Abstract

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

1. Introduction

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 [14]. We demonstrate that the signal can be encrypted to be equivalent to the cypher-text obtained using classical Double Random Phase Encryption (DRPE) [15]. 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 [14]:

E(r,φ,z)=A0θ002πE(θ,ϕ)exp(ikrsinθcos(φϕ))exp(ikzcosθ)sinθdθdϕ
where A is a proportionality constant, k is the wavenumber, r and ϕ are the polar coordinates at the focal plane, θ and φ are the polar and azimuthal angles at the exit pupil and θ0 is the semi-aperture angle, i.e. NA = sin θ0. See Fig. 1 for details. E is the field at the Gaussian sphere of reference, described as:
E(θ,ϕ)=P(θ)(ae1(ϕ)+be2(ϕ,θ)).
This field can also be understood as the vector angular spectrum of plane waves. In Eq. (2), P(θ) is the so-called apodization function; in particular, for isoplanatic optical systems following the sine condition P(θ)=cosθ; 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:
e1(ϕ)=(sinϕ,cosϕ,0)e2i(ϕ)=(cosϕ,sinϕ,0)e2(ϕ,θ)=(cosθcosϕ,cosθsinϕ,sinθ).
The wave-front vector s is defined as:
s=(α,β,γ)=(sinθcosϕ,sinθsinϕ,cosθ).
Here, e1, e2 and s form a triad of a mutually orthogonal right-handed system of unit vectors. Note that E is normal to the wavefront vector s, that is s · E = 0. Distributions a and b are the azimuthal and radial parts of input field E0 assumed transverse E0 = (E0x, E0y, 0):

 figure: Fig. 1

Fig. 1 Notation and coordinate systems at the Gaussian reference sphere and at the focal plane.

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a=E0e1=E0xsinϕ+E0ycosϕb=E0e2i=E0xcosϕ+E0ysinϕ.

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:

exp(ikrsinθcos(φϕ))=exp(i2πλ(αx+βy))
where x = r cosφ and y = r sinφ are the rectangular coordinates at the focal plane, and λ is the wavelength of the illumination source. Let (x∞, y∞, z) be the coordinates of a point on the Gaussian sphere of reference. According to Fig. 1, x = fα and y = fβ, andexp(i2πλ(αx+βy))=exp(i2πλf(xx+yy)), where f is the focal length of the objective lens. In addition, surface differentials are related by sinθdθdϕ=1f2cosθdxdy (see chapter 3 of reference [34] for details). Equation (1) at z = 0 becomes:
E(x,y,0)=FTλf[Ecosθ]=FTλf[((E0e1)e1+(E0e2ι)e2)cosθ],
where operator FTλf[] stands for the λf–scaled Fourier transform.

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 Iz

Iz=|Ez|2dxdy.
This longitudinal component is very small compared with the irradiance of the total field E:
IT=Ix+Iy+Iz=(|Ex|2+|Ey|2+|Ez|2)dxdy.
For instance, for a circularly polarized input beam used in combination with an objective lens NA = 0.9, Iz ≈0.003IT [35]. 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:

Ez=cosθbsinθ=cosθ(E0e2i)sinθ=cosθ(E0xcosϕ+E0ysinϕ)sinθ..
The longitudinal or z-component of the focused field is obtained using Eq. (7):
Ez(x,y,0)=FTλf[Ezcosθ]=FTλf[(E0xcosϕ+E0ysinϕ)sinθcosθ].
This equation provides a simple relationship between the longitudinal component of the focused field Ez and the input beam E0. Equivalently,
E0xcosϕ+E0ysinϕ=cosθsinθFTλf1[Ez(x,y,0)].
Depending on how the system is illuminated, E∞z is described differently. For instance, if the input beam is circularly polarized E0 = (E0, iE0, 0), thenEz=E0cosθsinθexp(iϕ)and
E0=exp(iϕ)cosθsinθFTλf1[Ez],Ez=FTλf[exp(iϕ)E0sinθcosθ].
Alternatively, if the system is illuminated with a radially polarized beam, then E0 = (E0 cosϕ, E0 sinϕ, 0) thenEz=E0cosθsinθ. In this case
E0=cosθsinθFTλf1[Ez],Ez=FTλf[E0sinθcosθ].
Interestingly, azimuthally polarized beams, that is E0 = (-E0 sinϕ, E0 cosϕ, 0), produce purely transverse distributions of light at the focal plane, i.e. E∞z = 0. This means that this polarization cannot be used since no longitudinal component is generated.

