All-optical &#x1D4AB;&#x1D4AF;-symmetric conversion of amplitude (phase) modulation to phase (amplitude) modulation

Oscar Ignacio Zaragoza Gutiérrez; Luis Felipe Salinas Mendoza; B. M. Rodríguez-Lara

doi:10.1364/OE.24.003989

1. Introduction

Optical communications are a necessity of modern life [1, 2]. All-optical systems based on monolithic photonic integrated circuits may prove a compact, robust and ultrafast alternative with low-dissipation to some optoelectronic devices [3]. Since the report of the optical directional coupler with attenuated waveguides [4] and the nonlinear coherent coupler [5], three-waveguide nonlinear directional couplers have played an important role in the field of integrated photonics; for example, devices based on the Kerr nonlinearity have been used to provide all-optical spatial switching [6–12] and logic gates [13–15].

Here, we are interested in providing another key element to optical communications, the all-optical conversion of amplitude (phase) modulation to phase (amplitude) modulation, using the well-developed planar three-waveguide coupler architecture with a gain mechanism different from the Kerr nonlinearity. The conversion from amplitude to phase modulation is important because phase shift keying formats in coherent optical communications allow for symbol distance transmissions that are twice as long as those provided by intensity modulation schemes [16]. In particular, we are going to study waveguides with complex effective refractive index, in other words, complex effective first-order susceptibilities, such that there is effective linear gain or loss in each waveguide. In coupled-mode theory [17–19], this system is described by the differential equation set,

- i \frac{d}{d z} ℰ_{0} (z) = n_{0} ℰ_{0} (z) + g ℰ_{1} (z),

- i \frac{d}{d z} ℰ_{1} (z) = n_{1} ℰ_{1} (z) + g [ℰ_{0} (z) + ℰ_{2} (z)],

- i \frac{d}{d z} ℰ_{2} (z) = n_{2} ℰ_{2} (z) + g ℰ_{1} (z),

where the complex field amplitude at the j-th waveguide is given by ℰ_j(z), the constant effective refractive index is n_j = ω_j + iγ_j, with real parameters ω_j and γ_j, and the constant effective coupling between waveguide modes is g. While it is possible to deal with this coupler in general, we will focus on the so-called 𝒫𝒯-symmetric optical systems [20–22]. In our device, this restriction translates into effective refractive indices of the extremal waveguides that are complex conjugates of each other and that of the central waveguide is real. The model can be realized by engineering a three-waveguide coupler where the real part of the effective refractive index is ω₀ = ω₂ = ω and the gain-loss at each waveguide fulfills γ₁ = (γ₀ + γ₂)/2 and γ = (γ₀ − γ₂)/2, such that a solution of the form,

ℰ_{j} (z) = e^{i {(ω_{1} + i γ_{1})}_{z}} E_{j} (z),

delivers an effective coupled-mode differential set,

- i \frac{d}{d z} E_{0} (z) = (ω - ω_{1} + i γ) E_{0} (z) + g E_{1} (z),

- i \frac{d}{d z} E_{1} (z) = g [E_{0} (z) + E_{2} (z)],

- i \frac{d}{d z} E_{2} = (ω - ω_{1} - i γ) E_{2} (z) + g E_{1} (z) .

Feasible experimental realizations for our device include but are not limited to lossy waveguides, Fig. 1(a), pumped waveguides, Fig. 1(b), non-Hermitian electronics, Fig. 1(c), or whispering-gallery mode microcavities, Fig. 1(d). Note that two-waveguide PT-symmetric couplers have already been demonstrated in all these architectures [23–26]. For lossy waveguides, it may be possible to fabricate a three-waveguide coupler on multilayer Al_xGa_1−xAs heteroestructures and induce controlled waveguide losses by chromium deposition over two of the three coupler arms [23], Fig. 1(a). If the three waveguide coupler is fabricated on a photorefractive LiNbO₃ substrate, and an amplitude mask is added on top of the waveguides such that the last waveguide is not optically pumped and the first two waveguides receive a different amount of pumping, then the first two waveguides will show an effective gain [24], Fig. 1(b). Furthermore, the effective gain idea can also be realized in RLC electronic systems where the traditional resistor is replaced by an effective negative resistor built from standard operational amplifiers [25], Fig. 1(c), or with three directly coupled whispering gallery mode resonators, where the first two are fabricated from Er³⁺ doped silica and optically pumped, while the last is a standard passive whispering gallery mode resonator built from silica [26], Fig. 1(d), please note that in the laboratory the array may not be collinear as separate input (output) waveguides must be included.

