Abstract

We demonstrate a new security feature for visual control of the authenticity of optical security features — the change of the images when the optical element is turned by 180 degrees (“switch-180°”). The diffractive optical element has an asymmetric microrelief structure resulting from the asymmetry of the scattering pattern. The phase function of the diffractive optical element is computed in terms of Fresnel's scalar wave model. We developed efficient algorithms for computing the structure of flat optical elements to produce the switch effect. A sample of flat optical element for the “switch-180” effect has been developed using electron-beam lithography. The effectiveness of the development is illustrated by the photos and the video captured from a real sample. The visual “switch-180°” effect is easy to control allowing secure anti-counterfeit protection of the optical security feature developed. The new security feature is already used to protect IDs and excise stamps.

© 2015 Optical Society of America

1. Introduction

Optical elements are currently widely used to protect documents, bank notes, and brands against counterfeit [1]. In this paper we discuss the problems of the synthesis of optical security elements where security features are formed as a result of light diffraction on the microreflief of the optical element. In the literature such security elements are referred to as Surface Relief Holograms. The technology of the synthesis of holographic security elements consists of origination (creation of the master hologram) and subsequent mass replication of security elements. Embossing the microrelief of security elements is an important part of the technology of mass replication. Mass replication allows the price of holographic elements to be kept low when producing large numbers of copies.

Most of the security features are formed during origination (creation of master copies) of optical elements. Various holographic technologies have been developed to this end (kinegrams, pixelgrams, KINEMAX technology, etc [1].). Despite the extensive selection of various technologies, they can be subdivided into two groups, which differ fundamentally at the stage of origination. The first group includes technologies that use optical radiation to record the master hologram. This group includes, in addition to the technologies just mentioned, the security holograms whose originals are recorded on optical benches using interference of laser radiation. The original of the famous «Dove» hologram on VISA card was recorded using this technology whose emergence marked the beginning of mass use of optical security elements. The second group includes optical elements whose originals are recorded using electron-beam technology. How does the optical origination technology differ from electron-beam technology?

First, the two technologies differ in the cost of the equipment employed. Modern electron-beam lithographs are several dozen and even several hundred times more expensive than the equipment used for optical origination of holograms limiting the application of electron-beam technology for synthesizing optical security elements. Currently only a few companies use it for producing master holograms [2]. Electron-beam technology is more knowledge intensive, requires more skilled personnel, special premises, etc. The question naturally arises — what fundamentally new features can we expect from the use of electron-beam technology for synthesizing optical security elements?

It goes without saying that electron-beam technology offers higher resolution compared to optical technology of hologram origination. Former-generation electron-beam lithographs have a resolution on the order of 0.1 μm. Modern electron-beam lithographs have a 10 times higher resolution reaching 0.01 μm. There are two types of electron-beam lithographs. Lithographs of the first type use a beam with a Gaussian distribution, whereas those of the second type assemble the image from rectangular fragments whose sizes can be varied [3]. The possibility of recording the microirelief of optical security elements with a higher resolution offered by electron-beam technology is an important but by no means the only advantage it provides. More importantly, the electron-beam technology enables the synthesis of asymmetric microrelief. All the optical technologies of hologram origination mentioned above are based on recording the interference pattern of two laser beams, which always result in a symmetric microirelief. In electron-beam technology the electron beam is shaped with high precision within a limited area whose size can be as small as several tenths of a micron. Microrelief depth can be arbitrarily varied by varying the dose (or, which is the same, the exposure time) allowing fragments with both symmetric and asymmetric microrelief to be produced.

Diffractive elements with asymmetric microrelief are widely used in optics. Blazed gratings belong to this category [4]. Several different technologies have been developed to form blazed gratings [5]. The asymmetric microreflief of blazed grating can be formed using either optical methods [6], X-ray lithography [7], or electron-beam lithography [8]. In this article we use kinoform [9] as a diffractive element with asymmetric microrelief. In contrast to blazed gratings a kinoform gives us possibility to form any scattering pattern. Asymmetric microrelief of a kinoform allows us to create a principally new security feature — the change of the images when the optical element is turned by 180 degrees. Any optical elements created by using widely spread optical origination methods give only identical images at 0 degree and 180 degree angles of view.

