Analysis of polarisation states at sharp focusing

Accepted Manuscript Title: Analysis of Polarisation States at Sharp Focusing Author: Svetlana Nikolaevna Khonina Dmitry Andreevich Savelyev Nikolay Lvovitch Kazanskiy PII: DOI: Reference:

S0030-4026(15)02017-3 http://dx.doi.org/doi:10.1016/j.ijleo.2015.12.108 IJLEO 57046

To appear in: Received date: Accepted date:

20-8-2015 10-12-2015

Please cite this article as: S.N. Khonina, D.A. Savelyev, N.L. Kazanskiy, Analysis of Polarisation States at Sharp Focusing, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.12.108 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

*Manuscript

Analysis of Polarisation States at Sharp Focusing Svetlana Nikolaevna Khoninaa,b , Dmitry Andreevich Savelyeva,b,* , Nikolay Lvovitch Kazanskiy1,2 1

Image Processing Systems Institute of the RAS, Molodogvardiskaya st. 151, 443001 Samara, Russia 2

Samara State Aerospace University, Moskovskoye sh. 34, 443086 Samara, Russia *

Corresponding author, e-mail: [email protected]

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Abstract

We investigate the inter-relation of the phase and polarisation singularities in systems with high numerical aperture. We

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consider sharp focusing by means of three types of systems: a micro-objective, a diffractive axicon and a microobjective combined with an axicon-type multichannel diffractive optical element. The complex transmission function of the element is matched to optical vortices. We analyse numerical results of Gaussian laser beam focusing with various

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polarisations and phase distributions in detail.

Keywords: Optical vortices, Polarisation state, Sharp focusing, Diffractive axicon, Multichannel diffractive optical

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element 1. Introduction

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Signal processing often involves the expansion of the signal in some way to reduce the number of features. One of the remarkable methods is optical vortices carrying orbital angular momentum (OAM). It finds applications in diverse areas including fibre-optic communications, free-space optical communications and RF communications [1].

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It is convenient to use multichannel optical systems [2-9] for simultaneous signal expansion into several basic functions. Telecommunications is a field of especially important application of multichannel optical systems. Owing to

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the stable volume increment of global traffic in telecommunications systems, interest in telecommunications systems with spatial division multiplexing (SDM) is increasing. The SDM technology in the optical fiber can be implemented

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using mode division multiplexing (MDM) [6]. However, when an MDM system is used, the problem of mode coupling should be solved [7]. One method of solving this problem is to use the “multiple-input, multiple-output” (MIMO) technology based on an electronic correction of inter-mode interference [7-9]. Polarisation multiplexing is also used [10-12]. The polarisation is exploited both in optical fibres and for antennas. Polarisation state affects the diffraction picture [13], so it can be used to analyse and detect the type of input beam [14, 15].

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Investigations of the mutual influence of the optical phase vortices and polarisation singularities, transformation one to another or enhancement of the angular momentum have a long history [16-28]. The vortex phase is used for the analysis of polarising properties of laser fields [29-31]. Overwhelmingly, visual observation of the interrelation of the vector (polarisation) and scalar (phase) optical vortices is possible only in a high numerical aperture mode, for example, at sharp focusing [30-34]. We have investigated the possibility of detecting the polarisation state of the focused incident beam using singular phase elements [35]. The complex transmission function of such elements can be described as a superposition of optical vortices. These elements can be realised by means of diffractive optics [36] and be used as additions to focusing systems [30-35]. Moreover, a singularity can be inserted in the structure of a focusing element [37].

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Sharp focusing can be implemented by means of a micro-objective [38], a parabolic mirror [39, 40], a diffractive lens [40-42] or an axicon [43-46]. It was suggested [40] that sharper focusing (comparing with a micro-objective) can be achieved using a parabolic mirror or a diffractive lens. For the parabolic mirror, this suggestion was confirmed experimentally [39]. For a diffractive lens, it was shown numerically [41, 42]. In addition, an improvement of focusing properties of an aplanatic lens by addition of an axicon structures was shown [42, 47]. We consider sharp focusing by means of three types of system: a micro-objective, a diffractive axicon and a microobjective combined with axicon-type multichannel diffractive optical element. The complex transmission function of

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the element is matched to optical vortices. We analyse the numerical results of Gaussian laser beam focusing with various polarisations and phase distributions in detail.