Fig. 1 Schematics for a PT-symmetric three waveguide coupler realized in four different architectures: (a) lossy waveguides, n₀ = ω₀ and n_j = ω_j + iγ_j with j=1,2, (b) pumped waveguides, (c) non-Hermitian electronics and (d) coupled whispering gallery microcavities, n_j = ω_j − iγ_j with j=0,1, and n₂ = ω₂.

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Our device is the first, to the best of our knowledge, all-optical solution to the conversion of amplitude (phase) modulation to phase (amplitude) modulation involving 𝒫𝒯-symmetric planar three-waveguide couplers. For this reason, we will provide a detailed discussion of propagation through the 𝒫𝒯-symmetric coupler for the case of waveguides with identical real part of the effective refractive index, provide a closed-form analytic propagator for the device, and explore its use as an amplitude to phase and phase to amplitude modulator in the following.

2. Field propagation

For the sake of simplicity, we will consider waveguides with identical real part of the effective refractive index, ω₁ = ω, such that we obtain the coupled-mode set,

- i \frac{d}{d ζ} E (ζ) = ℍ E (ζ),

where we have defined a scaled propagation distance, ζ = λz, in order to have a single control parameter in the form of the ratio between the imaginary part of the effective refractive index and the coupling parameter, ξ = γ/λ, henceforth named gain-to-coupling ratio, and used the shorthand notation,

ℍ = (\begin{matrix} i ξ & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & - i ζ \end{matrix}), E (ζ) = (\begin{matrix} E_{0} (ζ), \\ E_{1} (ζ), \\ E_{2} (ζ) \end{matrix}),

for the mode coupling matrix and the complex field amplitudes vector, in that order. Note that this coupled-mode matrix is a particular example of on-chip non-reciprocal devices inspired in quantum systems [27] that belongs to the class of finite optical representations of the Lorentz group [28]. It is straightforward to show that this mode coupling matrix has two nonzero eigenvalues, ±Ω, with

Ω = \sqrt{2 - ξ^{2}} .

The eigenvalues will be real when the gain to coupling ratio fulfills

ξ < \sqrt{2}

, completely degenerate for

ξ = \sqrt{2}

, and complex if

ξ > \sqrt{2}

. In other words, 𝒫𝒯-symmetry is broken in the latter case.

The mode coupling matrix, ℍ, does not depend on the scaled propagation distance, ζ, thus, we can calculate the propagated complex field amplitudes as [19],

E (ζ) = 𝕌 (ζ) E (0), 𝕌 (ζ) = e^{i ℍ ζ},

where the input complex field amplitudes are collected in the vector E(0). The propagator, 𝕌(ζ), can be calculated from its Taylor series,

𝕌 (ζ) = \sum_{j = 0}^{\infty} \frac{{(i ζ)}^{j}}{j!} ℍ^{j},

= {\begin{array}{l} 𝟙 + \frac{i}{Ω} sin Ω ζ ℍ + \frac{1}{2} (cos Ω ζ - 1) ℍ^{2}, & ξ \neq \sqrt{2}, \\ 𝟙 + i ξ ℍ - \frac{1}{2} ζ^{2} ℍ^{2}, & ξ = \sqrt{2} \end{array},

where the identity matrix is given by 𝟙 and we used the Cayley-Hamilton theorem [29], a square matrix satisfies its own characteristic equation, in this particular case,

ℍ^{3} = Ω^{2} ℍ .

After some algebra, it is possible to provide the propagation as a response to an impulse function, that is, the complex field amplitude at the n-th waveguide given that the input complex field amplitude impinged at just the m-th waveguide,

E_{n}^{(m)} (ζ) = 𝕌_{n, m} (ζ), n, m = 0, 1, 2,

= \frac{α^{m + n}}{α_{0}} \sqrt{(\begin{array}{l} 2 \\ m \end{array}) (\begin{array}{l} 2 \\ m \end{array})} {}_{2}{F_{1}} [- m, - n, - 2, - \frac{α_{0}}{α_{2}}],

with Gauss hypergeometric function [30], ₂F₁(a, b, c, d), and auxiliary functions,

α_{0} = {\begin{array}{l} Ω^{2} / β^{2}, & ξ \neq \sqrt{2} \\ 2 / {(\sqrt{2} - ζ)}^{2}, & ξ = \sqrt{2}, \end{array}

α = {\begin{array}{l} i \sqrt{2} / β, & ξ \neq \sqrt{2} \\ i ζ / (\sqrt{2} - ζ), & ξ = \sqrt{2}, \end{array}

β = Ω cos \frac{1}{2} Ω ζ - ξ sin \frac{1}{2} Ω ζ .

