Topological properties of the interaction between focusing regions kind cusped

G. Martínez-Niconoff; S. I. De Los Santos-García; M. A. Torres-Rodríguez; R. Suárez Xique; M. Vargas-Morales; P. Martinez Vara; A. Carbajal-Domínguez

doi:10.1364/OE.24.014648

1. Introduction

It is well known that energy is dissipated or stored around equilibrium points, also known as critical points, which constitute the support for the realization of contemporary physical models [1]. In an optical context, this means that an optical field is organized around focusing regions also known as caustic or singular regions, defined as the envelope of critical points to the amplitude function [2–4]. In this sense, the physical properties of the focusing regions are fundamental to understand the global structure of the optical field because in its neighborhood, important optical features are expected. For example, bifurcation effects [5, 6] consisting in the generation of different physical properties when some parameters implicit in the description of the optical fields change its values. The physical consequence is that the optical field is split in two or more optical branches whose time/spatial evolution may generate optical vortices [7, 8]. In addition, the focusing regions present similar features with the topological charges which becomes evident when it interacts with another optical field [9, 10], as it will be shown below.

The main point of this manuscript is supported by the property that the phase function of the optical field present adiabatic features in the neighborhood of the focusing regions. The physical meaning is that the optical field loses its wave behavior acquiring particle-like properties [11], in addition the focusing regions present the highest irradiance value, offering the possibility to transfer energy to other regions of the optical field, this fact is described in detail implementing the irradiance transfer equation. In this context, the objective of the present manuscript is to describe the interaction between two Pearcey functions where each one has associated a focusing region shape cusped [12]. The boundary condition to generate the Pearcey functions consists in two slit shape parabolas as it is sketched in Fig. 1(a), this configuration can be interpreted as a topological version of the Young interferometer. In Fig. 1(b) we show the corresponding cusped region which is generated by the diffraction field emerging from each slit shape parabola.

Fig. 1 a) Topological young interferometer and shape cusped focusing regions associated to each Pearcey function. The parabolic slits are contained in a square of 8 mm preside, the vertexes separation is in the interval $[0, 2]$ mm. The illumination source was a He-Ne laser with a wave length of 632.8 nm. b) Geometrical description of the focusing region.

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2. Irradiance interaction between focusing regions

By the adiabatic behavior of the focusing region, the interaction first must be described implementing an energy redistribution, where novel physical properties are identified, in particular we will show that each focusing region is capable to transfer energy to other regions of the optical field, this is a new feature that appears in the topological Young interferometer.

The interaction is performed using the irradiance transport equation [13], it is applied in the neighborhood of the cusped points generating an “irradiance channel” which constitutes an extension of the focusing regions as it will be shown in short. The other regions of the optical field must be analyzed by an amplitude interaction where the interference fringes are organized around the irradiance channel. The irradiance transport equation is given by

\nabla_{T} \cdot (I (x, y, z) \nabla_{T} L (x, y, z)) = - \frac{\partial}{\partial z} I (x, y, z),

where

\nabla_{⊥}

is the two-dimensional gradient operator acting in the x-y plane. The irradiance on a point

P (x, y, z)

is generated by the contribution of each focusing region represented by

I (x, y, z) = I_{1} + I_{2},

the phase term is given by

L = L_{1} - L_{2},

where

L_{1, 2} = k r_{1, 2},

k

is the wave number and

r_{i}

is the distance from a point on the boundary condition to the detection point P. Only the parameters associated to the upper parabola are sketched in Fig. 1(b).

Considering the coordinates $(ε, \pm a, 0)$ for the points on the boundary condition in the neighborhood of the parabola vertexes and an arbitrary point $(0, y, z)$ on the linear trajectory that connects the cusped points, the distance functions in the paraxial approximation are

r_{1} = \sqrt{{(y - a)}^{2} + z^{2} + ε^{2}} \approx z + \frac{{(y - a)}^{2}}{2 z},

r_{2} = \sqrt{{(y + a)}^{2} + z^{2} + ε^{2}} \approx z + \frac{{(y + a)}^{2}}{2 z},

where we have avoided the quadratic term

ε^{2}

because

| ε | ≪ 1.

The difference phase term takes the form

k r_{1} - k r_{2} = \frac{2 a y k}{z} .