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 αEx+βEy+γEz=0 withγ=1α2β2. Then, the longitudinal component Ez can be deduced from:

Ez=FTλf[αFTλf1[Ex]+βFTλf1[Ey]1α2β2].
Because the transverse components of the focused field Ex and Ey can be determined experimentally (see section 4), 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:

E'0x=exp(iϕ)cosθsinθM2FTλf[M1t],E'0y=iE'0x.
Interestingly, in this case the generated cypher-text E’z is identical to the one obtained with the classical double random phase encryption procedure (DRPE) [15,16]:
Ez'=FTλf[exp(iϕ)E'0xsinθcosθ]=FTλf[M2FTλf[M1t]].
As indicated in 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.

 figure: Fig. 2

Fig. 2 (a) Optical setup to implement for the proposed encryption procedure: L1: Fourier lens; LA and LB: relay optics lenses; f1, fA, fB and f2 are the focal lengths of lenses L1, LA, LB and the microscope objective respectively; SLM: spatial light modulator; HWP and QWP: half and quarter wave plates; LP: linear polarizer; CCD: camera.

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t=|FTλf1[M21αFTλf1[E'x]+βFTλf1[E'y]1α2β2]|.

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 [36]. The photon-counting binary version |E’x|ph of |E’x| is obtained according to:

|Ex'|ph(x,y)={0,ifrand(x,y)exp(np(x,y))1,otherwise
where rand(x, y) is a uniformly distributed random number within the range [0,1], NP is the predetermined number of photon counts in the entire scene, NxM is the total number of pixels and np(x,y) is the normalized irradiance at pixel (x, y):
np(x,y)=Np|Ex'(x,y)|2n,m=1N,M|Ex'(n,m)|2
m and n are the summation indices and |E’y|ph is generated using the same approach. The encrypted signal uses the photon-counting version of E’x and E’y but the phase remains the same:
Ex'ph=|Ex'|phEx'|Ex'|andEy'ph=|Ey'|phEy'|Ey|.
Finally, tph is estimated by means of Eq. (18). To determine whether tph contains information related with t or not, the correlation coefficient ρ calculated at pixel (x,y) is:
ρ(x,y)=n,m=1N,M[tph(m+x,n+y)tph][t(m,n)t]n,m=1N,M[tph(m,n)tph]2n,m=1N,M[t(m,n)t]2,
where tph and t are respectively the mean values of tph and t.

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 [37], 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 information

H=exp(iϕ)cosθsinθM2.
The SLM plane is imaged on the entrance pupil plane of the microscope objective using lenses LA and LB in a 4f configuration. The spatial filter in the back focal plane of lens LA is required to remove non-desired high order terms produced during the codification. Interestingly, the informative part of the beam is propagated on axis. Then, the field E’x is focused by means of an objective lens and the transverse part of field is recorded by a CCD camera. In order to preserve the phase of the encrypted field, the interference between the field E’x and a reference beam is recorded. E’y is accessed in a similar way, by rotating 90° polarizer LP. Using encrypted components E’x, and E’y, and the correct mask M2, plain-text t can be numerically accessed with the help of Eq. (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: Fig. 3

Fig. 3 Encrypted components: (a) |E’x|2 and (b) |E’z|2.

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Figure 4 shows photon counting versions of |E’x|ph, |E’y|ph. NP is set to 10% of the pixels of the image.

 figure: Fig. 4

Fig. 4 Photon counting encrypted components: (a) |E’x|ph, (b) |E’y|ph.

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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).

 figure: Fig. 5

Fig. 5 Recovered signals using the correct key M2: (a) decrypted plaintext t (b) photon-counting decrypted plaintext tph, (c) correlation signal ρ using the correct key M2 and NP = 0.1 photons/pixel, (d) ρ using an incorrect key M2 and NP = 0.1 photons/pixel, (e) correlation signal ρ using the correct key M2 and NP = 0.15 photons/pixel, (f) ρ using an incorrect key M2 and NP = 0.15 photons/pixel.

