We develop a method for encoding information in the longitudinal component of a focused field. Focused beams display a non-zero contribution of the electric field in the direction of propagation. However, the associated irradiance is very weak and difficult to isolate from the transverse part of the beam. For these reasons, the longitudinal component of a focused field could be a good choice for encoding and securing information. Using the Richards and Wolf formalism we show how to encrypt information in the longitudinal domain of the focal area. In addition, we use quantum imaging techniques to enhance the security and to prevent unauthorized access to the information. To the best of our knowledge, this is the first report on using the longitudinal component of the focused fields in optical security.
© 2016 Optical Society of America
Nowadays, highly focused beams are present in numerous research areas and technical applications [1–10]. However, to the best of our knowledge no research has been reported in the field of optical security [11–13], using focused fields. It is well known that the electric field associated to a plane wave is transverse to the direction of propagation. However, in general this is not true for converging beams since a non-zero contribution of the electric field in the direction of propagation appears. This fact is a consequence of the Maxwell’s Equations. Nevertheless, the irradiance of this longitudinal component is small compared to the energy associated to the transverse components, even when the beam is focused with a high numerical aperture objective lens. In fact, the transverse part of the field completely embeds the longitudinal irradiance. Furthermore, it is not possible to isolate the irradiance of the longitudinal component by using holographic techniques or by means of polarizers. Taking focused fields into account, it seems appropriate to hide and/or secure information in the longitudinal component. Thus, focused fields can be used for optical security provided an authorized user is able to access the encoded message. Despite the fact that the longitudinal component of the field cannot be easily accessed, the Gauss law provides a mean to numerically access the encoded information by using the transverse field distribution. This encoding procedure can be used in combination with a variety of optical encryption techniques, and providing an extra layer of security.
In this paper, we describe how to encode information in the longitudinal component of a focused field within the framework of the Richardson and Wolf vector propagation theory . We demonstrate that the signal can be encrypted to be equivalent to the cypher-text obtained using classical Double Random Phase Encryption (DRPE) . Consequently, additional improvements with optical security techniques [16–22] could be adapted to be used with focused beams. To avoid conventional attacks against the information encrypted in the longitudinal component [23–30], the use of quantum imaging techniques is suggested [31–33].
The paper is organized as follows: in section 2 we summarize basic concepts in vector diffraction theory, and in section 3 we introduce a method for encoding information in the longitudinal component of a focused field. In section 4, we present how the codification technique is adapted for obtaining encrypted signals in the longitudinal domain. Finally, the conclusions are presented in section 5.
2. Review on highly focused beams
The electric field E at the focal area of a high numerical aperture (NA) microscope lens following the sine condition is described by the so-called Richards-Wolf integral :Fig. 1 for details. E∞ is the field at the Gaussian sphere of reference, described as:Eq. (2), P(θ) is the so-called apodization function; in particular, for isoplanatic optical systems following the sine condition ; e1 and ei2 are unit vectors in the radial and azimuthal directions, and e2 is the projection of ei2 on the convergent wave-front surface:
The Richards-Wolf integral can be rewritten in a more compact way by using Fourier transforms. The first exponential term in Eq. (1) is developed as follows:Fig. 1, x∞ = fα and y∞ = fβ, and, where f is the focal length of the objective lens. In addition, surface differentials are related by (see chapter 3 of reference  for details). Equation (1) at z = 0 becomes:
It is worth to point out that even though the incident beam E0 is assumed to be purely transverse, the electric field at the focal plane E(x,y,0) shows a non-negligible longitudinal component Ez . Thus, the polarization has to be described as a 3D phenomenon. The radial part of E∞ generates the longitudinal component of the focused field since vector e2 is not transverse. On the other hand, an azimuthal beam with b = 0 produces a purely transverse focused field, i.e. with a longitudinal component Ez = 0.
