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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 3 — Feb. 6, 2006
  • pp: 1086–1093
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On the control of the spatial orientation of the transverse profile of a light beam

R. Martínez-Herrero and P. M. Mejías  »View Author Affiliations


Optics Express, Vol. 14, Issue 3, pp. 1086-1093 (2006)
http://dx.doi.org/10.1364/OE.14.001086


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Abstract

A first-order optical system (represented by its 4×4 ABCD matrix) is given in order to obtain a beam that preserves its spatial orientation of the transverse profile under free propagation from a beam with rotating irradiance distribution in free space. Within the formalism of the second-order irradiance moments, this transverse orientation is analyzed in terms of the evolution of the principal axes of the field irradiance distribution. It is shown that the spatial profile of the beam emerging from the proposed optical system does not rotate when light freely propagates. The improvement of the joint near-field and far-field beam spread product at the output of this optical system is also studied.

© 2006 Optical Society of America

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

01. R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988). [CrossRef]

] In fact, a number of measurable parameters (beam width, far-field divergence, etc) have been accepted as current ISO standards.[7

07. ISO 11146, Laser and laser related equipment-Test methods for laser beam widths, divergence angles and beam propagation ratios: ISO 11146-1:2005, Part 1: Stigmatic and simple astigmatic beams; ISO11146-2:2005, Part 2: General astigmatic beams; ISO/TR 11146-3:2004, Part 3: Intrinsic and geometrical laser beam classification, propagation, and details of test method; ISO/TR 11146-3:2004/Cor1:2005 (International Organization for Standardization, Geneva, Switzerland, 2005).

] 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

01. R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988). [CrossRef]

] [8

08. F. Encinas-Sanz, J. Serna, C. Martínez, R. Martínez-Herrero, and P. M. Mejías, “Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses,” IEEE J. Quantum Electron. 34, 1835–1838 (1998). [CrossRef]

,9

09. P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarized laser beams,” Prog. Quantum Electron. 26, 65–130 (2002), and references therein. [CrossRef]

]

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

10. J. A. Arnaud and H. Kogelnik, “Light beams with general astigmatism,” Appl. Opt. 8, 1687–1693 (1969). [CrossRef] [PubMed]

]

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

10. J. A. Arnaud and H. Kogelnik, “Light beams with general astigmatism,” Appl. Opt. 8, 1687–1693 (1969). [CrossRef] [PubMed]

] 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

05. J. Serna, R. Martínez-Herrero, and P. M. Mejías, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991). [CrossRef]

,14

14. J. Serna, P. M. Mejías, and R. Martínez-Herrero, “Rotation of partially coherent beams through free space,” Opt. Quantum Electron. 24, 873–880 (1992). [CrossRef]

] 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

15. G. Nemes and J. Serna, “Do not use spherical lenses and free spaces to characterize beams: a possible improvement of the ISO/DIS 11146 document,” in Proceedings of the Fourth Workshop on Laser Beam and Optics Characterization, A. Giesen and M. Morin, eds. (Verein Deutscher Ingenieure-Technologiezentrum, Düseldorf, Germany, 1997), pp. 29–49.

] These fields are usually called pseudo-ST (PST) and pseudo-SA (PSA) beams (also called pseudo-type fields).[15–17

15. G. Nemes and J. Serna, “Do not use spherical lenses and free spaces to characterize beams: a possible improvement of the ISO/DIS 11146 document,” in Proceedings of the Fourth Workshop on Laser Beam and Optics Characterization, A. Giesen and M. Morin, eds. (Verein Deutscher Ingenieure-Technologiezentrum, Düseldorf, Germany, 1997), pp. 29–49.

] 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.