Thus, even with amplification and losses, we expect periodic oscillations of the complex field amplitudes in the 𝒫𝒯-symmetric regime,

ξ < \sqrt{2}

with a period equal to 2πΩ⁻¹, Fig. 2, that will disappear as the 𝒫𝒯-symmetry is broken,

ξ \geq \sqrt{2}

, and gain dominates, Fig. 3. We conducted both a numerical solution of the differential equation set and propagation with the analytic results and obtained good agreement between both methods. This is well known and expected from 𝒫𝒯-symmetric devices [20–28].

Fig. 2 Squared response to impulse function, ${| E_{n}^{(m)} |}^{2}$ , at the zeroth (dotted blue), n = 0, first (solid black), n = 1, and second (dashed red), n = 2, waveguides for an initial field impinging at the (a) zeroth, m = 0, (b) first, m = 1, and (c) second, m = 2, waveguide of a coupler with gain-to-coupling ratio ξ = 0.5 that keeps 𝒫𝒯-symmetry.

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Fig. 3 Same as Fig. 2 but for a gain-to-coupling ratio $ξ = 1.25 \sqrt{2}$ that breaks 𝒫𝒯-symmetry.

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3. 𝒫𝒯-symmetric modulator

As mentioned before, a 𝒫𝒯-symmetric device is expected to behave as a directional oscillator or directional amplifier in the unbroken and broken symmetry phases, respectively. We want togo beyond oscillators and amplifiers and provide another use to 𝒫𝒯-symmetric devices. We will focus on just the 𝒫𝒯-symmetric regime, $0 \leq ξ < \sqrt{2}$ . We want to bring forward that the response to impulse function, Fig. 2, reveals that, at a certain scaled distance,

ζ_{f} = \frac{arccos (ξ^{2} - 1) + 2 n π}{Ω}, n = 0, 1, 2, \dots,

the output complex field amplitudes are quite simple,

E_{0} (ζ_{f}) = - E_{2} (0),

E_{1} (ζ_{f}) = - E_{1} (0) + i 2 ξ E_{2} (0),

E_{2} (ζ_{f}) = - E_{0} (0) + 2 ξ [i E_{1} (0) + ξ E_{2} (0)] .

Here, we can see the importance of the directionality provided by a 𝒫𝒯-symmetric device. In a standard passive optical device, the bidirectional symmetry would make it hard to find an adequate and simple set of initial complex field amplitudes that provide us with something beyond an optical oscillator. These simple expressions for the field propagation in our 𝒫𝒯-symmetric planar three-waveguide coupler allows us to propose the initial complex field amplitudes,

E_{0} (0) = 𝒜_{0} e^{i ϕ_{1}}, E_{1} (0) = 𝒜_{1} e^{i ϕ_{1}}, E_{2} (0 = 0),

with real amplitudes and phases, 𝒜_j ≥ 0 and 0 ≤ ϕ_j < 2π, that provide output fields,

E_{0} (ζ_{f}) = 0,

E_{1} (ζ_{f}) = 𝒜_{1} e^{i (ϕ_{1} + π)},

E_{2} (ζ_{f}) = 𝒜_{0} e^{i (ϕ_{0} + π)} + 2 ξ 𝒜_{1} e^{i (ϕ_{1} + \frac{π}{2})} .

These propagated field expressions show that it is possible to convert amplitude and phase differences between the signal, E₀(0), and reference, E₁(0), input fields to phase and amplitude differences at the output field,

| E_{2} (ζ_{f}) | = \sqrt{𝒜_{0}^{2} + 4 ξ^{2} 𝒜_{1}^{2} - 4 ξ 𝒜_{0} 𝒜_{1} sin δ},

arg [E_{2} (ζ_{f})] = arctan \frac{𝒜_{0} sin ϕ_{0} - 2 ξ 𝒜_{1} cos ϕ_{1}}{𝒜_{0} cos ϕ_{0} - 2 ξ 𝒜_{1} sin ϕ_{1}},

with the phase difference between signal and reference given by δ = ϕ₀ − ϕ₁. We will take the argument function as modulo 2π to have a phase in the range [0, 2π). Note, we will assign a null phase value to zero fields.

