On the control of the spatial orientation of the transverse profile of a light beam

R. Martínez-Herrero; P. M. Mejías

doi:10.1364/OE.14.001086

1. Introduction

The overall characterization of the spatial structure of light fields by means of the irradiance-moments formalism is a topic largely investigated in the past years.[1–6] In fact, a number of measurable parameters (beam width, far-field divergence, etc) have been accepted as current ISO standards.[7] Furthermore, a beam quality parameter (also called beam propagation factor) has revealed to be a useful tool to provide a joint description of the focusing and collimation capabilities of the light at the near and at the far field.[1–6] [8,9]

In characterizing optical beams, the spatial symmetry of their transverse irradiance profiles is another interesting property. Stigmatic (ST), simple astigmatic (SA) and general astigmatic (GA) beams (associated to rotational symmetry, orthogonal symmetry and other type of symmetry, respectively) are the main classes of fields with regard to this property. [10–13]

In the most generalized case, the irradiance distribution at the beam cross-section rotates as the field propagates in free space. This occurs, for instance, when a typical He-Ne TEM00 laser beam passes through two cylindrical non-parallel and non-orthogonal lenses separated by a free space. The resulting field would then exhibit a rotating elliptical spot at different transverse planes along the propagation axis.[10] But this behaviour will generate harmful effects in those applications requiring spatial accuracy in both, beam position and shape across the working region.

Based on the irradiance-moments formalism, the spatial orientation of a general partially coherent beam at each transverse plane can be described by introducing a pair of orthogonal axes (principal axes) defined in terms of the second-order coherence features of the field.[5,14] In the present paper we will show an optical system (represented by its 4×4 ABCD matrix) able to transform a rotating beam into a beam with no rotation of its irradiance profile along the beam axis upon free propagation. The matrix elements will be given in terms of the values of the measurable second-order irradiance-moments of the beam at the input plane of such system.

Before beginning the calculations, let us point out the following remark: Note that there are non-rotating GA beams which behave very similar to ST and SA fields: in fact, the internal features of these GA beams remain hidden if only rotationally-symmetric optics is used, and can only be revealed after propagation through cylindrical lenses.[15] These fields are usually called pseudo-ST (PST) and pseudo-SA (PSA) beams (also called pseudo-type fields).[15–17] The output beams emerging from our optical system could be either simple astigmatic or pseudo-SA. Both kinds of beams look similar in free space, but PSA fields carry orbital angular momentum, a characteristic that the SA beam does not have. In the present work, we are interested in those applications where the distinction between a SA and a PSA beam is not important, so that the attention will be focused on obtaining partially coherent fields having a non-rotating spot in free space.

In the next section the key parameters and definitions to be used are introduced. Section 3 shows the proposed optical system and proves that the principal axes of the output beam do not rotate. In addition, the improvement of the beam spread product is analysed in Section 4. Finally, the main conclusions are summarized in Section 5.

2. Formalism and key definitions

As is well known, the second-order coherence properties of a beam can be described by means of the cross-spectral density (CSD) function W(r ₁ r ₂), where r _j, j = 1, 2, represent the two-dimensional position vectors at two points over the beam cross-section, transverse to the propagation direction z. Since we will consider quasimonochromatic fields, explicit dependence on frequency ω will, for simplicity, be omitted in our expressions.

Instead of analysing the structure of the light field by means of the function W, here we are interested on the global behaviour of the beam, described by certain overall parameters that propagate according to simple laws. Let us then introduce the Wigner distribution function (WDF) associated with the CSD function through a Fourier transform relationship: [5,18]

h (r, η, z) = \int_{- \infty}^{+ \infty} W (\frac{r + s}{2}, \frac{r - s}{2}) \exp (i k η • s) d s,

where r (x, y) denotes again the two-dimensional position vector, the dot symbolizes the inner product, and kη = (ku, kv) = (k_x, k_y) provides the wavevector components along the x and y-axes (accordingly, u and v represent angles of propagation, without taking the evanescent waves into account). The WDF can be physically understood in Optics as the amplitude associated to a ray passing through a point along a certain direction. [19] In terms of the WDF, the so-called beam irradiance moments (denoted by sharp brackets) can be defined as follows