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

05. J. Serna, R. Martínez-Herrero, and P. M. Mejías, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991). [CrossRef]

,18

18. M. J. Bastiaans, “Wigner distribution function and its applications to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979). [CrossRef]

]

h(r,η,z)=+W(r+s2,rs2)exp(ikηs)ds,
(1)

where r (x, y) denotes again the two-dimensional position vector, the dot symbolizes the inner product, and kη = (ku, kv) = (kx, ky) 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

19. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968). [CrossRef]

] In terms of the WDF, the so-called beam irradiance moments (denoted by sharp brackets) can be defined as follows

<xmynupvq>1Io+xmynupvqhrηzdrdη,
(2)

where m, n, p and q are integer numbers and IO = ∫ h dr dη is proportional to the total beam power. As is well known, the four first-order moments, <x>, <y>, <u> 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 <u 2 + v 2> , 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

05. J. Serna, R. Martínez-Herrero, and P. M. Mejías, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991). [CrossRef]

]

Q=<x2+y2><u2+v2><xu+yv>2.
(3)

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

04. A. E. Siegman, “New developments in laser resonators” in Laser Resonators, Proc. SPIE 1224, 2–14 (1990). [CrossRef]

] 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

20. A. T. Friberg, E. Tervonen, and T. Turunen, “Interpretation and experimental demonstration of twisted Gauss-Schell-mode beams,” J. Opt. Soc. Am A 11, 1818–1826 (1994). [CrossRef]

] 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

05. J. Serna, R. Martínez-Herrero, and P. M. Mejías, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991). [CrossRef]

,14

14. J. Serna, P. M. Mejías, and R. Martínez-Herrero, “Rotation of partially coherent beams through free space,” Opt. Quantum Electron. 24, 873–880 (1992). [CrossRef]

] 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

14. J. Serna, P. M. Mejías, and R. Martínez-Herrero, “Rotation of partially coherent beams through free space,” Opt. Quantum Electron. 24, 873–880 (1992). [CrossRef]

]

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>, <u 2 + v 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,

Qx=<x2><u2><xu>2,
(4)
Qy=<y2><v2><yv>2.
(5)

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

tan2θ=<y2>f<v2>f<x2>f<u2>f<uv>f<x2+y2>f+<xy>f<u2+v2>f,
(6)

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=22=a01a00b01ba01a00b01b,
(7)

with a2=(<u2>i<x2>i)12 and b2=(<v2>i<y2>i)12, 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=a0000b00001a00001b.
(8)

The second one would represent a fractional Fourier-transform system in two dimensions,[22

22. R. Simon and K. Bernardo Wolf, “Fractional Fourier transform in two dimensions,” J. Opt. Soc. Am. A 17, 2368–2381 (2000). [CrossRef]

] whose matrix has the form

F=221010010110100101,
(9)

which corresponds to the values α = ½ and β = -½ in Ref. [22

22. R. Simon and K. Bernardo Wolf, “Fractional Fourier transform in two dimensions,” J. Opt. Soc. Am. A 17, 2368–2381 (2000). [CrossRef]

]. Matrix S would then be obtained from the matricial product S = FM.

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

05. J. Serna, R. Martínez-Herrero, and P. M. Mejías, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991). [CrossRef]

,14

14. J. Serna, P. M. Mejías, and R. Martínez-Herrero, “Rotation of partially coherent beams through free space,” Opt. Quantum Electron. 24, 873–880 (1992). [CrossRef]

]

tan2φ(z)=2<xy><x2><y2>,
(10)

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

tan2φ(z)=
=2<xy>o+2kz(<xv>o+<yu>o)+2k2z2<uv>o<x2>o<y2>o+2kz(<xu>o<yu>o)+k2z2(<u2>o<v2>o),
(11)

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:

<x2>o=12(a2<x2>i+<u2>ia2+2<xu>i)=<x2>i<u2>i+<xu>i
(12)
<y2>o=12(b2<y2>i+<v2>ib2+2<yv>i)=<y2>i<v2>i+<yv>i
(13)

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

<x2>o=<y2>o.
(14)

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

tan2φ(z)=,foranyz,
(15)

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

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

<x2>o+<y2>o=2<x2>i<u2>i+(<xu>i<yu>i).
(16)

In an analogous way,

<u2>o+<v2>o=2<x2>i<u2>i(<xu>i<yv>i),
(17)

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

Qo=4<x2>i<u2>i(<xu>i<yv>i)2,
(18)

where Qo 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

Qo=4<x2>i<u2>i4<xu>i2=4Qx=4Qy.
(19)