Then, the irradiance transport equation acquires the form

\frac{\partial}{\partial y} ({I_{1} + I_{2}} \frac{\partial}{\partial y} (k r_{1} - k r_{2})) = - \frac{\partial (I_{1} + I_{2})}{\partial z},

introducing Eq. (4) into Eq. (5), the partial differential equation for the irradiance interaction is

\frac{- 2 a k}{z} \frac{\partial}{\partial y} (I_{1} + I_{2}) = \frac{\partial (I_{1} + I_{2})}{\partial z},

and the corresponding system of characteristic equations is

\frac{d y}{2 a k} = \frac{d z}{z} d (I_{1} + I_{2}) = 0,

whose solution is given by

I_{1} + I_{2} = c_{2}, \frac{y}{2 a k} = \ln (z) + c_{1} .

Note that $z \neq 0,$ because $z = 0$ corresponds to the transmittance plane and no focusing region is yet generated. The physical meaning of Eq. (8) is the generation of an irradiance channel connecting the two cusped points. We remark that Eq. (8) is valid in the neighborhood of the cusped points and the irradiance channel geometry is nondependent on relative separation between the parabola slits. It agrees with computer simulations for the irradiance distribution shown in Figs. 2(e) and 2(h). These results are in agreement with the experimental results shown in Figs. 2(f) and 2(i). When the separation between the cusped regions is large enough, no irradiance interaction exists and the geometry of the interference fringes resembles the classical Young interferometer, as it is shown in the experimental result shown in Fig. 2(c).

Fig. 2 In a) we show the parabolic boundary condition to generate the Pearcey function shown in b). In d) and g) we show the boundary condition to generate two Pearcey functions, this configuration allows to control the interaction between cusped regions. In e) and h) we show the computer simulation associated to the irradiance distribution for the each optical field. In c) we show the experimental result when no irradiance interaction exist between the cusped regions. In f) and i) we show the experimental results for the generation of the irradiance channel.

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3. Amplitude interaction

The next point of the study describes the amplitude interaction between the Pearcey optical fields. Considering the two slits as a topological interferometer, the problem consists in to describe the structure of the interference fringes also as its relation with the focusing region. The optical fields under study is generated by the diffraction emerging from each slit whose transmittance function is given by

t (x, y) = δ (y - (x^{2} + a)) + δ (y + (x^{2} + a)),

Where $δ$ is the Dirac delta function. The diffraction field for the amplitude function in the paraxial approximation is

\begin{array}{l} φ (x_{0}, y_{0}, z) = \int \int_{- \infty}^{\infty} [δ (y - (x^{2} + a)) + δ (y + (x^{2} + a))] \\ \times \exp [\frac{i π}{λ z} (x^{2} + y^{2})] \exp [- \frac{i 2 π}{λ z} (x x_{0} + y y_{0})] d x d y . \end{array}

Making the integration respect the variable

y,

thus obtaining

\begin{matrix} φ (x_{0}, y_{0}, z) = φ_{1} + φ_{2} \\ = \int \exp \frac{4 i π}{λ z} (\frac{x^{4}}{4} - \frac{x^{2}}{2} (y_{0} - a - \frac{1}{2}) - x x_{0}) d x \\ + \int \exp \frac{4 i π}{λ z} (\frac{x^{4}}{4} - \frac{x^{2}}{2} (y_{0} + a + \frac{1}{2}) - x x_{0}) d x, \end{matrix}

the phase term in the first integral is of the form

L (x, α, β) = \frac{x^{4}}{4} - α \frac{x^{2}}{2} + β x .

The structure of the phase function can be obtained from its critical points given by

\frac{\partial L}{\partial x} = 0 = x^{3} - α x + β,

where the envelope of the critical points satisfy the equation

\frac{\partial^{2} L}{\partial x^{2}} = 0 = 3 x^{2} - α .

In order for the critical points to take real values, it is necessary that $α > 0,$ which occurs when

y_{0} - a - \frac{1}{2} > 0.

From Eqs. (13) and (14), the geometry of the focusing region is then

β = \pm 2 {(\frac{α}{3})}^{\frac{3}{2}} .