< x^{m} y^{n} u^{p} v^{q} > \equiv \frac{1}{I_{o}} \int_{- \infty}^{+ \infty} x^{m} y^{n} u^{p} v^{q} h (r, η, z) d r d η,

where m, n, p and q are integer numbers and I_O = ∫ h dr dη is proportional to the total beam power. As is well known, the four first-order moments, <x>, <y>, and <v>, characterize the centre of the beam and its mean direction. For simplicity, in what follows we assume that these moments vanish (this is not a restriction, since it is equivalent to a shift of the Cartesian coordinate system). On the other hand, the (squared) beam width at a plane z = constant and the (squared) far-field divergence are represented by <x ² + y ²> and , respectively. In addition, the crossed moment <xu + yv> gives the position of the beam waist through the condition <xu + yv> = 0. Finally, it would also be useful to introduce an easy-to-measure figure of merit, Q, that provides a joint information about the near and far-field behaviour of the beam. For brevity, we will refer to this parameter as the beam spread product, which has been defined for partially coherent fields as follows [5]

Q = < x^{2} + y^{2} > < u^{2} + v^{2} > - {< {xu}^{} + yv >}^{2} .

For stigmatic fields, the beam spread product is, a k ² factor apart, the beam propagation factor (M ²)² first introduced by Siegman.[4] The interpretation of Q as a 3D quality parameter relies on its invariance under propagation through rotationally-symmetric ABCD systems. In particular, Q remains constant under free propagation. Note, however, that Q is not constant in the general case.[20–21] In other words, Q is a kind of “partial invariant”, which can be modified by using non-rotationally-symmetric systems. Consequently, this class of optical systems could then improve this parameter: a lower value of Q would mean better simultaneous focusing and collimation capabilities for freely propagating light. It should also be noted that Q has a lower limit, Q ≥ 1/k ², where the equality is only reached by the idealized Gaussian beam.

In the present work attention will be devoted to the spatial orientation of the irradiance profile of a general partially coherent beam at each transversal plane. This feature has been characterized in the literature[5,14] by the orientation of two orthogonal axes (the so-called principal axes) for which the crossed x-y moment vanishes, i.e. <xy> = 0. It can also be shown that the beam widths <x ²>^1/2 and <y ²>^1/2 reach their extreme values along these axes. [14]

Since, in general, the spatial profile rotates as the field propagates in free space, the principal axes can be used to describe this rotation: it would be determined from the angle that the principal axes make with some fixed laboratory coordinate system. In the next section we will introduce an optical system that transform a rotating general astigmatic beam into a non-rotating field.

3. Optical system after which the beam profile is non-rotating

Let us here define, for convenience, the transverse position vector r as the product of k = 2π/λ by the conventional Cartesian coordinates of the point at which the field is evaluated. Consequently, from now on, x and y should be considered dimensionless variables, and all the irradiance moments (e.g., <x ² + y ²>, , <xu + yv>, <xy>), given by Eq. (2), will become dimensionless functions.

For simplicity, we will consider that the beam waist is placed at the input plane of the system. In addition, we will choose, for the sake of convenience, a reference coordinate system with respect to which Q _x = Q _y, where Q _j, j = x, y, would represent the 2D beam spread products associated with each transverse field component, namely,

Q_{x} = < x^{2} > < u^{2} > - {< xu >}^{2},

Q_{y} = < y^{2} > < v^{2} > - {< yv >}^{2} .

It can then be shown that the angle θ between this new coordinate system and the former one (laboratory Cartesian axes) would be given by the formula

\tan 2 θ = \frac{{< y^{2} >}_{f} {< v^{2} >}_{f} - {< x^{2} >}_{f} {< u^{2} >}_{f}}{{< {uv}^{} >}_{f} {< x^{2} + y^{2} >}_{f} + {< {xy}^{} >}_{f} {< u^{2} + v^{2} >}_{f}},

where the subscript f denotes the irradiance moments referred to the former laboratory coordinate system. Remember that <x ²>, <y ²> and <xy> are here dimensionless parameters.