On the other hand, since Q x = Q y , then < x 2 >i <u 2 >i = < y 2>i <v 2>i , and the beam spread product Qi at the input plane would read

Qi=2<x2>i<u2>i+<y2>i<u2>i+<x2>i<v2>i=
=2<x2>i<u2>i+<x2>i<u2>i2<v2>i+<x2>i<v2>i.
(20)

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

QoQi=2<x2>i<u2>i<x2>i<u2>i2<v2>i<x2>i<v2>i4<xu>i2=
=<x2>i<v2>i(<u2>i<v2>i)24<xu>i20,
(21)

so that Q/oQi , Q.E.D.

It is interesting to note that, in order to reach the equality Qo = Qi, in Eq. (21), the input beam should fulfil two conditions, namely, < u 2>i ,=< v 2>i and< xu >i = 0. But, since Qx = Qy, 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 Qo < Qi 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

15. G. Nemes and J. Serna, “Do not use spherical lenses and free spaces to characterize beams: a possible improvement of the ISO/DIS 11146 document,” in Proceedings of the Fourth Workshop on Laser Beam and Optics Characterization, A. Giesen and M. Morin, eds. (Verein Deutscher Ingenieure-Technologiezentrum, Düseldorf, Germany, 1997), pp. 29–49.

], 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.

Reference and links

01.

R. Simon, N. Mukunda, and E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988). [CrossRef]

02.

S. Lavi, R. Prochaska, and E. Keren, “Generalized beam parameters and transformation law for partially coherent light,” Appl. Opt. 27, 3696–3703 (1988). [CrossRef] [PubMed]

03.

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik 82, 173–181 (1989).

04.

A. E. Siegman, “New developments in laser resonators” in Laser Resonators, Proc. SPIE 1224, 2–14 (1990). [CrossRef]

05.

J. Serna, R. Martínez-Herrero, and P. M. Mejías, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991). [CrossRef]

06.

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1049 (1992). [CrossRef]

07.

ISO 11146, Laser and laser related equipment-Test methods for laser beam widths, divergence angles and beam propagation ratios: ISO 11146-1:2005, Part 1: Stigmatic and simple astigmatic beams; ISO11146-2:2005, Part 2: General astigmatic beams; ISO/TR 11146-3:2004, Part 3: Intrinsic and geometrical laser beam classification, propagation, and details of test method; ISO/TR 11146-3:2004/Cor1:2005 (International Organization for Standardization, Geneva, Switzerland, 2005).

08.

F. Encinas-Sanz, J. Serna, C. Martínez, R. Martínez-Herrero, and P. M. Mejías, “Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses,” IEEE J. Quantum Electron. 34, 1835–1838 (1998). [CrossRef]

09.

P. M. Mejías, R. Martínez-Herrero, G. Piquero, and J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarized laser beams,” Prog. Quantum Electron. 26, 65–130 (2002), and references therein. [CrossRef]

10.

J. A. Arnaud and H. Kogelnik, “Light beams with general astigmatism,” Appl. Opt. 8, 1687–1693 (1969). [CrossRef] [PubMed]

11.

J. Serna and G. Nemes, “Decoupling of coherent Gaussian beams with general astimatism,” Opt. Lett. 18, 1174–1176 (1993). [CrossRef]

12.

G. Nemes and A. E. Siegman, “Measurement of all ten second-order moments of an astigmatic beam by use of rotating simple astigmatic (anamorphic) optics,” J. Opt. Soc. Am A 11, 2257–2264 (1994). [CrossRef]

13.

G. Nemes, “Synthesis of general astigmatic optical systems, the detwisting procedure and the beam quality factors for general astigmatic laser beams,” in Proceedings of the Second Workshop on Laser Beam CharacterizationH. Weber, N. Reng, J. Ludtke, and P. M. Mejías, eds., Festkorper-Laser Institut Berlin GmbH, Berlin, Germany, 1994, pp. 93–104.

14.