The main consequence of Eq. (15) is that the phase function can be considered as a catastrophe function. If this condition is not fulfilled the phase term will not correspond to a catastrophe function because no singular points are present, and it can be represented as a quadratic function. This occurs when

y_{0} \leq a + \frac{1}{2},

and no envelope of singular points are present. In this case, the quartic term can be removed and the integral is reduced to

\int \exp \frac{2 i π}{λ z} (x^{2} (y_{0} - a - \frac{1}{2})) \exp (\frac{2 i π}{λ z} x x_{0}) d x .

The last expression corresponds to the Fourier transform of a Gaussian function with complex variance. Calculating explicitly the integral we obtain

φ_{1} = \exp \frac{2 i π}{λ z} (\frac{x_{0}^{2}}{{(y_{0} - a - \frac{1}{2})}^{2}}) .

Based on the previous analysis, we partially conclude that the amplitude function $φ_{1}$ has two possible representations for its phase function. One of them corresponds to a catastrophe function, another is a quadratic function, separated by the focusing region. For each representation we expect different physical properties of the optical field as it is shown below.

From similar analysis for the second integral in Eq. (11), the condition to the phase term takes the form of a catastrophe function is

y_{0} < - a - \frac{1}{2} .

If the previous condition is not fulfilled, the phase term is again a quadratic function and the amplitude is given by

φ_{1} = \exp \frac{2 i π}{λ z} (\frac{x_{0}^{2}}{{(y_{0} + a + \frac{1}{2})}^{2}}),

which is valid in regions defined by

y_{0} \geq - a - \frac{1}{2} .

From Eqs. (19) and (21), it is easy to show that the interference term is then

2 Re φ_{1} φ_{2}^{*} = 2 \cos [\frac{2 π}{λ z} (\frac{x_{0}^{2}}{{(y_{0} - a - \frac{1}{2})}^{2}}) - \frac{2 π}{λ z} (\frac{x_{0}^{2}}{{(y_{0} + a + \frac{1}{2})}^{2}})],

where the maximum of irradiance occurs when

(\frac{x_{0}^{2}}{{(y_{0} - a - \frac{1}{2})}^{2}}) - (\frac{x_{0}^{2}}{{(y_{0} + a + \frac{1}{2})}^{2}}) = m λ z,

and the geometry of the interference fringes is a set of hyperbolae having the

x_{0}

-axis as an asymptote.

The fact that the phase terms have two possible representations implies that the interference fringes emerge from the focusing region “flowing” toward the phase quadratic region having an asymptotic behavior, these sentences are according to Eqs. (23) and (24). The interference occurs only in regions where the phase function is single value. Then we have that the sources for the interference fringes are placed on the focusing region justifying the topological charge features. This behavior can be observed in the computer simulations shown in Figs. 2(e) and 2(f) also as in the experimental results shown in Figs. 2(h) and 2(i).

4. Conclusions

We have presented the analysis for the interaction between two optical fields emerging from two slits parabolic shape, this configuration corresponds to a topological version of the Young interferometer. The diffraction field emerging from each parabola has associated a cusped shape focusing region. The study of the optical fields was described analyzing two kinds of optical interactions. First of them, was the irradiance transfer between the focusing regions using the irradiance transport equation, we showed that focusing regions are capable to transfer energy to other regions of the optical field. For the proposed geometry, we showed the generation of an irradiance channel connecting the cusped points, constituting an extension of the focusing regions, offering interesting applications, in particular, it can be implemented as a guide light also as tunable optical tweezers. The second effect was the amplitude interaction, where interesting interference effects were generated, the analysis demonstrate that the interference fringes starts on the focusing regions having an asymptotic behavior toward $x_{0}$ -axis and do not cross the irradiance channel, these effects were manifested in regions with quadratic phase functions. On regions where phase functions correspond to a catastrophe function, no interference effects are manifested. The computer simulations and the experimental results are in very well agreement with the analysis performed. Finally, the topological Young interferometer, where the slits have an arbitrary shape, can be used to predict and control other optical effects such as bifurcations, vortices, etc., this analysis will be presented in a forthcoming paper.

Acknowledgments

The authors S. I. D.L. S, M. A. T. R., and M. V. M. are thankful for the support of CONACyT.

References and links

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Topological properties of the interaction between focusing regions kind cusped

Abstract

1. Introduction

2. Irradiance interaction between focusing regions

3. Amplitude interaction

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (2)

Equations (24)

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