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References and links

1. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef]   [PubMed]  

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]  

3. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14(23), 11277–11291 (2006). [CrossRef]   [PubMed]  

4. Y. Kozawa and S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A 24(6), 1793–1798 (2007). [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]  

6. G. M. Lerman and U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express 16(7), 4567–4581 (2008). [CrossRef]   [PubMed]  

7. X. Hao, C. Kuang, T. Wang, and X. Liu, “Phase encoding for sharper focus of the azimuthally polarized beam,” Opt. Lett. 35(23), 3928–3930 (2010). [CrossRef]   [PubMed]  

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]  

10. R. Martínez-Herrero, I. Juvells, and A. Carnicer, “On the physical realizability of highly focused electromagnetic field distributions,” Opt. Lett. 38(12), 2065–2067 (2013). [CrossRef]   [PubMed]  

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]  

15. P. Réfrégier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995). [CrossRef]   [PubMed]  

16. B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33(6), 1752–1756 (1994). [CrossRef]  

17. O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24(11), 762–764 (1999). [CrossRef]   [PubMed]  

18. G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. 25(12), 887–889 (2000). [CrossRef]   [PubMed]  

19. X. Tan, O. Matoba, Y. Okada-Shudo, M. Ide, T. Shimura, and K. Kuroda, “Secure optical memory system with polarization encryption,” Appl. Opt. 40(14), 2310–2315 (2001). [CrossRef]   [PubMed]  

20. O. Matoba and B. Javidi, “Secure holographic memory by double-random polarization encryption,” Appl. Opt. 43(14), 2915–2919 (2004). [CrossRef]   [PubMed]  

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]  

22. W. Chen, X. Chen, and C. J. R. Sheppard, “Optical image encryption based on diffractive imaging,” Opt. Lett. 35(22), 3817–3819 (2010). [CrossRef]   [PubMed]  

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]  

24. X. Peng, P. Zhang, H. Wei, and B. Yu, “Known-plaintext attack on optical encryption based on double random phase keys,” Opt. Lett. 31(8), 1044–1046 (2006). [CrossRef]   [PubMed]  

25. Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15(16), 10253–10265 (2007). [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]  

27. K. Nakano, M. Takeda, H. Suzuki, and M. Yamaguchi, “Evaluations of phase-only double random phase encoding based on key-space analysis,” Appl. Opt. 52(6), 1276–1283 (2013). [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]  

29. T. J. Naughton, B. M. Hennelly, and T. Dowling, “Introducing secure modes of operation for optical encryption,” J. Opt. Soc. Am. A 25(10), 2608–2617 (2008). [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]  

31. E. Pérez-Cabré, M. Cho, and B. Javidi, “Information authentication using photon-counting double-random-phase encrypted images,” Opt. Lett. 36(1), 22–24 (2011). [CrossRef]   [PubMed]  

32. M. Cho and B. Javidi, “Three-dimensional photon counting double-random-phase encryption,” Opt. Lett. 38(17), 3198–3201 (2013). [CrossRef]   [PubMed]  

33. D. Maluenda, A. Carnicer, R. Martínez-Herrero, I. Juvells, and B. Javidi, “Optical encryption using photon-counting polarimetric imaging,” Opt. Express 23(2), 655–666 (2015). [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]  