3. Information encoding
A remarkable property of focused fields is the irradiance associated to the longitudinal component Iz35]. The z-component of the focal electric field cannot be separated from the other two components using linear polarizers and/or holographic recording. Thus, it cannot be accessed by direct observation using conventional optical equipment. For this reason, the use of highly focused fields in optical security enables the possibility to securely encode information in the longitudinal component Ez. Since the energy associated with the longitudinal component is very weak, the information is embedded by the transverse part of the focused field. This encoding approach can be understood as a way of implementing steganography using the physical properties of focused light beams.
According to Eqs. (2) and (3), the longitudinal component of the vector angular spectrum reads:Eq. (7):
Despite the fact that Iz is very weak compared with the total irradiance of the focused field and Ez cannot be isolated from the transverse part of the beam, the longitudinal component can be accessed using the condition s · E∞ = 0, or with. Then, the longitudinal component Ez can be deduced from:Eq. (15) indicates a practical way to estimate the component Ez.
4. Encryption and validation
In the previous section, we have demonstrated a method for encoding information in the longitudinal field of a highly focused beam. Among the different possibilities for encrypting information in the longitudinal component Ez using the present approach, we have selected the simplest one. Let M1 and M2 random phase masks (keys), and t the plain-text to be encoded [Fig. 2]. Using circularly polarized light [Eq. (13)], the encrypted components of the input beam E’0x and E’0y are:15,16]:Eq. (15) and (16), t can be determined from the information contained in the focused encoded components E’x and E’y, provided M2 is known.
Note that the present encoding system shares the same weakness of the DRPE method. Despite the fact that a plurality of attacks have been designed to break DRPE systems [23–30], different approaches were suggested to improved security in DRPE. For instance, it has been demonstrated that quantum encryption systems that works with few photons are very secure [31,32]. In this case, the encrypted signal is no longer accessible but it can be authenticated. Moreover, note that other non-linear encryption procedures can be implemented in the longitudinal domain as well.
If a system works in low light illumination conditions, irradiance is recorded according to the photon-counting model. It is assumed that, in these conditions the image is statistically modeled by the Poisson distribution . The photon-counting binary version |E’x|ph of |E’x| is obtained according to:Eq. (18). To determine whether tph contains information related with t or not, the correlation coefficient ρ calculated at pixel (x,y) is:
The presented encryption procedure could be implemented in practice using an optical setup able to generate highly focused fields using only conventional components. To provide more insight, we suggest a possible design for a practical implementation. This system is sketched in Fig. 2(a). First, image t is phase-encoded using a random code such as a diffuser (mask M1) and illuminated by a circularly polarized coherent source. The set M1t is located in the front focal plane of lens L1. This distribution is optically Fourier transformed using lens L1. A transmission type modulator is placed in the back focal plane of L1. Half-wave plate HWP and quarter-wave plate QWP are used to set up a twisted nematic modulator in order to achieve a phase-mostly configuration. Using Arrizon’s cell-oriented codification method , it is possible to achieve full complex modulation: a certain value in the complex plane can be accessed as a combination of two points that belong to the modulation curve. A detailed explanation of the implementation of this procedure can be found in references [38,39]. This device displays hologram H containing the following informationEq. (18).
Note that misalignment is a very serious problem that can jeopardize the encryption procedure. Since the matching procedure can be a laborious task, instead of using lens L1 to produce the Fourier transform of the input signal, distribution H FT[M1 t] can be displayed directly on the SLM, being lens L1 no longer necessary.
5. Numerical tests
Some calculations were carried out to demonstrate how the system works. A 512x512 pixels image of Lena is used as the plaintext t to be encoded in the longitudinal domain Ez. Figure 3 displays |E’x|2 and |E’z|2. Note that |E’x|2 = |E’y|2 whereas |E’z|2 is a random distribution.
Figure 4 shows photon counting versions of |E’x|ph, |E’y|ph. NP is set to 10% of the pixels of the image.