Let us now introduce an optical system whose 4×4 ABCD matrix, S, takes the form

S = \frac{\sqrt{2}}{2} = (\begin{matrix} a & 0 & \frac{1}{a} & 0 \\ 0 & b & 0 & \frac{- 1}{b} \\ - a & 0 & \frac{1}{a} & 0 \\ 0 & b & 0 & \frac{1}{b} \end{matrix}),

with $a^{2} = {(\frac{{< u^{2} >}_{i}}{{< x^{2} >}_{i}})}^{\frac{1}{2}}$ and $b^{2} = {(\frac{{< v^{2} >}_{i}}{{< y^{2} >}_{i}})}^{\frac{1}{2}}$ , where the subscript i refers to the values of the beam irradiance moments at the input plane of this system. Note again that, according with the normalization introduced at the beginning of this section, the parameters a and b are dimensionless quantities.

Matrix S can be synthesized as the combination of two systems: the first one can be understood as a magnifier M whose ABCD matrix reads

M = (\begin{matrix} a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \\ 0 & 0 & \frac{1}{a} & 0 \\ 0 & 0 & 0 & \frac{1}{b} \end{matrix}) .

The second one would represent a fractional Fourier-transform system in two dimensions,[22] whose matrix has the form

F = \frac{\sqrt{2}}{2} (\begin{matrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & - 1 \\ - 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \end{matrix}),

which corresponds to the values α = ½ and β = -½ in Ref. [22]. Matrix S would then be obtained from the matricial product S = FM.

We next show that, at the output of the optical system S, the spatial profile of the emerging beam does not rotate when it propagates freely.

To study the z-evolution of the spatial orientation of the beam profile, we will analyze the angle φ between the principal axes of the output field and our (fixed) Cartesian coordinate axes (for which Q _x = Q _y). At each transverse plane z, the angle φ is given in terms of the intensity moments at such plane by the formula [5,14]

\tan 2 φ (z) = \frac{2 < xy >}{< x^{2} > - < y^{2} >},

where <x ²>, <y ²> and <xy> are evaluated at plane z. Taking into account the free-propagation law of the irradiance moments, this expression can also be written in the form

\tan 2 φ (z) =

where the subscript o denotes the values at the output plane z = 0 of the system S. Note that, in Eq. (11), kz is a dimensionless product.

From the application of the ABCD matrix S, it can be shown that the (squared) transverse beam sizes of the emerging beam, namely, < x ² >_oand <y ²>_o, are connected with their values at the input plane of the system by means of the following expressions:

{< x^{2} >}_{o} = \frac{1}{2} (a^{2} {< x^{2} >}_{i} + \frac{{< u^{2} >}_{i}}{a^{2}} + 2 {< xu >}_{i}) = \sqrt{{< x^{2} >}_{i} {< u^{2} >}_{i}} + {< xu >}_{i}

{< y^{2} >}_{o} = \frac{1}{2} (b^{2} {< y^{2} >}_{i} + \frac{{< v^{2} >}_{i}}{b^{2}} + 2 {< yv >}_{i}) = \sqrt{{< y^{2} >}_{i} {< v^{2} >}_{i}} + {< yv >}_{i}

But< xu + yv >_i = 0 (beam waist at the input plane) and <x ²>_i_i = <y ²>_i<v ²>_i (since Q _x = Qy). Consequently, from Eqs. (12) and (13) it follows at once

{< x^{2} >}_{o} = {< y^{2} >}_{o} .

In a similar way, it can be shown that _o =<v ²>_o. Moreover, <xu>_o =<yv>^o = 0. Therefore, after substitution of these expressions in Eq. (9), we obtain

\tan 2 φ (z) = \infty, for any z,

In other words, φ (z) = constant = π/4. Accordingly, the spatial profile of the output beam does not rotate after travelling through the optical system described by matrix S, Q.E.D.

It should finally be noted that there might be other solutions to solve the problem of preserving the orientation of the beam profile. The solution proposed in this section, however, is particularly simple.