In the article we developed a new flat optical element for the security purpose. The accuracy of the microrelief forming is 10 nm in depth. A sample of flat optical element for the “switch-180°” effect has been developed using electron-beam lithography. The feature can be easily controlled visually and is safely protected against counterfeit. Electron beam technology is a knowledge-intensive and expensive method of microrelief formation and it is not so widely spread. It is the way the new security feature and optical security elements are safely protected against counterfeiting.

2. Formulation of the problem

The most important task of optical security technology is to develop security features for visual control. Holographic security technology offers a wide range of visual security features. Let us mention one of such widely used security features consisting in the image switch effect when the optical security element is turned.

Figure 1 shows a scheme of the observation of the effect of image switching when the optical element is turned. Let a flat optical security element be located in the z = 0 plane. We further assume, for the sake of simplicity, that the source of daylight is located along the Oz axis. The left and right eyes of the observer are located in the z = f plane. The angle θ is counted from the Oz axis and φ is the DOE turn angle in the Oξη plane about the Oz axis.

 

Fig. 1 Scheme of the observation of image switching in the case of the turn of the optical element.

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Holographic security technology makes extensive use of the image switch effect when the element is turned by 90°. The observer sees one color image when the hologram is in the normal position (angle φ = 0°) and a different image when the hologram is turned by φ = 90°. To demonstrate the switch effect in the case of a 90° turn, we made an optical diffractive element containing fragments with symmetric microrelief exclusively. Figure 2 shows photographic images obtained using a real diffractive element. The visual effect of image switching in the case of 90° turn is easily controlled.

 

Fig. 2 Image switch effect in the case of the 90° turn: (a) φ = 0°, (b) φ = 90°, (c) φ = 180°.

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The technology of the origination of optical elements with visual features of the type of the 90°- turn switch-effect is well known and easy to implement using optical methods [1]. The visual security feature of the 90°-turn switch-effect can be produced, e.g., in the following way. First, let us partition the area of the optical element into elementary areas Gi, i = 1, … N as shown in Fig. 3.

 

Fig. 3 Partitioning of the optical element into elementary areas.

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We now subdivide each elementary area Gi, i = 1, … N into two parts, one responsible for the formation of the image seen in the case of the φ = 0° turn, and another, for the image seen in the case of the φ = 90° turn. By creating diffraction gratings of different orientation and period in each of the parts of areas Gi we can produce different images that the observer sees when the element is turned by φ = 0° and φ = 90°. Figure 2 shows an example of the 90° turn switch-effect based visual security feature. The microrelief of the flat optical element is shaped using fragments of diffraction gratings with symmetric microreliefs. As a result, in the case of the 180° turn the observer sees the same image (only turned upside down). The identical appearance of the images seen with the element turned by 0° and 180° (Fig. 2) is a consequence of the symmetric shape of the microreflief recorded using optical techniques. Note that the images seen when the element is turned by 0° and 180° are identical irrespectively of the optical technology used for the origination of the hologram.

Electron-beam technology offers optical security features that are in principle impossible to imitate or counterfeit using optical techniques of origination. We present one of such features in this paper. The fundamental difference consists in incorporating fragments with asymmetric microrelief, which can be produced using electron-beam technology, into the security element. Electron-beam technology allows creating security elements that produce different images when the element is turned by 0° and 180°. We further refer to such security features as the “180° turn switch effect”. Let us show how such an element can be synthesized.

We use the scheme of partitioning the optical element area into elementary areas Gi, i = 1,…N as shown in Fig. 3. Elementary areas Gi have the size on the order of 40 × 40 μm2 and cannot be discerned by the human eye. To form DOE microrelief we use fragments with both symmetric and asymmetric microrelief. Let us select an asymmetric microrelief area on the diffractive optical element. It is the area where the “hummingbird” is formed, which corresponds to white color in Fig. 4. Throughout the remaining part of the diffractive optical element (shaded in black in Fig. 4) fragments with symmetric microirelief exclusively are usedThe main problem with origination is to compute and form the microrelief in the area where it is asymmetric. Elementary areas with asymmetric microrelief should produce asymmetric scattering pattern as shown in Fig. 5. The width of rectangle Q1 in the case of the normal position of the diffractive optical element at φ = 0° is chosen so as to cover the view from both eyes of the observer. When the diffractive optical element is turned by 180° the view from the eyes of the observer is covered by area Q2 whose size is much greater than that of area Q1, thereby ensuring substantial reduction of the light flux registered by the observer.

