “Stationary-phase analysis of generalized axicons,” J. Opt. Soc. Am. A **13**, 743 (1996). [CrossRef]

5. J. H. McLeod, “The axicon: A new type of optical element,” J. Opt. Soc. Am. **44**, 592 (1954). [CrossRef]

## 2. Theory

### 2.1 Lambertian evaporation: cosine law

_{v}= 1 in Hertz-Knudsen equation [1]. This results in the following mathematical relation for the evaporated mass about an exit angle of θ into the forward solid angle

*d*ω

*dM*=

_{c}*dM*, and the projection of the elemental area ρ

_{e}^{2}

*dω*onto the substrate at an angle of Ψ (see Fig. 1).

*a*/ρ =

*a*/(

*a*

^{2}+

*r*

^{2})

^{1/2}, and Eq. (2) can be written as

*r*as given in Eq. (3),

### 2.2 Optical function analysis

*n*, the amplitude transmission function of such a refractive optical structure would be, from Eq. (4) [2,4

4. A. T. Friberg, “Stationary-phase analysis of generalized axicons,” J. Opt. Soc. Am. A **13**, 743 (1996). [CrossRef]

*r*’ and

*z*on the image plane behind this refractive optical structure, can be analysed by solving Fresnel diffraction integral [2,4

4. A. T. Friberg, “Stationary-phase analysis of generalized axicons,” J. Opt. Soc. Am. A **13**, 743 (1996). [CrossRef]

*r*and ϕ at the object plane, into the surface

*S*.

*U*(

*r*,

*z*) is the incident light field, λ the wavelength, and

*k*= 2π/λ.

*U*(

*r*,

*z*) can be determined from the amplitude transmission function φ(

*r*) of Eq. (3). If gaussian intensity distribution with width σ at

*I*

_{0}/e

^{2}is further considered for the input light,

*U*(

*r*,

*z*) can be eventually written as

*f*(

*r*) =

*r*

^{2}/(2

*z*) - φ(

*r*). In the case of the amplitude transmission function given by Eq. (3), the integral of Eq. (8) can be solved by the stationary-phase approximation [2, 4

**13**, 743 (1996). [CrossRef]

#### 2.3.1 Lateral resolution

*w*, behind the refractive optical structure under study, can be found solving equation

*w*(

*z*) for a gaussian incident laser beam with σ = 2 mm, and values for the design parameters

*A*= 5 μm,

*a*= 1 mm and

*n*= 2 , is shown in Fig. 3.

*w*

_{min}, is observed in Fig. 3, which appears at distance of

*f*

_{0}from the refractive optical structure. This distance can be determined from the design parameters a,

*A*and

*n*, solving the equation

*w*

_{min}can be derived numerically by solving Eq. (12) at

*f*

_{0}, and it can be probed to give the following approximated relationship

#### 2.3.2 Focal-region intensity

*f*

_{0},

*I*(0,

*z*)/

*I*(0,

*f*

_{0}), for a gaussian incident laser beam with σ = 2 mm, and values for the design parameters

*A*= 5 μm,

*a*= 1 mm and

*n*= 2, is shown in Fig. 3

*I*(0,

*z*) with respect

*z*, at

*f*

_{0}, from Eq. (15), which is given by

*w*(

*z*) and the minimum for

*I*(0,

*z*) can be made concurrent at

*f*

_{0}, in similar fashion as logarithmic axicons [10

10. J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, and S. Bará, “Nonparaxial designing of generalized axicons,” Appl. Opt. **31**, 5326 (1992). [CrossRef]

## 3. Discussion

### 3.1 Design parameters

*f*, of about 40 mm would be attained for the refractive optical structure whose results are shown in Fig. 3.

*w*(

*z*) and

*I*(0, z)/

*I*(0,

*f*

_{0}) for two different values of the parameter

*A*are shown in Figs. 4(a) and 4(b). As expected according to Eq. (14), the beam width decreases at the focal region with increasing the A value, and concurrently, the focal region makes shorter. This result is consistent with the typical behaviour observed in lenses when increasing the lens curvature, where the Airy spot makes smaller when increasing the numerical aperture, and depth of focus similarly makes shorter [2]. Also in consistence with the well-known behaviour occurring in spherical lenses, the increase of the refractive-index value leads to a higher lateral resolution, as observed in Figs. 4(c) and 4(d). In all cases illustrated in Figs. 3 and 4, a positive slope is observed in the intensity along

*z*, within the focal region.

### 3.2 Intensity control

_{0}, change the slope of the intensity within the focal region from positive to negative, respectively, with no influence on the lateral resolution and focal region location. Such effect has also been reported for axicons and lens axicons fabricated by different means [8

8. B. P. S. Ahluwalia, W. C. Cheong, X.-C. Yuan, L.-S. Zhang, S.-H. Tao, J. Bu, and H. Wang, “Design and fabrication of a double-axicon for generation of tailored self-imaged three-dimensional intensity voids,” Opt. Lett. **31**, 987 (2006). [CrossRef] [PubMed]

9. J. X. Pu, H. H. Zhang, and S. Nemoto, “Lens axicons illuminated by Gaussian beams for generation of uniform-axial intensity Bessel fields,” Opt. Eng. **39**, 803 (2000). [CrossRef]

*w*(

*z*) and

*I*(0,

*z*)/

*I*(0,

*f*

_{0}) for fixed values of the design parameters

*a*,

*A*and

*n*, and values of σ at and around σ

_{0}are shown in Fig. 6. From a technological point of view, such a kind of actuation on the input laser beam allows getting control on the intensity slope over the focal region. This result provides extra functionalities to the refractive optical structure.