3.1. Amplitude to phase conversion

Let us consider first a case with the following input fields,

E_{0} (0) = \sqrt{1 + 𝒜_{1}^{2}}, E_{1} (0) = \pm 𝒜_{1}, E_{2} (0) = 0 .

Note that we have added a binary phase on the reference field, E₁(0), that will help us in the future. The output field amplitude and phase are given by the following,

| E_{2} (ζ_{f}) | = \sqrt{1 - (4 ξ^{2} - 1) 𝒜_{1}^{2}},

arg [E_{2} (ζ_{f})] = arctan \mp 2 ξ η,

where the phase is a function of the reference-to-signal amplitude ratio,

η = \frac{𝒜_{1}}{\sqrt{1 - 𝒜_{1}^{2}}} .

For any given gain-to-coupling ratio, each of the two binary phase options provides a π/2 modulation range but in different phase regions,

𝒱_{ϕ} = {[arg E_{2} (ζ_{f})]}_{max} - {[arg E_{2} (ζ_{f})]}_{min},

= {\begin{array}{l} π - π / 2 & E_{1} (0) = 𝒜_{1}, \\ 3 π / 2 - π, & E_{1} (0) = - 𝒜_{1}, \end{array}

= \frac{π}{2},

Thus, the introduction of a binary phase allows for continuous modulation of phase in a π range. Figure 4 shows the conversion from an initial reference field amplitude including a binary phase, E₁(0) = ±𝒜₁ with 0 ≤ 𝒜₁ ≤ 1, to an output field phase, argE₂(ζ_f), through a device with gain-to-coupling ratio ξ = 0.5,

Fig. 4 Conversion from the initial reference field amplitude, 𝒜₁, to the output field phase, argE₂(ζ_f), for an initial reference field (a) E₁(0) = 𝒜₁ and (b) E₁(0) = −𝒜₁ in a 𝒫𝒯-symmetric amplitude to phase converter with gain-to-coupling parameter ξ = 0.5.

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3.2. Phase to amplitude conversion

The converse is provided when the input fields have the same amplitude but different phase,

E_{0} (0) = e^{i δ} 𝒜_{1} E_{1} (0) = 𝒜_{1}, E_{2} (0) = 0 .

Here the output field amplitude and phase are a function of the phase difference,

\begin{array}{l} | E_{2} (ζ_{f}) | & = & \sqrt{1 + 4 ξ (ξ - 1) sin δ} 𝒜, \end{array}

\begin{array}{l} arg [E_{2} (ζ_{f})] & = & arctan \frac{2 ξ - sin δ}{cos δ} \end{array} .

Thus, we are mapping the phase difference to both the amplitude and phase of the output field. The range of the phase to amplitude modulation can be measured by an equivalent of the interferometric visibility,

𝒱_{a} = \frac{{| E_{2} (ζ_{f}) |}_{max}^{2} - {| E_{2} (ζ_{f}) |}_{min}^{2}}{{| E_{2} (ζ_{f}) |}_{max}^{2} + {| E_{2} (ζ_{f}) |}_{min}^{2}},

= \frac{8 ξ 𝒜_{0} 𝒜_{1}}{𝒜_{0}^{2} + 4 ξ^{2} 𝒜_{1}^{2}},

where we can see that the gain-to-coupling ratio plays a fundamental role. Figure 5 shows the conversion of an initial phase diference, δ, between signal, E₀(0), and reference, E₁(0), fields with equal intensities, |E₀(0)|² = |E₁(0)|², to an amplitude of the output field, |E₂(ζ_f)|, through a device with gain-to-coupling ration ξ = 0.5.

Fig. 5 Conversion from the initial phase difference, δ, between signal, $E_{0} (0) = e^{i δ} / \sqrt{2}$ , and reference, $E_{1} (0) = 1 / \sqrt{2}$ , to the output field amplitude, |E₂(ζ_f)|, in a 𝒫𝒯-symmetric phase to amplitude converter with gain-to-coupling parameter ξ = 0.5. The squared output field amplitude, |E₂(ζ_f)|², is shown.