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2. Theoretical Models 2.1. Focusing by a Micro-objective

E(, , z )  

if 

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To simulate an aplanatic focusing optical system, we can use the Debye approximation [48]:  2

  B (, ) T () P(, ) exp ik  sin  cos(  )  z cos   sin  d d, 0 0

(1)

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where (, , z ) are cylindrical coordinates of the focal area, (, ) are the spherical angular coordinates of the output pupil of the focusing system, B(, ) is the transmission function, T () is the pupil apodisation function (for an

cos  ), P(, ) is the polarisation vector, sin   NA / n , n is the refractive index

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aplanatic micro-objective T () 

of the media, k  2 /  is the wavenumber,  is the wavelength, f is the focal length and P(, ) is polarisation vector. If the transmission function is given by:

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B(, )  R() d m exp(im)

(2)

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m

we can simplify Eq. (1) for the majority of polarisation states [33-35, 42]: 

m

where vector

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E(, , z )  ikf  d m i m eim  Qm (, , ) R () T ()sin  exp (ikz cos ) d ,

(3)

0

Qm (, , ) depends on the input field polarisation.

2.2. Focusing by a Diffractive Axicon

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To describe focusing by an axicon, we can use the plane wave expansion [49, 50]:

E(, , z ) 

1 2

2 2 

 Fx (, ) 

  M(, )  F (, )  exp ikz

1 0

y

1  2  exp ik  cos(  )   d d , 

 Fx (, )  r0 2   E0 x (r , )     exp  ikr cos       r dr d,  Fy (, )  0 0  E0 y (r , ) 

(4)

(5)

where M(, ) is the polarisation transform matrix. If the input field components can be given as vortice superpositions, as in Eq. (4), the relations (4) and (5) can be simplified [45, 46] as follows:

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



 Px ()  R  E0 x (r )       J m (kr) r dr ,  Py ()  0  E0 y (r ) 

(6)

(7)

where Sm (, , ) is the polarisation transform matrix for vortex phase of m-order.

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When we use the diffractive phase axicon as a focusing element, Eqs. (6) and (7) are simplified [46]. 2.3. Focusing with multichannel diffractive optical element

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Complex transmission function of a multichannel diffractive optical element [4, 36, 51] is as follows:

B  ,     d mn  mn  ,   exp i mn cos    mn   ,

(8)

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m,n

where  mn  ,   are functions of interest, for example, such as in Eq. (2),

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of spatial frequency, respectively.

 mn and mn are the radial and angular parts

To design the binary phase, which is convenient for manufacturing, it is necessary to use complex-conjugated orders in Eq. (8). Axicon-type addition can be realised by radial phase jumps [34, 42].

Figure 1 shows the axicon-type binary phase (Fig. 1a) of a 4-order diffractive optical element matched with

M

exp(i) (Fig. 1b).

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combinations of the first order vortices

Fig. 1. 4-order diffractive optical element: (a) binary phase, (b) accordance of diffractive orders to combinations of optical vortices

3. Numerical Simulation

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1  Px ()  2 E(, , z )  k 2  d mi 2 m exp (im)  S m (, , )   exp ikz 1    d , P (  ) m  y  0

We simulate the sharp focusing of a Gaussian laser beam with various polarisations and phase distributions with the different systems discussed above. Table 1 shows a comparison of the focal intensity pictures calculated by means of the micro-objective and the diffractive axicon with the same numerical aperture (NA) = 0.95. The parameters of the calculation were: wavelength of incident radiation  = 1 m, focal length f = 101 m, the numerical aperture of the micro-objective NA = 0.99, and the radius of the Gaussian beam is 50 m. As can be seen from Table 1, when m = 0 it is very easy to recognise the azimuthal polarisation; opposite to other types of polarisation, we can detect it by zero value of intensity in the central focal point. The linear polarisation is

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detected by elongation of a focal spot along the direction of polarisation. It becomes more obvious with the use of a high-NA diffractive axicon. Table 1 Gaussian laser beam Focusing by the micro-objective and the diffractive axicon with NA = 0.95 (picture size is 2  2). x-linear