References

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  1. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
    [Crossref] [PubMed]
  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]
  3. M. Leutenegger, R. Rao, R. A. Leitgeb, and T. Lasser, “Fast focus field calculations,” Opt. Express 14(23), 11277–11291 (2006).
    [Crossref] [PubMed]
  4. Y. Kozawa and S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A 24(6), 1793–1798 (2007).
    [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]
  6. G. M. Lerman and U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express 16(7), 4567–4581 (2008).
    [Crossref] [PubMed]
  7. X. Hao, C. Kuang, T. Wang, and X. Liu, “Phase encoding for sharper focus of the azimuthally polarized beam,” Opt. Lett. 35(23), 3928–3930 (2010).
    [Crossref] [PubMed]
  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]
  10. R. Martínez-Herrero, I. Juvells, and A. Carnicer, “On the physical realizability of highly focused electromagnetic field distributions,” Opt. Lett. 38(12), 2065–2067 (2013).
    [Crossref] [PubMed]
  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]
  15. P. Réfrégier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995).
    [Crossref] [PubMed]
  16. B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33(6), 1752–1756 (1994).
    [Crossref]
  17. O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24(11), 762–764 (1999).
    [Crossref] [PubMed]
  18. G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. 25(12), 887–889 (2000).
    [Crossref] [PubMed]
  19. X. Tan, O. Matoba, Y. Okada-Shudo, M. Ide, T. Shimura, and K. Kuroda, “Secure optical memory system with polarization encryption,” Appl. Opt. 40(14), 2310–2315 (2001).
    [Crossref] [PubMed]
  20. O. Matoba and B. Javidi, “Secure holographic memory by double-random polarization encryption,” Appl. Opt. 43(14), 2915–2919 (2004).
    [Crossref] [PubMed]
  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]
  22. W. Chen, X. Chen, and C. J. R. Sheppard, “Optical image encryption based on diffractive imaging,” Opt. Lett. 35(22), 3817–3819 (2010).
    [Crossref] [PubMed]
  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]
  24. X. Peng, P. Zhang, H. Wei, and B. Yu, “Known-plaintext attack on optical encryption based on double random phase keys,” Opt. Lett. 31(8), 1044–1046 (2006).
    [Crossref] [PubMed]
  25. Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15(16), 10253–10265 (2007).
    [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]
  27. K. Nakano, M. Takeda, H. Suzuki, and M. Yamaguchi, “Evaluations of phase-only double random phase encoding based on key-space analysis,” Appl. Opt. 52(6), 1276–1283 (2013).
    [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]
  29. T. J. Naughton, B. M. Hennelly, and T. Dowling, “Introducing secure modes of operation for optical encryption,” J. Opt. Soc. Am. A 25(10), 2608–2617 (2008).
    [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]
  31. E. Pérez-Cabré, M. Cho, and B. Javidi, “Information authentication using photon-counting double-random-phase encrypted images,” Opt. Lett. 36(1), 22–24 (2011).
    [Crossref] [PubMed]
  32. M. Cho and B. Javidi, “Three-dimensional photon counting double-random-phase encryption,” Opt. Lett. 38(17), 3198–3201 (2013).
    [Crossref] [PubMed]
  33. D. Maluenda, A. Carnicer, R. Martínez-Herrero, I. Juvells, and B. Javidi, “Optical encryption using photon-counting polarimetric imaging,” Opt. Express 23(2), 655–666 (2015).
    [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]

2015 (1)

2014 (3)

2013 (4)

2012 (1)

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]

2011 (1)

2010 (4)

2009 (3)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

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]

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]

2008 (3)

2007 (2)

2006 (3)

2005 (2)

2004 (2)

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

2001 (1)

2000 (1)

1999 (1)

1995 (1)

1994 (1)

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33(6), 1752–1756 (1994).
[Crossref]

1959 (1)

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]

Arcos, S.

Arrizón, V.

Barrera, J. F.

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]

Bokor, N.

Bolognini, N.

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]

Carnicer, A.

Castro, A.

Chen, W.

W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photonics 6(2), 120–155 (2014).
[Crossref]

W. Chen, X. Chen, and C. J. R. Sheppard, “Optical image encryption based on diffractive imaging,” Opt. Lett. 35(22), 3817–3819 (2010).
[Crossref] [PubMed]

Chen, X.

W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photonics 6(2), 120–155 (2014).
[Crossref]

W. Chen, X. Chen, and C. J. R. Sheppard, “Optical image encryption based on diffractive imaging,” Opt. Lett. 35(22), 3817–3819 (2010).
[Crossref] [PubMed]

Cho, M.

Chong, C. T.

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]

Davidson, N.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Dowling, T.

Frauel, Y.

González, L. A.

Hao, X.

Henao, R.

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]

Hennelly, B. M.

Horner, J. L.

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33(6), 1752–1756 (1994).
[Crossref]

Ide, M.

Javidi, B.

D. Maluenda, A. Carnicer, R. Martínez-Herrero, I. Juvells, and B. Javidi, “Optical encryption using photon-counting polarimetric imaging,” Opt. Express 23(2), 655–666 (2015).
[Crossref] [PubMed]

W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photonics 6(2), 120–155 (2014).
[Crossref]

M. Cho and B. Javidi, “Three-dimensional photon counting double-random-phase encryption,” Opt. Lett. 38(17), 3198–3201 (2013).
[Crossref] [PubMed]

E. Pérez-Cabré, M. Cho, and B. Javidi, “Information authentication using photon-counting double-random-phase encrypted images,” Opt. Lett. 36(1), 22–24 (2011).
[Crossref] [PubMed]

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]

Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15(16), 10253–10265 (2007).
[Crossref] [PubMed]

O. Matoba and B. Javidi, “Secure holographic memory by double-random polarization encryption,” Appl. Opt. 43(14), 2915–2919 (2004).
[Crossref] [PubMed]

O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24(11), 762–764 (1999).
[Crossref] [PubMed]

P. Réfrégier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995).
[Crossref] [PubMed]

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33(6), 1752–1756 (1994).
[Crossref]

B. Javidi and A. Carnicer, “Roadmap in optical encryption and security,” J. Opt. (accepted for publication).

Joseph, J.