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4. Conclusion

In summary, we have studied the propagation of electromagnetic fields through three coupled waveguides with gain or loss. We focus in the particular case of effective refractive indices with identical real parts, and imaginary parts chosen to provide an optical analog of quantum 𝒫𝒯-symmetry. Then, we provide an analytic propagator and show that, as expected, the device can be used as a directional optical oscillator or directional amplifier inside and outside the 𝒫𝒯-symmetric regime, in that order.

We use the directionality in the system to show that there exist characteristic lengths that deliver all-optical conversion of amplitude (phase) modulation to phase (amplitude) modulation. The amplitude to phase configuration provides a phase conversion range of π/2 that can be increased to π if a binary phase is introduced into the reference signal. The phase to amplitude configuration provides an amplitude conversion range that depends linearly on the gain-to-coupling ratio of the waveguide array. Our device may be used to implement phase shift keying formats for coherent optical communication, where phase modulation provides longer transmission distances than intensity modulation schemes.

Of course, experimental realizations may deviate from our proposed ideal model but they can be treated as perturbations around these particular results as long as the deviations are small. In the case of large deviations, the algebraic method provided through the first and second sections can be used to derive the field propagation of the experimental model for any given effective refractive index and coupling.

Acknowledgments

OIZG acknowledges financial support from the Academia Mexicana de Ciencias through the Verano de la Investigación Científica program and LFSM from the Instituto Tecnológico de Lázaro Cárdenas through the Verano de la Investigación Científica y Tecnológica del Pacífico program. BMRL acknowledges fruitful discussion with J. D. Sánchez de la Llave.

References and links

1. H. Liu, C. F. Lam, and C. Johnson, “Scaling optical interconnects in datacenter networks: Opportunities and challenges for wdm,” 2010 18th IEEE Symp. High Perf. Interconnects pp. 113–116 (2010). [CrossRef]

2. A. F. Benner, D. M. Kuchta, P. K. Pepeljugoski, R. A. Budd, G. Hougham, B. V. Fasano, K. Marston, H. Bagheri, E. J. Seminaro, H. Xu, D. Meadowcroft, M. H. Fields, L. McCullogh, M. Robinson, F. W. Miller, R. Kaneshiro, R. Granger, D. Childers, and E. Childers, “Optics for high-performance servers and supercomputers,” OFC ’10 p. OTuH1 (2010).

3. X. Yu, E. Arbabi, L. L. Goddard, X. Li, and X. Chen, “Monolithically integrated self-rolled-up microtube-based vertical coupler for threedimensional photonic integration,” Appl. Phys. Lett. 107, 031102 (2015). [CrossRef]

4. S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22, 46–47 (1973). [CrossRef]

5. S. M. Jensen, “The nonlinear coherent coupler,” IEEE J. Quantum Elect. 18, 1580–1583 (1982). [CrossRef]

6. N. Finlayson and G. I. Stegeman, “Spatial switching, instabilities, and chaos in a three-waveguide nonlinear directional coupler,” Appl. Phys. Lett. 56, 2276–2278 (1990). [CrossRef]

7. G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quant. Electron. 22, 95–122 (1990). [CrossRef]

8. D. Artigas, J. Olivas, F. Dios, and F. Canal, “Supermode analysis of the three-waveguide nonlinear directional coupler: the critical power,” Opt. Commun. 131, 53–60 (1996). [CrossRef]

9. Y. Chen, “Fiber and integrated optics,” Fiber and Integrated Optics 16, 287–301 (1997). [CrossRef]

10. G. J. Liu, B. M. Lang, Q. Li, and G. L. Jin, “Three-core nonlinear directional coupler with variable coupling coefficient,” Opt. Eng. 42, 2930–2935 (2003). [CrossRef]

11. K. R. Khan, T. X. Wu, D. N. Christodoulides, and G. I. Stegeman, “Soliton switching and multi-frequency generation in a nonlinear photonic crystal fiber coupler,” Opt. Express 16, 9417–9428 (2008). [CrossRef] [PubMed]

12. Q. Tao, F. Luo, L. Cao, J. Hu, D. Cai, Z. Wan, J. Zhang, and Z. Yu, “Thermal stability analysis of nonlinear optical switch based on assymetric three-core nonlinear directional coupler with variable Gaussian coupling coefficient,” Opt. Eng. 50, 071104 (2011). [CrossRef]