Figs. 5(a) and 5(b) show the recovered signals t and tph. Correlation ρ when the correct key mask M2 and an incorrect key mask M2 are used are presented in Figs. 5(c) and 5(d). As expected, tph does not provide any visual information of the plain-text image, but correlation ρ between tph and t shows a clear peak when the proper key mask is used. Figures 5(e) and 5(f) display correlation ρ with the true and false phase masks but using a higher number of photons (NP = 0.15 photons/pixel).
References and links
2. N. Davidson and N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29(12), 1318–1320 (2004). [CrossRef] [PubMed]
5. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008). [CrossRef]
8. S. N. Khonina and S. G. Volotovsky, “Controlling the contribution of the electric field components to the focus of a high-aperture lens using binary phase structures,” J. Opt. Soc. Am. A 27(10), 2188–2197 (2010). [CrossRef] [PubMed]
9. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009). [CrossRef]
11. O. Matoba, T. Nomura, E. Pérez-Cabré, M. S. Millan, and B. Javidi, “Optical Techniques for Information Security,” Proc. IEEE 97(6), 1128–1148 (2009). [CrossRef]
12. B. Javidi and A. Carnicer, “Roadmap in optical encryption and security,” J. Opt. (accepted for publication).
13. W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photonics 6(2), 120–155 (2014). [CrossRef]
14. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” P. Royal Soc. London A Mater. 253(1274), 358–379 (1959). [CrossRef]
16. B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33(6), 1752–1756 (1994). [CrossRef]
21. J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encrypted data by using polarized light,” Opt. Commun. 260(1), 109–112 (2006). [CrossRef]
23. A. Carnicer, M. Montes-Usategui, S. Arcos, and I. Juvells, “Vulnerability to chosen-cyphertext attacks of optical encryption schemes based on double random phase keys,” Opt. Lett. 30(13), 1644–1646 (2005). [CrossRef] [PubMed]
26. H. Tashima, M. Takeda, H. Suzuki, T. Obi, M. Yamaguchi, and N. Ohyama, “Known plaintext attack on double random phase encoding using fingerprint as key and a method for avoiding the attack,” Opt. Express 18(13), 13772–13781 (2010). [CrossRef] [PubMed]
28. P. Kumar, A. Kumar, J. Joseph, and K. Singh, “Impulse attack free double-random-phase encryption scheme with randomized lens-phase functions,” Opt. Lett. 34(3), 331–333 (2009). [CrossRef] [PubMed]
30. K. Nakano, M. Takeda, H. Suzuki, and M. Yamaguchi, “Security analysis of phase-only DRPE based on known-plaintext attack using multiple known plaintext-ciphertext pairs,” Appl. Opt. 53(28), 6435–6443 (2014). [CrossRef] [PubMed]
34. L. Novotni and B. Hecht, Principles of Nano-Optics (Cambridge University, 2012)
35. A. Carnicer, I. Juvells, D. Maluenda, R. Martínez-Herrero, and P. M. Mejías, “On the longitudinal component of paraxial fields,” Eur. J. Phys. 33(5), 1235–1247 (2012). [CrossRef]
36. J. W. Goodman, Statistical Optics (John Wiley & Sons, 1985)
37. V. Arrizón, L. A. González, R. Ponce, and A. Serrano-Heredia, “Computer-generated holograms with optimum bandwidths obtained with twisted-nematic liquid-crystal displays,” Appl. Opt. 44(9), 1625–1634 (2005). [CrossRef] [PubMed]
38. D. Maluenda, R. Martínez-Herrero, I. Juvells, and A. Carnicer, “Synthesis of highly focused fields with circular polarization at any transverse plane,” Opt. Express 22(6), 6859–6867 (2014). [CrossRef] [PubMed]
39. D. Maluenda, I. Juvells, R. Martínez-Herrero, and A. Carnicer, “Reconfigurable beams with arbitrary polarization and shape distributions at a given plane,” Opt. Express 21(5), 5432–5439 (2013). [CrossRef] [PubMed]