Juvells, I.

Khonina, S. N.

Kozawa, Y.

Kuang, C.

Kumar, A.

Kumar, P.

Kuroda, K.

Lasser, T.

Leitgeb, R. A.

Lerman, G. M.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Leutenegger, M.

Levy, U.

Liu, X.

Lukyanchuk, B.

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]

Maluenda, D.

Martínez-Herrero, R.

Matoba, O.

Mejías, P. M.

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]

Millan, M. S.

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]

Montes-Usategui, M.

Nakano, K.

Naughton, T. J.

Nomura, T.

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]

Obi, T.

Ohyama, N.

Okada-Shudo, Y.

Peng, X.

Pérez-Cabré, E.

E. Pérez-Cabré, M. Cho, and B. Javidi, “Information authentication using photon-counting double-random-phase encrypted images,” Opt. Lett. 36(1), 22–24 (2011).
[Crossref] [PubMed]

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]

Ponce, R.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Rao, R.

Réfrégier, P.

Richards, B.

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]

Sato, S.

Serrano-Heredia, A.

Sheppard, C.

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]

Sheppard, C. J. R.

Shi, L.

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]

Shimura, T.

Singh, K.

Suzuki, H.

Takeda, M.

Tan, X.

Tashima, H.

Tebaldi, M.

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]

Torroba, R.

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]

Unnikrishnan, G.

Volotovsky, S. G.

Wang, H.

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]

Wang, T.

Wei, H.

Wolf, E.

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]

Yamaguchi, M.

Yu, B.

Zhan, Q.

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

Zhang, P.

Adv. Opt. Photonics (2)

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photonics 6(2), 120–155 (2014).
[Crossref]

Appl. Opt. (5)

Eur. J. Phys. (1)

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]

J. Opt. Soc. Am. A (3)

Nat. Photonics (1)

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]

Opt. Commun. (1)

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]

Opt. Eng. (1)

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33(6), 1752–1756 (1994).
[Crossref]

Opt. Express (7)

Opt. Lett. (12)

P. Réfrégier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995).
[Crossref] [PubMed]

E. Pérez-Cabré, M. Cho, and B. Javidi, “Information authentication using photon-counting double-random-phase encrypted images,” Opt. Lett. 36(1), 22–24 (2011).
[Crossref] [PubMed]

M. Cho and B. Javidi, “Three-dimensional photon counting double-random-phase encryption,” Opt. Lett. 38(17), 3198–3201 (2013).
[Crossref] [PubMed]

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]

W. Chen, X. Chen, and C. J. R. Sheppard, “Optical image encryption based on diffractive imaging,” Opt. Lett. 35(22), 3817–3819 (2010).
[Crossref] [PubMed]

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]

X. Peng, P. Zhang, H. Wei, and B. Yu, “Known-plaintext attack on optical encryption based on double random phase keys,” Opt. Lett. 31(8), 1044–1046 (2006).
[Crossref] [PubMed]

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]

X. Hao, C. Kuang, T. Wang, and X. Liu, “Phase encoding for sharper focus of the azimuthally polarized beam,” Opt. Lett. 35(23), 3928–3930 (2010).
[Crossref] [PubMed]

O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24(11), 762–764 (1999).
[Crossref] [PubMed]

G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. 25(12), 887–889 (2000).
[Crossref] [PubMed]

R. Martínez-Herrero, I. Juvells, and A. Carnicer, “On the physical realizability of highly focused electromagnetic field distributions,” Opt. Lett. 38(12), 2065–2067 (2013).
[Crossref] [PubMed]

P. Royal Soc. London A Mater. (1)

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]

Phys. Rev. Lett. (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

Proc. IEEE (1)

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]

Other (3)

B. Javidi and A. Carnicer, “Roadmap in optical encryption and security,” J. Opt. (accepted for publication).

L. Novotni and B. Hecht, Principles of Nano-Optics (Cambridge University, 2012)

J. W. Goodman, Statistical Optics (John Wiley & Sons, 1985)

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Figures (5)