 

Fig. 4 Symmetric and asymmetric microrelief areas.

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Fig. 5 Scattering pattern for elementary areas with asymmetric microrelief.

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In the case of the normal position of the optical security element (φ = 0°) both the right and left eyes of the observer see the color image shown in Fig. 6(а). The observer sees the “hummingbird” image as colored. The asymmetric structure of the scattering pattern shown in Fig. 5 allows the optical element to be made so that when it is turned by 180° the “hummingburd” image loses color and becomes gray (see Fig. 6(b)). Photographic images shown in Fig. 6 and the video were obtained using a real sample of the flat optical element.

 

Fig. 6 The 180° image switch-effect: (a) φ = 0, (b) φ = 180° (see Visualization 1).

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Let us now discuss in more detail how the microrelief is computed for elementary areas where it is asymmetric. For each elementary area Gi in the asymmetric microrelief domain the phase function has to be computed that forms the scattering pattern shown in Fig. 5. In the normal position the observer sees the color image including the color image of the “hummingbird”. Let us now assign to each elementary area one of the RGB colors assuming that R, G, and B correspond to the wavelengths λ of 0.65, 0.547, and 0.47 μm, respectively. The color of each elementary area is determined by the color image seen at φ = 0°. We compute for each of these three colors the microrelief in each elementary area Gi. We now write out the equation that relates the scalar wave function in the z = 0 and z = f planes in terms of Fresnel’s wave model [2, 10].

γGiu(ξ,η,00)exp[ikφ(ξ,η)]exp[ik(xξ)2+(yη)22f]dξdη=u(x,y,f).
Here u(x,y,f) is the scalar wave function in the z = f plane and u(ξ,η,00), the scalar wave function of the radiation incident onto the flat optical element; φ(ξ,η), the phase function of the flat optical element; k = 2π/λ; Gi, an elementary area, and γ = 2πk/f, a predefined constant. A distinguishing feature of the inverse problem of the computation of the phase function φ (ξ, η ) is that only the absolute value of the scalar wave functionu(x,y,f) is known in the right-hand side of Eq. (1). We thus arrive at the following operator equation
Aφ=F(x,y).
Here F(x,y)=|u(x,y,f)| is the given function, which is determined by the asymmetric scattering pattern and consists of two nonintersecting domains Q1 and Q2 in the upper and lower half-planes, respectively. We further assume that
F(x,y)={C1,(x,y)Q1C2,(x,y)Q2,C1>>C2.
Operator A is defined by the following relation:
Aφ=|γGiu(ξ,η,00)exp[ikφ(ξ,η)]exp[ik(xξ)2+(yη)22f]dξdη|.
Equation (2) is a nonlinear operator equation for the desired function φ(ξ,η) - the corresponding problem is ill posed [11, 12]. Efficient numerical algorithms have been developed for solving linear and nonlinear ill-posed problems [13, 14]. However, it was the method proposed by Lesem et al. [9] that proved to be one of the most efficient for solving problem (2). The optical elements developed in that paper are well known as “kinoforms”. Many studies [15–17] have been dedicated to analyzing and modifying the iterative methods proposed in [9]. Kinoform diffractive optical elements are widely used in optics for tasks that involve the forming of X-ray, optical, and infrared beams [18–20]. In this paper we use kinoforms with asymmetric microrelief to form visual features for protecting banknotes, IDs, and documents

3. Computation of the phase function of the DOE

We use the classical method proposed in [9]. We solve problem (2) for φ(ξ,η) for the given image F(x, y) and fixed wavelength λ to obtain the phase function of the optical element that determines the depth of the microrelief at point (ξ,η) of elementary area Gi. The computation of phase function φ(ξ,η) simplifies substantially if the size of the optical element is much smaller than the distance between the optical element and the observer. If the size of the optical element and its distance from the observer are of about 2 and 40 cm, respectively, it is sufficient to compute the microrelief for each wavelength only for one elementary area. In this case we can assume that other elementary areas Gi have the same microrelief, which is determined by wavelength λ exclusively. Below we describe an iterative algorithm for computing the phase function. We introduce the following notation

