Advertisement

Applied Physics A

, Volume 107, Issue 3, pp 525–529 | Cite as

Fabrication of three-focal diffractive lenses by two-photon polymerization technique

  • Vladimir Osipov
  • Leonid L. Doskolovich
  • Evgeni A. Bezus
  • Wei Cheng
  • Arune Gaidukeviciute
  • Boris Chichkov
Rapid communication

Abstract

Fabrication of submicron-height relief of three-focal diffractive lenses using two-photon polymerization is studied. Optical properties of the designed lenses are investigated theoretically and experimentally. The proposed design of the combined diffractive–refractive lenses is promising for the realization of three-focal optical ophthalmological implants with predetermined light intensity distribution between the foci. The realized three-focal optical element has a diameter size of 2.7 mm with the focal distances in the range of 27–34 mm.

Keywords

Phase Function Diffraction Efficiency Zone Plate Paraxial Approximation Diffractive Optical Element 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1 Introduction

Diffractive optical elements (DOEs) provide many interesting possibilities for the design of novel optical systems and efficient light transformations [1, 2, 3]. They are particularly important for ophthalmological applications where the main challenge is to create optical implants providing non-aberrated optical imagery at all focal distances. Available ophthalmological implants are usually based on bi-focal optics [4, 5]. The design and fabrication of three-focal and multi-focal DOEs, which can be applied as contact and intraocular lenses for vision improvement at intermediate distances, remain challenging.

Earlier it has been demonstrated that the two-photon polymerization (2PP) technique is a powerful instrument for the fabrication of arbitrary three-dimensional (3D) structures with sub-wavelength resolution (see, for example, [6, 7, 8, 9, 10, 16] and references therein). The high resolution and relatively low cost of 2PP make it attractive for the fabrication of DOEs with continuous [11] and binary [12] microreliefs. These DOEs can be applied for the generation of any desired longitudinal intensity distributions (coaxial lines, sets of sequential axial foci, etc.) required for different laser technologies.

The present paper is devoted to the fabrication of submicron-height reliefs of three-focal diffractive lenses by 2PP. The realized three-focal lens consists of a conventional thin lens with a fixed focus and a three-focal zone plate. In principle, the same DOE structure can be produced using diamond turning or electron lithography systems. Compared to these systems, 2PP allows cost-effective fabrication of complex 3D structures and is much more flexible.

2 Design of diffractive lenses for three-focus longitudinal intensity distributions

It is well known that a zoned phase microrelief with a limited height can perform phase retardation equivalent to a wide-range-varying phase function φ of a thin optical element. In this case the microrelief height is proportional to the phase function φ modulo 2π:
$$ h = \frac{\bmod_{2\pi} ( \varphi )}{k( n - 1 )},$$
(1)
where k=2π/λ is the wavenumber and n is the refractive index of the material. The general method applied in this work for designing multi-focal diffractive lenses was described in [3, 13, 14]. For convenience, we briefly describe this method for the case of radial symmetry. Let us assume that the complex amplitude of the incident beam is given by
$$ W_{0}( \rho) = \sqrt{I_{0}( \rho )} \exp\bigl[\mathrm{i}\varphi_{0}( \rho) \bigr],$$
(2)
where ρ is the radial coordinate in the lens plane. In this case the phase function of the multi-focal lens takes the form [13, 14]
$$ \varphi_{\mathrm{mf}}( \rho) = \varphi_{1}( \rho) + \varPhi \bigl[ \varphi_{2}( \rho) \bigr] - \varphi_{0}( \rho),\quad \rho\in[ 0, R ],$$
(3)
where R denotes the lens radius and φ 1(ρ) and φ 2(ρ) are the phase functions of lenses with the focal lengths f 1 and f 2. In the paraxial approximation, these functions are given by
$$ \varphi_{1}( \rho) = - \frac{k\rho^{2}}{2f_{1}},\qquad \varphi_{2}( \rho) = \bmod_{2\pi} \biggl( - \frac{k\rho^{2}}{2f_{2}}\biggr).$$
(4)

The function Φ[φ 2(ρ)]∈[0,2π) in Eq. (3) describes a nonlinear transformation of the phase function φ 2(ρ). In the case when the incident beam has a plane wavefront (i.e. \(\varphi_{0}( \rho) \equiv\operatorname{const}\)), the multi-focal lens can be represented as a superposition of a conventional lens and a zone plate with the phase functions φ 1(ρ) and Φ[φ 2(ρ)], respectively.