^{2}of σ = 2 mm (Coherent, Verdi V6), a refractive optical structure as those introduced here with focal region around 50 mm and

*uniform*intensity, could be fabricated setting the design parameter with values

*a*= 3.65 mm,

*A*= 0.115 mm and

*n*= 2. In this case, a minimum lateral resolution

*w*

_{min}≈ 12 μm, and a depth of focus Δ

*f*≈ 37 mm, according to the 5 % of tolerance agreed above, can be achieved.

## 4. Concluding remarks

## Acknowledgments

## References and links

1. | R. Glang, in: |

2. | M. Born and E. Wolf, Principles |

3. | J. M. González-Leal, R. Prieto-Alcón, J. A. Ángel, D. A. Minkov, and E. Márquez, “Influence of the substrate absorption on the optical and geometrical characterization of thin dielectric films,” Appl. Opt. |

4. | A. T. Friberg, “Stationary-phase analysis of generalized axicons,” J. Opt. Soc. Am. A |

5. | J. H. McLeod, “The axicon: A new type of optical element,” J. Opt. Soc. Am. |

6. | A. Burvall, K. Kolacz, Z. Jaroszewicz, and A. T. Friberg, “Simple lens axicon,” Appl. Opt. |

7. | Z. Jaroszewicz, A. Burvall, and A. T. Friberg, “Axicon: the most important optical element,” Opt. Photon. News |

8. | B. P. S. Ahluwalia, W. C. Cheong, X.-C. Yuan, L.-S. Zhang, S.-H. Tao, J. Bu, and H. Wang, “Design and fabrication of a double-axicon for generation of tailored self-imaged three-dimensional intensity voids,” Opt. Lett. |

9. | J. X. Pu, H. H. Zhang, and S. Nemoto, “Lens axicons illuminated by Gaussian beams for generation of uniform-axial intensity Bessel fields,” Opt. Eng. |

10. | J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, and S. Bará, “Nonparaxial designing of generalized axicons,” Appl. Opt. |

11. | R. Grunwald, U. Neumann, A. Rosenfeld, J. Li, and P. R. Herman, “Scalable multichannel micromachining with pseudo-nondiffracting vacuum ultraviolet beam arrays generated by thin-film axicons,” Opt. Lett. |

12. | J. W. Goodman, |

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

**OCIS Codes**

(220.1250) Optical design and fabrication : Aspherics

(310.1860) Thin films : Deposition and fabrication

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: February 6, 2007

Revised Manuscript: February 27, 2007

Manuscript Accepted: March 24, 2007

Published: April 20, 2007

**Citation**

J. M. González-Leal, "Optical functionalities of dielectric material deposits obtained from a Lambertian evaporation source," Opt. Express **15**, 5451-5459 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-9-5451

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### References

- R. Glang, in Handbook of Thin Film Technology, L. I. Maissel and R. Glang, eds., (McGraw-Hill, New York, 1983).
- M. Born, E. Wolf, Principles of Optics: Electromagnetic theory of propagation, interference and diffraction of light, 6th ed. (Pergamon Press, Oxford, 1989) p. 752.
- J. M. González-Leal, R. Prieto-Alcón, J. A. Ángel, D. A. Minkov, E. Márquez, "Influence of the substrate absorption on the optical and geometrical characterization of thin dielectric films," Appl. Opt. 41, 7300 (2002). [CrossRef] [PubMed]
- A. T. Friberg, "Stationary-phase analysis of generalized axicons," J. Opt. Soc. Am. A 13, 743 (1996). [CrossRef]
- J. H. McLeod, "The axicon: A new type of optical element," J. Opt. Soc. Am. 44, 592 (1954). [CrossRef]
- A. Burvall, K. Kolacz, Z. Jaroszewicz, A. T. Friberg, "Simple lens axicon," Appl. Opt. 43, 4838 (2004). [CrossRef]
- Z. Jaroszewicz, A. Burvall, A. T. Friberg, "Axicon: the most important optical element," Opt. Photon. News 16, 34 (2005). [CrossRef]
- B. P. S. Ahluwalia, W. C. Cheong, X.-C. Yuan, L.-S. Zhang, S.-H. Tao, J. Bu, H. Wang, "Design and fabrication of a double-axicon for generation of tailored self-imaged three-dimensional intensity voids," Opt. Lett. 31, 987 (2006). [CrossRef] [PubMed]
- J. X. Pu, H. H. Zhang, and S. Nemoto, "Lens axicons illuminated by Gaussian beams for generation of uniform-axial intensity Bessel fields," Opt. Eng. 39, 803 (2000). [CrossRef]
- J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, and S. Bará, "Nonparaxial designing of generalized axicons," Appl. Opt. 31, 5326 (1992). [CrossRef]
- R. Grunwald, U. Neumann, A. Rosenfeld, J. Li, P. R. Herman, "Scalable multichannel micromachining with pseudo-nondiffracting vacuum ultraviolet beam arrays generated by thin-film axicons," Opt. Lett. 31, 1666 (2006). [CrossRef] [PubMed]
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, Singapore, 1996).
- H. Kogelnik, "Coupled-wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909 (1969).

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