w(x,y)=Φ{v}(x,y)=k2πfGv(ξ,η)exp[ik(xξ)2+(yη)22f]dξdη.
Here Φ{v}(x,y) is the Fresnel transform of function v. The iterative process constructing the approximate solution for the phase function, which is an approximate solution for inverse problem (2), is arranged as follows. Four steps have to be performed to complete a single iteration in the iterative algorithm used to solve problem (2). Let v(k)(x, y) and w(k)(x, y) at k-th iteration be already known. We write function v(k)(x, y) and w(k)(x, y) in the form v(k)(x, y) = A0 exp[ikφ0(k)(x, y)] and w(k)(x, y) = A1 exp[ikφ1(k)(x, y)], respectively. Let A0(x, y) be the given amplitude distribution of incident light in the z = 0 plane and A1(x, y), the amplitude distribution in the focal plane z = f (A0 and A1 are known real functions). The algorithm of solving the inverse problem consists of the following four consecutive steps:

Step 1.

v˜(k)=Φ{v(k)}(x,y)=Wk(x,y)exp[ikφ1(k)(x,y)].

Step 2.

w(k)(x,y)=A1exp[ikφ1(k)(x,y)].

Step 3.

w˜(k)(x,y)=Φ1{w(k)}(x,y)=Vk(x,y)exp[ikφ0(k+1)(x,y)].

Step 4.

v(k+1)(x,y)=A0(x,y)exp[ikφ0(k+1)(x,y)].
The resulting v(k + 1) quantities are used as the v(k) values at step 1 of the next iteration. The constant phase distribution can be adopted as the initial approximation. The above iterative process is known to be of relaxation type [2]. The latter means that Rn + 1 ≤ Rn, where Rn=Aφ(n)F2. Hence this iterative process allows an approximation φ(k)(x, y) to be obtained or the phase function φ(x, y) that forms the given image F(x, y) in the focal plane. A characteristic feature of this iterative method is that it produces a sufficiently good approximate solution after a few – one or two dozen – iterations with further iterations resulting in a much slower decrease of the residual functional Rn = R(φ(n)). Figure 7 shows a typical dependence of residual functional Rn on the number of iterations n.

 

Fig. 7 Dependence of functional Rn on the number n of iterations.

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Functional Rn decreased by a factor of 10 by the 20th iteration compared to its value for the initial approximation. The phase function φ(ξ,η) computed using the iterative process described in the previous paragraph can be used to form the microrelief of the flat optical element [2].

4. Electron-beam technology for making the DOE

Given function φ(ξ,η), the microrelief depth can be determined at each point (ξ,η) of the optical element. The technology of electron-beam lithography was used to record the original and produce its microrelief. The electron-lithography facility employed had a minimum pixel size of 0.1 × 0.1 μm2 and allowed exposure by rectangles of various configurations with sizes of up to 6 × 6 μm2. Positive electron resist was used to form the microrelief, which was shaped with an accuracy to within 10 nm in height.

Figure 8(а) shows an example of a fragment of a kinoform with asymmetric microrelief computed for the wavelength of λ = 0.65 μm. Kinoforms are the building blocks that form the “hummingbird” image. All the remaining part of the optical element is made up of symmetric-microrelief fragments: an example of such a microrelief is shown in Fig. 8(b). The maximum microrelief depth is on the order of 0.3 μm. The microrelief fragments shown here have the sizes of about 8 × 8 μm2. To form quality visual switch-180° effect kinoforms with characteristic periods of 1.23, 1.06, and 0.9 μm for the R,G, and B colors, respectively, were used.

 

Fig. 8 Fragments of the microrelief of the optical element: (a) asymmetric, (b) symmetric

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Figure 6 shows the 180° turn image switch effect. The photos shown were taken from the nickel master hologram used for replicating the optical security element. As is evident from Fig. 6, the observer sees color image at φ = 0°. When turned by φ = 180° the “hummingbird” image loses its colors and becomes gray. This visual feature can be easily controlled safely protecting the optical security element against counterfeit.

5. Discussion and conclusion

The principally new security feature “switch-180°” and optical elements which protect against the counterfeiting of bank notes, IDs and brands are developed in this article. Asymmetric microrelief of the nano-optical elements allows us to obtain different visible images at 0 degree and at 180 degree angle of view.

There are a lot of companies working in the field of optical security all over the world. Most of them use optical origination technology based on interference of laser beams, but counterfeiters of security optical elements use the same equipment and methods. Any optical elements created by using optical origination technology gives only identical images at 0 degrees and at 180 degrees angle of view. So, the new visual security feature “switch-180°” can be easily controlled.