Substituting Eq. (4) into Eq. (3), the following expression for the complex amplitude of the transmitted beam can be obtained:
$$ W(\rho) = \sqrt{I_{0}( \rho )} \exp\biggl( -\frac{\mathrm{i}k\rho^{2}}{2f_{1}} + \mathrm{i}\varPhi\biggl[ \bmod_{2\pi} \biggl( -\frac{k\rho^{2}}{2f_{2}} \biggr) \biggr] \biggr).$$
(5)
Expanding the function exp[iΦ(ξ)] into Fourier series in the interval ξ∈[0,2π) yields
$$ \exp\bigl[ \mathrm{i}\varPhi( \xi) \bigr] = \sum _{n} c_{n}\exp( \mathrm{i}n\xi) ,$$
(6)
where
$$ c_{n} = \frac{1}{2\pi} \int_{0}^{2\pi}\exp\bigl[ \mathrm{i}\varPhi( \xi) - \mathrm{i}n\xi\bigr]\,\mathrm{d}\xi $$
(7)
and ∑ n |c n |2=1. Substituting Eq. (6) into Eq. (5), we obtain
$$ W( \rho) = \sqrt{I_{0}( \rho )} \sum _{n} c_{n} \exp\biggl[ - \frac{\mathrm{i}k\rho^{2}}{2}\biggl(\frac{1}{f_{1}} + \frac{n}{f_{2}} \biggr) \biggr].$$
(8)
In the paraxial approximation, Eq. (8) corresponds to the superposition of spherical beams with the foci
$$ F_{n} = \frac{f_{1}f_{2}}{f_{2} + nf_{1}}$$
(9)
and the energy distribution between the beams is given by the values |c n |2, nZ.
For a three-focal lens, the nonlinear transformation function Φ(ξ) can be defined in the following form [3]:
$$ \varPhi( \xi) = \left \{ \begin{array}{l} 0,\quad \xi\in[ 0,\pi), \\2\tan^{ - 1}(\pi / 2 ),\quad \xi\in[ \pi,2\pi). \end{array} \right .$$
(10)
Indeed, in this case the corresponding Fourier coefficients defined in Eq. (7) are given by
$$ c_{n} = \left \{ \begin{array}{l} \frac{1 - ( - 1 )^{n}}{2\pi \mathrm{i}n}[ 1- \exp( 2\mathrm{i}\tan^{ - 1}\frac{\pi}{2} ) ],\quad n \ne 0,\\\noalign{\vspace{3pt}}\frac{1}{2}[ 1 + \exp( 2\mathrm{i}\tan^{ - 1}\frac{\pi}{2} ) ],\quad n = 0.\end{array} \right .$$
(11)
According to Eq. (11), |c −1|2=|c 0|2=|c 1|2≈0.2884, which means that over 86 % of the incident energy will be concentrated in three beams with the foci
$$ F_{ - 1} = \frac{f_{1}f_{2}}{f_{2} - f_{1}},\qquad F_{0} =f_{1},\qquad F_{1} = \frac{f_{1}f_{2}}{f_{2} + f_{1}}.$$
(12)
Note that the phase function Φ[φ 2(ρ)] corresponds to a binary zone plate with the zone radii ρ m,l found from the equation
$$ \bmod_{2\pi} \biggl( - \frac{k\rho_{m,l}^{2}}{2f_{2}} \biggr) = \pi l,\quad l = 0,1.$$
(13)
Let us consider another expression for the nonlinear transformation function corresponding to a sinusoidal zone plate:
$$ \varPhi( \xi) = a\sin( \xi).$$
(14)
Substituting Eq. (14) into Eq. (7), we obtain
$$ c_{n} = \frac{1}{2\pi} \int_{0}^{2\pi}\exp\bigl[ \mathrm{i}a\sin( \xi) - \mathrm{i}n\xi\bigr]\,\mathrm{d}\xi.$$
(15)
The right-hand side of Eq. (15) is the integral representation of a Bessel function of the first kind [15]; thus,
$$ c_{n} = J_{n}( a ).$$
(16)
At a≈1.435, |c −1|2=|c 0|2=|c 1|2=0.3. Hence, if the nonlinear transformation function is defined by Eq. (14), 90 % of the incident beam energy will be concentrated in the desired three beams.

The multi-focal lens with the phase function given by Eq. (3) can be implemented either as a single DOE with complex diffractive microrelief or as a combination of a conventional thin lens with the focus f 1 and a binary or continuous-relief zone plate Φ[φ 2(ρ)]. In the present work, the latter approach was chosen.