“+”-circular

y-linear

“”-circular

radial

azimuthal

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m=0

Axicon

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m=1

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Objective

M

Objective

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Axicon

Objective

Axicon

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m = 1, m = 1

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m=2

Objective

Axicon

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To distinguish the direction of circular polarisation, it is necessary to add a vortex phase of the first order (m = 1) or a combination of vortices of opposite signs (m = 1, m = 1). In the latter case, the diffractive axicon provides good recognition between orthogonal states of linear and cylindrical polarisations. Comparing pictures in Table 1, we come to the conclusion that a diffractive axicon provides sharper focusing than a micro-objective with the same NA. The differences in the pictures are caused by the strong longitudinal component of the electric field. The above results indicate that we need multiple tests, even with sharp focusing, to analyse the polarisation state and

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phase distribution of an incident beam. Thus, simultaneously monitoring the effects of several optical vortices and their superposition is desirable. Combinations of optical vortices that can be generated by the multichannel binary phase elements are practically convenient to this purpose [34, 42].

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Tables 2 and 3 show simulations results of focusing a Gaussian laser beam with various polarisations using a microobjective combined with axicon-type 4-order diffractive optical element (fig. 1). The amplitude of the beam is a

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Gaussian function multiplied by the radius. The phase of the beam is constant or a vortex of the first order. Table 2 Simulation results for a linearly polarised beam.

Ex

2

Ey

2

2

Ex  E y  E z 2

2

d

[0, 0, 0, 0]

[0, 1, 1, 1]

[x, z, z, z]

[1, 0, 0, 0]

[0, 1, 1, 0]

[y, z, z, 0]

[0, 0, 1, 1]

[0, 1, 0, 1]

[1, 0, 0, 0]

[z, y, x, xy]

[0, 1, 0, 1]

[0, 0, 1, 1]

[1, 0, 0, 0]

[z, x, y, xy]

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[1, 0, 0, 0]

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y-linear

[0, 0, 0, 0]

Vortex x-linear

Ez

M

x-linear

2

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Intensity distribution in focal plane

Analysed beam: polarisation, phase

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Vortex y-linear

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We show the intensity distributions of different components of electric field in a focal plane (15  15 m) for different components of the electric field for detailed analysis. We indicate zero (0) and nonzero (1) values in according diffractive orders: [central (m = 0), left (m = −1), right (m = 1), top (m = ±1)]. From the modelling results, the presence or absence of a vortex phase of the first order is easily identified by the presence of a correlation peak in the corresponding diffractive order in the focal plane; the absence of an optical vortex corresponds to high intensity in the centre of the focal plane. When the phase vortex is absent, we can determine the direction of linear polarisation from the z-component

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intensity in the top or the bottom orders (first and second rows of Table 2). In this case, sharp focusing is required to strengthen the z-component intensity.

The presence of the first vortex phase in an incident beam is observed as a bright correlation peak in the right (for m

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= 1) or in the left (for m = −1) in the x- or y-component of electric field intensity, respectively (third and fourth rows of Table 2). In this case we can use a focusing system with a low NA.

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Table 3 Simulation results for a circularly polarised beam.

Intensity distribution in focal plane

Ex

2

Ey

Ez

2

2

Ex  E y  E z 2

2

M

«+»-circular

2

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Analysed beam: polarisation, phase

[0, 0, 1, 1]

[xy, 0, z, z]

[1, 0, 0, 0]

[0, 1, 0, 1]

[xy, z, 0, z]

[0, 0, 1, 1]

[0, 0, 1, 1]

[0, 0, 0, 0]

[0, 0, xy, xy]

[0, 0, 1, 1]

[0, 0, 1, 1]

[1, 0, 0, 0]

[z, 0, xy, xy]

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«−»-circular

[1, 0, 0, 0]

Vortex «+»-circular

[1, 0, 0, 0]

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[1, 0, 0, 0]

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Vortex «−»-circular

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Since the pictures of the transverse components’ intensities are practically identical, the direction of circular polarisation is difficult to distinguish in a paraxial case. Therefore, it is necessary to apply sharp focusing to strengthen the z-component intensity. When the phase vortex is absent, we can determine the direction of circular polarisation by z-component intensity in the left and in the right orders (first and second rows of Table 3). The presence of the first vortex phase in an incident beam is observed as a correlation peak of z-component intensity in the centre of the focal plane if the vortex direction is opposite to the polarisation direction (fourth row of Table 3).