Electron beam lithography can be used for microrelief fabrication of new optical security elements. Electron beam technology is a knowledge-intensive and expensive method of microrelief formation and it is not so widely spread. It is the way the new security feature and optical security elements are safely protected against counterfeiting.

New optical security elements recorded by e-beam lithography allow mass replication with equipment employed to replicate standard security holograms. The new security feature “switch-180°” is already used to protect ID documents and excise stamps.

References and links

1. R. L. Van Renesse, Optical Document Security (Artech House, 2005).

2. A. V. Goncharsky and A. A. Goncharsky, Computer Optics and Computer Holography (Moscow University, 2004).

3. P. Rai-Choudhury, Handbook of Microlithography, Micromachining, and Microfabrication: Microlithography, (SPIE Optical Engineering Press, 1997), Ch. 2.5.

4. C. Palmer, Diffraction Grating Handbook, 6th ed. (Newport Corporation, 2005).

5. T. Suleski and R. Kolste, “Fabrication trends for free-space microoptics,” J. Lightwave Technol. 23(2), 633–646 (2005). [CrossRef]  

6. B. Sheng, X. Xu, Y. Liu, Y. Hong, H. Zhou, T. Huo, and S. Fu, “Vacuum-ultraviolet blazed silicon grating anisotropically etched by native-oxide mask,” Opt. Lett. 34(8), 1147–1149 (2009). [CrossRef]   [PubMed]  

7. P. Mouroulis, F. Hartley, D. Wilson, V. White, A. Shori, S. Nguyen, M. Zhang, and M. Feldman, “Blazed grating fabrication through gray-scale Xray lithography,” Opt. Express 11(3), 270–281 (2003). [CrossRef]   [PubMed]  

8. T. Shiono, T. Hamamoto, and K. Takahara, “High-efficiency blazed diffractive optical elements for the violet wavelength fabricated by electron-beam lithography,” Appl. Opt. 41(13), 2390–2393 (2002). [CrossRef]   [PubMed]  

9. L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: a new wavefront reconstruction device,” IBM J. Res. Develop. 13(2), 150–155 (1969). [CrossRef]  

10. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

11. A. N. Tikhonov, “The solution of ill-posed problems and the regularization method,” Dokl. Akad. Nauk SSSR 151(3), 501–504 (1963).

12. C. W. Groetsch and A. Neubauer, “Regularization of ill-posed problems: Optimal parameter choice in finite dimensions,” J. Approx. Theory 58(2), 184–200 (1989). [CrossRef]  

13. A. N. Tikhonov, A. V. Goncharsky, and V. V. Stepanov, “Inverse problems in image processing,” in Ill-Posed Problems in the Natural Sciences, pp. 220–232 (Mir Publishers, 1987).

14. A. Bakushinsky and A. Goncharsky, Ill-Posed Problems: Theory and Applications (Springer Netherlands, 1994).

15. D. C. O’Shea, Diffractive Optics: Design, Fabrication, and Test (SPIE Press, 2004).

16. T. Shimobaba, T. Kakue, Y. Endo, R. Hirayama, D. Hiyama, S. Hasegawa, Y. Nagahama, M. Sano, M. Oikawa, T. Sugie, and T. Ito, “Random phase-free kinoform for large objects,” Opt. Express 23(13), 17269–17274 (2015). [CrossRef]   [PubMed]  

17. H. Zhang, H. Liu, Z. Lu, and H. Zhang, “Modified phase function model for kinoform lenses,” Appl. Opt. 47(22), 4055–4060 (2008). [CrossRef]   [PubMed]  

18. H. C. Hunt and J. S. Wilkinson, “Kinoform microlenses for focusing into microfluidic channels,” Opt. Express 20(9), 9442–9457 (2012). [CrossRef]   [PubMed]  

19. L. Alianelli, K. J. S. Sawhney, R. Barrett, I. Pape, A. Malik, and M. C. Wilson, “High efficiency nano-focusing kinoform optics for synchrotron radiation,” Opt. Express 19(12), 11120–11127 (2011). [CrossRef]   [PubMed]  

20. C. Romero, R. Borrego-Varillas, A. Camino, G. Mínguez-Vega, O. Mendoza-Yero, J. Hernández-Toro, and J. R. Vázquez de Aldana, “Diffractive optics for spectral control of the supercontinuum generated in sapphire with femtosecond pulses,” Opt. Express 19(6), 4977–4984 (2011). [CrossRef]   [PubMed]  