As an example, we designed a three-focal lens for the following parameters: λ=630 nm, F 0=30 mm, F −1=34 mm, lens radius R=1.35 mm, with the refractive index of the lens material n=1.6. For the listed parameters the values of f 1 and f 2 are 30 mm and 255 mm, respectively, and the location of the third focal spot is F 1≈26.8 mm. The zone plates corresponding to binary and sine transformation functions Φ[φ 2(ρ)] are shown in Fig. 1. The intensity distribution generated by the lens with the phase function φ mf(r) can be calculated using the Fresnel–Kirchhoff diffraction integral [14]:
$$ \begin{aligned}[b]I( r,z ) &= \biggl| \frac{k}{z}\int_{0}^{R}W( \rho)\exp\biggl\{ \mathrm{i}\biggl[ \varphi_{\mathrm{mf}}( \rho) +\frac{k\rho^{2}}{2z} \biggr] \biggr\}\\&\quad {}\times J_{0}\biggl( \frac{kr\rho}{z}\biggr)\rho\,\mathrm{d} \rho\biggr|^{2},\end{aligned}$$
(17)
where z denotes the coordinate along the optical axis.
Fig. 1

Radial profiles of the trifocal zone plates Φ[φ 2(ρ)]/k(n−1) with binary (a) and sine (b) transformation functions

The normalized intensity distribution along the optical axis for the binary zone plate (Fig. 1a) is shown in Fig. 2 and demonstrates high-quality focusing into three spots.
Fig. 2

Calculated normalized intensity distribution along the optical axis

A decrease in the focal peak intensities with increasing focal length is due to the fact that the intensity in the geometric focus is inversely proportional to the squared focal length \(( I( 0,F_{i} )\sim 1/F_{i}^{2})\). At the same time, the illuminating beam energy portions focused in the diffraction vicinity of radius Δ=0.61λF i /R, i=−1,0,1 are nearly constant in the planes z=F i , i=−1,0,1.

The diffraction efficiency of the three-focal lens with the sine transformation function given by Eq. (14) is only 4 % higher than for the binary zone plate. Therefore, the corresponding intensity distribution generated by such a lens is very close to that shown in Fig. 2. Due to only a minor improvement of the diffraction efficiency by the zone plate with the sine transformation function, the simpler binary zone plate was chosen for experimental realization.

3 Experimental setup and materials

For the fabrication of diffractive microreliefs by the 2PP technique, we apply femtosecond laser pulses from a mode-locked Yb:glass laser (pulse width 220 fs, wavelength λ=520 nm, maximal power 8 mW, and repetition rate 800 kHz). The 2PP process is initiated in a highly localized area around the focal spot so that any computer-generated 3D structure can be directly written into the volume of a photosensitive material. The laser beam is focused by the oil immersion objective lens (100×, 1.4 NA, Zeiss).

Samples with 3-μm-thick polymer layers (SU-8 photoresist from Gersteltec Engineering Solutions) were prepared by spin coating of glass substrates with the dimensions of 18×18×0.15 mm3. After deposition, the polymer layers were dried at 95 C to increase mechanical stability. The fabrication of the diffractive structure was performed by translating the sample using a computer-controlled 3D XYZ-positioning stage. The accuracy of the 2PP writing is ∼100 nm, and over the complete travel range it is better than 400 nm.

The processing characteristics and writing laser power were controlled using A3200 AEROTECH software according to the designed 3D models of the calculated microrelief. The designed binary three-focal zone plate with the diameter of 2.7 mm was fabricated using the following parameters: 2 mW writing laser power; writing speed of 0.25 mm/s; and laser writing hatch step of 0.3 μm. After polymerization, the unexposed monomer was removed by a standard SU-8 developer.

4 Experimental results

Optical and SEM images of the fabricated zone plate are shown in Fig. 3. For characterization of optical properties of this zone plate, the optical setup depicted in Fig. 4 was used.
Fig. 3

Optical (a) and SEM (b) views of the recorded binary DOE

Fig. 4

The optical setup for longitudinal intensity distribution registration

As an illuminating source, a cw He–Ne laser (Thorlabs) with a beam diameter of 1 mm was applied. The illuminating laser beam passed consecutively through the neutral filter, spatial filter consisting of two lenses with 125-mm focus and a 100-μm pinhole, 5× telescope, and 2.7-mm diaphragm with the aim to form a plane wave transverse intensity distribution. The experimentally realized three-focal lens consists of a 150-μm-thick glass substrate with the diffraction microrelief shown in Fig. 1 and a commercial glass lens with 30-mm focus length. The laser intensity distributions formed at different distances from the three-focal lens were registered by a CCD camera (uEye UI-1450C), which was installed on the optical platform and could be moved along the optical axis.

The longitudinal intensity distribution along the optical axis, generated by this lens, is shown in Fig. 5 together with the results of theoretical calculations. Three sharp foci located at designed positions are clearly visible in Fig. 5. Note that the calculations were performed taking into account a typical 50-nm error in the microrelief height, which is specific for the used fabrication technique. Thus, the measurement results are in good agreement with the theoretical data.
Fig. 5

The recorded (solid line) and calculated (dashed line) longitudinal intensity distributions along the optical axis

These results demonstrate that with the 2PP technique three-focal DOEs, providing desired intensity distributions with characteristics required for ophthalmological applications, can be fabricated.