4. Beam spread product at the output of the system

For complementary purposes, let us now analyse the behaviour of the beam spread product Q, defined in Eq. (3), when the beam travels through the optical system represented by matrix S (cf Eq. (7)). More specifically, we will next show that the value of the parameter Q decreases after propagation through the proposed optical system.

Note first that, from Eqs. (12), (13) and (14), it follows

{< x^{2} >}_{o} + {< y^{2} >}_{o} = 2 \sqrt{{< x^{2} >}_{i} < {u^{2} >}_{i}} + ({< xu >}_{i} - {< yu >}_{i}) .

In an analogous way,

{< u^{2} >}_{o} + {< v^{2} >}_{o} = 2 \sqrt{{< x^{2} >}_{i} {< u^{2} >}_{i}} - ({< xu >}_{i} - {< yv >}_{i}),

along with < xu>_o =< yv >_o = 0 . Then

Q_{o} = 4 {< x^{2} >}_{i} {< u^{2} >}_{i} - {({< xu >}_{i} - {< yv >}_{i})}^{2},

where Q_o denotes the beam spread product at the output plane of the system S. But, since the waist plane is placed at its input plane, we have < xu >_i = - < yv >_i, and one gets

Q_{o} = 4 {< x^{2} >}_{i} {< u^{2} >}_{i} - 4 < xu >_{i}^{2} = 4 Q_{x} = 4 Q_{y} .

On the other hand, since Q _x = Q _y , then < x ² >_i _i = < y ²>_i <v ²>_i , and the beam spread product Q_i at the input plane would read

Q_{i} = {2 < x^{2} >}_{i} {< u^{2} >}_{i} + {< y^{2} >}_{i} {< u^{2} >}_{i} + {< x^{2} >}_{i} {< v^{2} >}_{i} =

Therefore we can write (cf. Eqs. (18) and (20))

Q_{o} - Q_{i} = 2< x^{2} >_{i} < u^{2} >_{i} - \frac{< x^{2} >_{i} < u^{2} >_{i}^{2}}{< v^{2} >_{i}} - < x^{2} >_{i} < v^{2} >_{i} - 4 < xu >_{i}^{2} =

so that Q/_o ≤ Q_i , Q.E.D.

It is interesting to note that, in order to reach the equality Q_o = Q_i, in Eq. (21), the input beam should fulfil two conditions, namely, _i ,=< v ²>_i and< xu >_i = 0. But, since Q_x = Q_y, we would also have < x ²> _i =< y ²> _i, and this input beam would not rotate under free propagation. In other words, use of system S would not be required for such beam. As a consequence, we finally get that Q_o < Q_i for any input rotating beam.

5. Conclusions

On the basis of the irradiance moments (up to second-order) formalism, a rotating general field has been transformed into a non-rotating beam by means of certain ABCD device. The proposed optical system can be implemented from the combination of a magnifier and a fractional Fourier transform. As an additional property, the beam spread product at the output of the proposed arrangement becomes improved with respect to its value at the input plane.

It should finally be remarked that, according with the classification scheme considered, for example, in Ref. [15], the final output field can be either a simple astigmatic (SA) or a pseudo-SA (PSA) beam. Both type of beams might have the same Q, and the same near- and far-field behaviour under free propagation, but the differences would appear after travelling, for example, through a cylindrical lens following by a free space. We do not proceed further into these properties because the present work does not concern with the full detwisting problem. These aspects, however, deserve attention in the future.

By means of certain first-order optical system, we have shown the possibility of maintaining the spatial orientation of the transverse profile of a beam. More specifically, from the measurable values of the second-order intensity moments of the light field, an optical system has been designed in such a way that the spatial profile of the emerging beam of the system does not rotate under free propagation. The proposed device could be implemented from the combination of a magnifier and a fractional Fourier-transform optical system. As a complementary advantage, the beam quality parameter at the output of our optical device could be improved with respect to its value at the input plane.

Acknowledgments

This work has been supported by the Ministerio de Educación y Ciencia of Spain, under project FIS2004-1900, within the framework of EUREKA project EU-2359.

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On the control of the spatial orientation of the transverse profile of a light beam

Abstract