References

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  1. R. L. Van Renesse, Optical Document Security (Artech House, 2005).
  2. A. V. Goncharsky and A. A. Goncharsky, Computer Optics and Computer Holography (Moscow University, 2004).
  3. P. Rai-Choudhury, Handbook of Microlithography, Micromachining, and Microfabrication: Microlithography, (SPIE Optical Engineering Press, 1997), Ch. 2.5.
  4. C. Palmer, Diffraction Grating Handbook, 6th ed. (Newport Corporation, 2005).
  5. T. Suleski and R. Kolste, “Fabrication trends for free-space microoptics,” J. Lightwave Technol. 23(2), 633–646 (2005).
    [Crossref]
  6. B. Sheng, X. Xu, Y. Liu, Y. Hong, H. Zhou, T. Huo, and S. Fu, “Vacuum-ultraviolet blazed silicon grating anisotropically etched by native-oxide mask,” Opt. Lett. 34(8), 1147–1149 (2009).
    [Crossref] [PubMed]
  7. P. Mouroulis, F. Hartley, D. Wilson, V. White, A. Shori, S. Nguyen, M. Zhang, and M. Feldman, “Blazed grating fabrication through gray-scale Xray lithography,” Opt. Express 11(3), 270–281 (2003).
    [Crossref] [PubMed]
  8. T. Shiono, T. Hamamoto, and K. Takahara, “High-efficiency blazed diffractive optical elements for the violet wavelength fabricated by electron-beam lithography,” Appl. Opt. 41(13), 2390–2393 (2002).
    [Crossref] [PubMed]
  9. L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: a new wavefront reconstruction device,” IBM J. Res. Develop. 13(2), 150–155 (1969).
    [Crossref]
  10. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  11. A. N. Tikhonov, “The solution of ill-posed problems and the regularization method,” Dokl. Akad. Nauk SSSR 151(3), 501–504 (1963).
  12. C. W. Groetsch and A. Neubauer, “Regularization of ill-posed problems: Optimal parameter choice in finite dimensions,” J. Approx. Theory 58(2), 184–200 (1989).
    [Crossref]
  13. A. N. Tikhonov, A. V. Goncharsky, and V. V. Stepanov, “Inverse problems in image processing,” in Ill-Posed Problems in the Natural Sciences, pp. 220–232 (Mir Publishers, 1987).
  14. A. Bakushinsky and A. Goncharsky, Ill-Posed Problems: Theory and Applications (Springer Netherlands, 1994).
  15. D. C. O’Shea, Diffractive Optics: Design, Fabrication, and Test (SPIE Press, 2004).
  16. T. Shimobaba, T. Kakue, Y. Endo, R. Hirayama, D. Hiyama, S. Hasegawa, Y. Nagahama, M. Sano, M. Oikawa, T. Sugie, and T. Ito, “Random phase-free kinoform for large objects,” Opt. Express 23(13), 17269–17274 (2015).
    [Crossref] [PubMed]
  17. H. Zhang, H. Liu, Z. Lu, and H. Zhang, “Modified phase function model for kinoform lenses,” Appl. Opt. 47(22), 4055–4060 (2008).
    [Crossref] [PubMed]
  18. H. C. Hunt and J. S. Wilkinson, “Kinoform microlenses for focusing into microfluidic channels,” Opt. Express 20(9), 9442–9457 (2012).
    [Crossref] [PubMed]
  19. L. Alianelli, K. J. S. Sawhney, R. Barrett, I. Pape, A. Malik, and M. C. Wilson, “High efficiency nano-focusing kinoform optics for synchrotron radiation,” Opt. Express 19(12), 11120–11127 (2011).
    [Crossref] [PubMed]
  20. C. Romero, R. Borrego-Varillas, A. Camino, G. Mínguez-Vega, O. Mendoza-Yero, J. Hernández-Toro, and J. R. Vázquez de Aldana, “Diffractive optics for spectral control of the supercontinuum generated in sapphire with femtosecond pulses,” Opt. Express 19(6), 4977–4984 (2011).
    [Crossref] [PubMed]

2015 (1)

2012 (1)

2011 (2)

2009 (1)

2008 (1)

2005 (1)

2003 (1)

2002 (1)

1989 (1)

C. W. Groetsch and A. Neubauer, “Regularization of ill-posed problems: Optimal parameter choice in finite dimensions,” J. Approx. Theory 58(2), 184–200 (1989).
[Crossref]

1969 (1)

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: a new wavefront reconstruction device,” IBM J. Res. Develop. 13(2), 150–155 (1969).
[Crossref]

1963 (1)

A. N. Tikhonov, “The solution of ill-posed problems and the regularization method,” Dokl. Akad. Nauk SSSR 151(3), 501–504 (1963).