5 Conclusions

In the present work, the ability of the two-photon polymerization technique to fabricate three-focus 2.7-mm binary diffractive optical elements has been demonstrated. It has been shown that the fabrication of concentric binary rings with heights of about 300 nm using commercially available photoresists is possible. Optical properties of the realized three-focal lenses have been investigated. The obtained experimental results are in good agreement with the numerical design.

It is particularly important that the 2PP technique can be used for the fabrication of DOEs forming multi-focal longitudinal intensity distributions. Such DOEs can be applied in metrology and medicine.

Notes

Acknowledgements

We would like to gratefully acknowledge the EC FP7-Marie Curie-IIF Program (Proposal No. 235969), the Russian Federation state contract no. 07.514.11.4060, and the German project REMEDIS for support of this work.

References

  1. 1.
    B.C. Kress, P. Meyrueis, Applied Digital Optics: From Micro-optics to Nanophotonics (Wiley, New York, 2009) Google Scholar
  2. 2.
    D.C. O’Shea, T.J. Suleski, A.D. Kathman, D.W. Pratner, Diffractive Optics: Design, Fabrication, and Test (SPIE, Bellingham, 2004) Google Scholar
  3. 3.
    L.L. Doskolovich, D.L. Golovashkin, N.L. Kazanskiy, S.N. Khonina, V.V. Kotlyar, V.S. Pavelyev, R.V. Skidanov, V.A. Soifer, V.S. Solovyev, G.V. Uspleniev, A.V. Volkov, in Methods for Computer Design of Diffractive Optical Elements, ed. by V.A. Soifer (Wiley, New York, 2002) Google Scholar
  4. 4.
    A.L. Cohen, Appl. Opt. 31(19), 3750 (1992) ADSCrossRefGoogle Scholar
  5. 5.
    M.J. Simpson, Appl. Opt. 31(19), 3621 (1992) ADSCrossRefGoogle Scholar
  6. 6.
    S. Kawata, H.B. Sun, T. Tanaka, K. Takada, Nature 412(6848), 4675 (1997) Google Scholar
  7. 7.
    B.H. Cumpston, S.P. Ananthavel, S. Barlow, D.L. Dyer, J.E. Ehrlich, L.L. Erskine, A.A. Heikal, S.M. Kuebler, I.-Y. Sandy Lee, D. McCord-Maughon, J. Qin, H. Röskel, M. Rumi, X.-L. Wu, S.R. Marder, J.W. Perry, Nature 398, 51 (1999) ADSCrossRefGoogle Scholar
  8. 8.
    J. Serbin, A. Egbert, A. Ostendorf, B.N. Chichkov, R. Houbertz, G. Domann, J. Schulz, C. Cronauer, L. Fröhlich, M. Popall, Opt. Lett. 28(5), 301 (2003) ADSCrossRefGoogle Scholar
  9. 9.
    J. Serbin, A. Ovsianikov, B. Chichkov, Opt. Express 12(21), 5221 (2004) ADSCrossRefGoogle Scholar
  10. 10.
    J. Fischer, G. von Freymann, M. Wegener, Adv. Mater. 22, 3578 (2010) CrossRefGoogle Scholar
  11. 11.
    B. Jia, J. Serbin, H. Kim, B. Lee, J. Li, M. Gu, Appl. Phys. Lett. 90, 073503 (2007) ADSCrossRefGoogle Scholar
  12. 12.
    V. Osipov, V. Pavelyev, D. Kachalov, A. Žukauskas, B. Chichkov, Opt. Express 18(25), 25808 (2010) ADSCrossRefGoogle Scholar
  13. 13.
    M.A. Golub, L.L. Doskolovich, N.L. Kazanskiy, S.I. Kharitonov, V.A. Soifer, J. Mod. Opt. 39(6), 1245 (1992) ADSCrossRefGoogle Scholar
  14. 14.
    V. Soifer, V. Kotlyar, L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis, London, 1997) Google Scholar
  15. 15.
    F.W.J. Olver, Bessel functions of integer order, in Handbook of Mathematical Functions, with Formulas, Graphs and Tables, ed. by M. Abramowitz, I. Stegun (National Bureau of Standards, Washington, D.C., 1964), pp. 355–434 Google Scholar
  16. 16.
    K. Obata, J. Koch, U. Hinze, B.N. Chichkov, Opt. Express 18(16), 17193 (2010) ADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Vladimir Osipov
    • 1
  • Leonid L. Doskolovich
    • 2
    • 3
  • Evgeni A. Bezus
    • 2
    • 3
  • Wei Cheng
    • 1
  • Arune Gaidukeviciute
    • 1
  • Boris Chichkov
    • 1
  1. 1.Laser Zentrum Hannover e.V.HannoverGermany
  2. 2.Image Processing Systems Institute of RASSamaraRussia
  3. 3.Technical Cybernetics DepartmentSamara State Aerospace UniversitySamaraRussia

Personalised recommendations