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When the directions of phase and polarisation coincide, there is a zero intensity value in the centre of the focal plane (third row of Table 3). Orthogonal cylindrical polarisation states are easily recognised in sharp focusing mode because azimuthal polarization has no z-components.

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In the case of the absence of the vortex phase in an incident beam, Table 4 shows a zero intensity value just in the centre of the focal plane for azimuthal polarisation (compare the first and second rows of Table 4).

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Table 4 Simulation results for cylindrically polarised beam.

Ex

2

Ey

Ez

2

2

Ex  E y  E z 2

2

[1, 0, 0, 0]

[z, xy, xy, x]

[0, 1, 1, 1]

[0, 0, 0, 0]

[0, xy, xy, y]

[1, 0, 0, 0]

[1, 0, 0, 0]

[0, 0, 1, 1]

[xy, 0, z, z]

[1, 0, 0, 0]

[1, 0, 0, 0]

[0, 0, 0, 0]

[xy, 0, 0, 0]

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azimuthal

[0, 1, 1, 0]

Vortex radial

[0, 1, 1, 0]

te

[0, 1, 1, 1]

d

M

radial

2

an

Intensity distribution in focal plane

Analysed beam: polarisation, phase

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Vortex azimuthal

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When the phase vortex is present, we can detect a zero intensity value just in the left or right orders (according to the sign of the vortex) for radial polarisation. For azimuthal polarisation, we will see a nonzero intensity value just in the centre of the focal plane. 4. Conclusions It has been shown that a diffractive axicon provides sharper focusing than a micro-objective with the same NA. The

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differences in the pictures are caused by the strong longitudinal component of the electric field. It has been demonstrated that the azimuthal polarisation is easily recognised because of the zero value of intensity in the central focal point, the opposite to other types of polarisation. The linear polarisation is detected by elongation of a

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focal spot along the direction of polarisation. To distinguish the direction of circular polarisation, it is necessary to add an additional vortex phase of the first order (m = 1) or a combination of vortices of opposite signs (m = 1, m = −1).

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To simultaneously monitor the effects of several optical vortices and their superposition, we used a micro-objective combined with an axicon-type 4-order diffractive optical element. The simulation results of Gaussian laser beam focusing with various polarisations have been discussed.

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For a linearly polarised beam, the presence or absence of a vortex phase of the first order is easily identified by the presence of a correlation peak in the corresponding diffractive order in the focal plane; the absence of an optical vortex corresponds to high intensity in the centre of the focal plane. When the phase vortex is absent, we can determine

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direction of linear polarisation from the z-component intensity in the top or the bottom diffractive orders. In this case, sharp focusing is required to strengthen the z-component intensity. The presence of the first vortex phase in an incident beam is observed as a bright correlation peak in the right (for m = 1) or in the left (for m = −1) of the diffraction picture

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in the x- or y-component of electric field intensity, respectively. In this case, we can use a focusing system with low NA.

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For circularly polarised beams, sharp focusing is necessary to strengthen the z-component intensity. When the phase vortex is absent, we can determine the direction of circular polarisation from the z-component intensity in the left and in

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the right diffractive orders. The presence of the first vortex phase in an incident beam is observed as a correlation peak of the z-component intensity in the centre of the focal plane if the vortex direction is opposite to the polarisation direction. When the directions of phase and polarisation coincide, there is a zero intensity value in the centre of the focal plane.

Orthogonal cylindrical polarisation states are easy recognised in sharp focusing mode because azimuthal

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polarisation has no z-components. When the phase vortex is present, we can detect a zero intensity value just in the left or right orders (according to the sign of the vortex) for radial polarisation. For azimuthal polarisation, we will see a nonzero intensity value just in the centre of the focal plane. In the absence of the vortex phase in an incident beam, this research shows that there will be a zero intensity value just in the centre of the focal plane for azimuthal polarisation. Thus, a micro-objective combined with an axicon-type 4-order diffractive optical element and axicons can be used as detectors of the various polarisation states of an incident beam. Acknowledgements The work was financially supported by the Russian Foundation for Basic Research (grants 13-07-00266, 14-0731079 mol_а), by the Ministry of Education and Science of Russian Federation.

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