Fig. 1
Fig. 1 Notation and coordinate systems at the Gaussian reference sphere and at the focal plane.
Fig. 2
Fig. 2 (a) Optical setup to implement for the proposed encryption procedure: L1: Fourier lens; LA and LB: relay optics lenses; f1, fA, fB and f2 are the focal lengths of lenses L1, LA, LB and the microscope objective respectively; SLM: spatial light modulator; HWP and QWP: half and quarter wave plates; LP: linear polarizer; CCD: camera.
Fig. 3
Fig. 3 Encrypted components: (a) |E’x|2 and (b) |E’z|2.
Fig. 4
Fig. 4 Photon counting encrypted components: (a) |E’x|ph, (b) |E’y|ph.
Fig. 5
Fig. 5 Recovered signals using the correct key M2: (a) decrypted plaintext t (b) photon-counting decrypted plaintext tph, (c) correlation signal ρ using the correct key M2 and NP = 0.1 photons/pixel, (d) ρ using an incorrect key M2 and NP = 0.1 photons/pixel, (e) correlation signal ρ using the correct key M2 and NP = 0.15 photons/pixel, (f) ρ using an incorrect key M2 and NP = 0.15 photons/pixel.

Equations (23)

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E(r,φ,z)=A 0 θ 0 0 2π E ( θ,ϕ )exp( ikrsinθcos( φϕ ) )exp( ikzcosθ )sinθdθdϕ
E ( θ,ϕ )=P( θ )( a e 1 ( ϕ )+b e 2 ( ϕ,θ ) ).
e 1 ( ϕ )=(sinϕ,cosϕ,0) e 2 i ( ϕ )=(cosϕ,sinϕ,0) e 2 ( ϕ,θ )=(cosθcosϕ,cosθsinϕ,sinθ).
s=(α,β,γ)=( sinθcosϕ,sinθsinϕ,cosθ ).
a= E 0 e 1 = E 0x sinϕ+ E 0y cosϕ b= E 0 e 2 i = E 0x cosϕ+ E 0y sinϕ.
exp( ikrsinθcos( φϕ ) )=exp( i 2π λ ( αx+βy ) )
E(x,y,0)= FT λf [ E cosθ ]= FT λf [ ( ( E 0 e 1 ) e 1 +( E 0 e 2 ι ) e 2 ) cosθ ],
I z = | E z | 2 dxdy.
I T = I x + I y + I z = ( | E x | 2 + | E y | 2 + | E z | 2 ) dxdy.
E z = cosθ bsinθ= cosθ ( E 0 e 2 i )sinθ= cosθ ( E 0x cosϕ+ E 0y sinϕ )sinθ..
E z (x,y,0)= FT λf [ E z cosθ ]= FT λf [ ( E 0x cosϕ+ E 0y sinϕ ) sinθ cosθ ].
E 0x cosϕ+ E 0y sinϕ= cosθ sinθ FT λf 1 [ E z (x,y,0) ].
E 0 =exp( iϕ ) cosθ sinθ FT λf 1 [ E z ], E z = FT λf [ exp( iϕ ) E 0 sinθ cosθ ].
E 0 = cosθ sinθ FT λf 1 [ E z ], E z = FT λf [ E 0 sinθ cosθ ].
E z = FT λf [ α FT λf 1 [ E x ]+β FT λf 1 [ E y ] 1 α 2 β 2 ].
E ' 0x =exp( iϕ ) cosθ sinθ M 2 FT λf [ M 1 t ], E ' 0y =iE ' 0x .
E z ' = FT λf [ exp( iϕ )E ' 0x sinθ cosθ ]= FT λf [ M 2 FT λf [ M 1 t ] ].
t=| FT λf 1 [ M 2 1 α FT λf 1 [ E ' x ]+β FT λf 1 [ E ' y ] 1 α 2 β 2 ] |.
| E x ' | ph ( x, y )={ 0, if rand( x, y )exp( n p ( x,y ) ) 1, otherwise
n p ( x,y )= N p | E x ' ( x,y ) | 2 n,m=1 N,M | E x ' ( n,m ) | 2
E x 'ph = | E x ' | ph E x ' | E x ' | and E y 'ph = | E y ' | ph E y ' | E y | .
ρ( x,y ) = n,m=1 N,M [ t ph ( m+x,n+y ) t ph ][ t( m,n ) t ] n,m=1 N,M [ t ph ( m,n ) t ph ] 2 n,m=1 N,M [ t( m,n ) t ] 2 ,
H=exp( iϕ ) cosθ sinθ M 2 .

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