J. Serna, P. M. Mejías, and R. Martínez-Herrero, “Rotation of partially coherent beams through free space,” Opt. Quantum Electron. 24, 873–880 (1992). [CrossRef]

15.

G. Nemes and J. Serna, “Do not use spherical lenses and free spaces to characterize beams: a possible improvement of the ISO/DIS 11146 document,” in Proceedings of the Fourth Workshop on Laser Beam and Optics Characterization, A. Giesen and M. Morin, eds. (Verein Deutscher Ingenieure-Technologiezentrum, Düseldorf, Germany, 1997), pp. 29–49.

16.

G. Nemes and J. Serna, “Laser beam characterization with use of second order moments: an overview,” in Diode Pumped Solid State Lasers: Applications and IssuesM. W. Dowley, ed., OSA TOPS17, 200–207 (1998).

17.

J. Serna, F. Encinas, and G. Nemes, “Complete spatial characterization of a pulsed doughnut-type beam by use of spherical optics and a cylindrical lens,” J. Opt. Soc. Am. A 18, 1726–1733 (2001). [CrossRef]

18.

M. J. Bastiaans, “Wigner distribution function and its applications to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979). [CrossRef]

19.

A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256–1259 (1968). [CrossRef]

20.

A. T. Friberg, E. Tervonen, and T. Turunen, “Interpretation and experimental demonstration of twisted Gauss-Schell-mode beams,” J. Opt. Soc. Am A 11, 1818–1826 (1994). [CrossRef]

21.

A. T. Friberg, C. Gao, B. Eppich, and H. Weber, “Generation of partially coherent fields with twist,” Proc. SPIE 3110, 317–318 (1997). [CrossRef]

22.

R. Simon and K. Bernardo Wolf, “Fractional Fourier transform in two dimensions,” J. Opt. Soc. Am. A 17, 2368–2381 (2000). [CrossRef]

OCIS Codes
(140.3300) Lasers and laser optics : Laser beam shaping
(350.5500) Other areas of optics : Propagation

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: October 31, 2005
Revised Manuscript: December 27, 2005
Manuscript Accepted: January 27, 2006
Published: February 6, 2006

Citation
R. Martínez-Herrero and P. Mejías, "On the control of the spatial orientation of the transverse profile of a light beam," Opt. Express 14, 1086-1093 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-3-1086


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References

  1. R. Simon, N. Mukunda and E. C. G. Sudarshan, "Partially coherent beams and a generalized ABCD-law," Opt. Commun. 65, 322-328 (1988). [CrossRef]
  2. S. Lavi, R. Prochaska and E. Keren, "Generalized beam parameters and transformation law for partially coherent light," Appl. Opt. 27, 3696-3703 (1988). [CrossRef] [PubMed]
  3. M. J. Bastiaans, "Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems," Optik 82, 173-181 (1989).
  4. A. E. Siegman, "New developments in laser resonators" in Laser Resonators, Proc. SPIE 1224, 2-14 (1990). [CrossRef]
  5. J. Serna, R. Martínez-Herrero and P. M. Mejías, "Parametric characterization of general partially coherent beams propagating through ABCD optical systems," J. Opt. Soc. Am. A 8, 1094-1098 (1991). [CrossRef]
  6. H. Weber, "Propagation of higher-order intensity moments in quadratic-index media," Opt. Quantum Electron. 24, 1027-1049 (1992). [CrossRef]
  7. ISO 11146, Laser and laser related equipment-Test methods for laser beam widths, divergence angles and beam propagation ratios: ISO 11146-1:2005, Part 1: Stigmatic and simple astigmatic beams; ISO11146-2:2005, Part 2: General astigmatic beams; ISO/TR 11146-3:2004, Part 3: Intrinsic and geometrical laser beam classification, propagation, and details of test method; ISO/TR 11146-3:2004/Cor1:2005 (International Organization for Standardization, Geneva, Switzerland, 2005).
  8. F. Encinas-Sanz, J. Serna, C. Martínez, R. Martínez-Herrero and P. M. Mejías, "Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses," IEEE J. Quantum Electron. 34, 1835-1838 (1998). [CrossRef]
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