13. J. W. M. Menezes, W. B. de Fraga, A. C. Ferreira, K. D. A. Saboia, A. F. G. F. Filho, G. F. Guimaraes, J. R. R. Sousa, H. H. B. Rocha, and A. S. B. Sombra, “Logic gates based in two- and three-modes nonlinear optical fiber couplers,” Opt. Quant. Electron. 39, 1191–1206 (2007). [CrossRef]

14. J. W. M. Menezes, W. B. de Fraga, G. F. Guimaraes, A. C. Ferreira, H. H. B. Rocha, M. G. da Silva, and A. S. B. Sombra, “Optical switches and all-fiber logical devices based on triangular and planar three-core nonlinear optical fiber couplers,” Opt. Commun. 276, 107–115 (2007). [CrossRef]

15. J. A. G. Coelho, M. B. Costa, A. C. Ferreira, M. G. da Silva, M. L. Lyra, and A. S. B. Sombra, “Realization of all-optical logic gates in a triangular triple-core photonic crystal fiber,” J. Lightwave Technol. 31, 731–739 (2013). [CrossRef]

16. K. Kikuchi, “Coherent Optical Communications: Historical Perspectives and Future Directions,” in High Spectral Density Optical Communication Technologies vol. 6 (Springer, 2010), p. 11–49. [CrossRef]

17. G. S. Agarwal and K. Qu, “Spontaneous generation of photons in transmission of quantum fields in 𝒫𝒯-symmetric optical systems,” Phys. Rev. A 85, 031802 (2012). [CrossRef]

18. B. M. Rodríguez-Lara, H. M. Moya-Cessa, and D. N. Christodoulides, “Propagation and perfect transmission in three-waveguide axially varying couplers,” Phys. Rev. A 89, 013802 (2014). [CrossRef]

19. B. M. Rodríguez-Lara, F. Soto-Eguibar, and D. N. Christodoulides, “Quantum optics as a tool for photonic lattice design,” Phys. Scr. 90, 068014 (2015). [CrossRef]

20. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical 𝒫𝒯-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007). [CrossRef] [PubMed]

21. S. Longhi, “Bloch oscillations in complex crystals with 𝒫𝒯 symmetry,” Phys. Rev. Lett. 103, 123601 (2009). [CrossRef]

22. S. Longhi, “Bound states in the continuum in 𝒫𝒯-symmetric optical lattices,” Opt. Lett. 39, 1697–1700 (2014). [CrossRef] [PubMed]

23. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of 𝒫𝒯-symmetric breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009). [CrossRef]

24. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and D. Kip, “Observation of parity-time symmetry in optics,” Nature Phys. 6, 192–195 (2010). [CrossRef]

25. J. Schindler, Z. Lin, J. M. Lee, H. Ramezani, F. M. Ellis, and T. Kottos, “𝒫𝒯-symmetric electronics,” J. Phys A: Math. Theor. 45, 444029 (2012). [CrossRef]

26. B. Peng, S. K. Ozdemir, F. C. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. H. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nature Phys. 10, 394–398 (2014). [CrossRef]

27. R. El-Ganainy, A. Eisfeld, M. Levy, and D. N. Christodoulides, “On-chip non-reciprocal optical devices based on quantum inspired photonic lattices,” Appl. Phys. Lett. 103, 161105 (2013). [CrossRef]

28. B. M. Rodríguez-Lara and J. Guerrero, “Optical finite representation of the Lorentz group,” Opt. Lett. 40, 5682–5685 (2015). [CrossRef] [PubMed]

29. B. M. Rodríguez-Lara, H. M. Moya-Cessa, and S. M. Viana, “Por qué y cómo encontramos funciones de matrices: entropía en mecánica cuántica,” Rev. Mex. Fis. 51, 87–98 (2005).

30. N. N. Lebedev, Special Functions and Their Applications (Prentice-Hall, 1965).

All-optical 𝒫𝒯-symmetric conversion of amplitude (phase) modulation to phase (amplitude) modulation

Abstract

1. Introduction

2. Field propagation

3. 𝒫𝒯-symmetric modulator

3.1. Amplitude to phase conversion

3.2. Phase to amplitude conversion

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (41)

Optics Express