Alianelli, L.

Barrett, R.

Borrego-Varillas, R.

Camino, A.

Endo, Y.

Feldman, M.

Fu, S.

Groetsch, C. W.

C. W. Groetsch and A. Neubauer, “Regularization of ill-posed problems: Optimal parameter choice in finite dimensions,” J. Approx. Theory 58(2), 184–200 (1989).
[Crossref]

Hamamoto, T.

Hartley, F.

Hasegawa, S.

Hernández-Toro, J.

Hirayama, R.

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: a new wavefront reconstruction device,” IBM J. Res. Develop. 13(2), 150–155 (1969).
[Crossref]

Hiyama, D.

Hong, Y.

Hunt, H. C.

Huo, T.

Ito, T.

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: a new wavefront reconstruction device,” IBM J. Res. Develop. 13(2), 150–155 (1969).
[Crossref]

Kakue, T.

Kolste, R.

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: a new wavefront reconstruction device,” IBM J. Res. Develop. 13(2), 150–155 (1969).
[Crossref]

Liu, H.

Liu, Y.

Lu, Z.

Malik, A.

Mendoza-Yero, O.

Mínguez-Vega, G.

Mouroulis, P.

Nagahama, Y.

Neubauer, A.

C. W. Groetsch and A. Neubauer, “Regularization of ill-posed problems: Optimal parameter choice in finite dimensions,” J. Approx. Theory 58(2), 184–200 (1989).
[Crossref]

Nguyen, S.

Oikawa, M.

Pape, I.

Romero, C.

Sano, M.

Sawhney, K. J. S.

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Supplementary Material (1)

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» Visualization 1: MP4 (774 KB)      Visualization 1

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Figures (8)

Fig. 1
Fig. 1 Scheme of the observation of image switching in the case of the turn of the optical element.
Fig. 2
Fig. 2 Image switch effect in the case of the 90° turn: (a) φ = 0°, (b) φ = 90°, (c) φ = 180°.
Fig. 3
Fig. 3 Partitioning of the optical element into elementary areas.
Fig. 4
Fig. 4 Symmetric and asymmetric microrelief areas.
Fig. 5
Fig. 5 Scattering pattern for elementary areas with asymmetric microrelief.
Fig. 6
Fig. 6 The 180° image switch-effect: (a) φ = 0, (b) φ = 180° (see Visualization 1).
Fig. 7
Fig. 7 Dependence of functional Rn on the number n of iterations.
Fig. 8
Fig. 8 Fragments of the microrelief of the optical element: (a) asymmetric, (b) symmetric

Equations (9)

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γ G i u(ξ,η,00)exp[ ikφ(ξ,η) ]exp[ ik (xξ) 2 + (yη) 2 2f ]dξdη =u( x,y,f ).
Aφ=F( x,y ).
F( x,y )={ C 1 ,(x,y) Q 1 C 2 ,(x,y) Q 2 , C 1 >> C 2 .
Aφ=| γ G i u(ξ,η,00)exp[ ikφ(ξ,η) ]exp[ ik (xξ) 2 + (yη) 2 2f ]dξdη |.
w(x,y)=Φ{v}(x,y)= k 2πf G v(ξ,η)exp[ ik (xξ) 2 + (yη) 2 2f ]dξdη .
v ˜ (k) =Φ{ v (k) }(x,y)= W k (x,y)exp[ik φ 1 (k) (x,y)].
w (k) ( x,y ) = A 1 exp[ ik φ 1 (k) ( x,y ) ].
w ˜ (k) (x,y)= Φ 1 { w (k) }(x,y)= V k (x,y)exp[ik φ 0 (k+1) (x,y)].
v (k+1) ( x,y )= A 0 ( x,y ) exp[ ik φ 0 (k+1) ( x,y ) ].

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