quality improvement of partially polarized beams
TRANSCRIPT
Quality improvement of partially polarized beams
Jesus M. Movilla, Rosario Martınez-Herrero, and Pedro M. Mejıas
Improvement of the beam-quality parameter of partially polarized beams is investigated. We focus onthe use of a Mach–Zehnder-type interferometric arrangement with crossed polarizers. The analysis hasbeen carried out within the framework of the intensity moment formalism. Conditions are given underwhich the beam-quality parameter is optimized. © 2001 Optical Society of America
OCIS codes: 260.5430, 350.5500.
1. Introduction
The intensity moment formalism has been shown tobe a useful tool to describe the spatial features ofscalar beams.1–5 In fact, the standard definitions ofbeam width and beam divergence are given in termsof specific second-order beam moments.6 In partic-ular, the beam-quality parameter �also called beampropagation factor6� has been proved to be especiallyrelevant, and a number of conditions and optical sys-tems to improve this parameter have been reportedin the literature.7–9
Until now, however, most of these studies werecarried out without regard to polarization effects.The problem arises when the beam undergoes depo-larization after propagation along optical systemsthat contain polarizing elements or exhibit a nonuni-form spatial distribution of its polarization state.10–13
Several papers have been published14–16 that de-scribe the overall spatial characteristics of thesekinds of beam. So far, little effort has been devotedto describing optical systems that would enable us toimprove the beam quality of partially polarizedbeams.
In Section 2 we extend the intensity moment def-initions to the vectorial case. In section 3 we in-troduce the Mach–Zehnder-type �MZT� opticalarrangement, and study the beam-quality changesgenerated by this kind of system. Attention is fo-cused on the conditions under which the quality
parameter could be optimized. Finally, the con-cluding remarks are outlined in Section 4.
2. Second-Order Intensity Moments of PartiallyPolarized Beams
We consider quasi-monochromatic fields that propa-gate along the z axis and are represented by the Jonesvector
E�r, z� � �Es�r; z�, Ep�r; z��, (1)
where s and p refer to the axes of an arbitraryCartesian coordinate system �orthogonal to z� and r� �x, y� denotes the position vector. In terms of theelectric-field vector, the second-order intensitymoments associated with the j component � j � s, p�of a partially polarized beam can be defined as fol-lows:
���j �1Ij
k2
42 � � � ���Ej*�r � s�2, z�
� Ej�r � s�2, z�exp�iks��dsdrd�, (2)
where the asterisk indicates complex conjugation, theangle brackets denote an ensemble average, � � �u,v� is a vector whose components represent angles ofpropagation �without taking the evanescent wavesinto account�, �, � � x, y, u, v, and
Ij � � � ��Ej�r��2dr, j � s, p (3)
is the total power associated with the j component.The well-known second-order moments of the globalbeam can be expressed in terms of the second-order
The authors are with the Departamento de Optica, Facultad deCiencias Fisicas, Universidad Complutense de Madrid, 28040 Ma-drid, Spain. The e-mail address for P. M. Mejias is [email protected].
Received 26 June 2000; revised manuscript received 21 May2001.
0003-6935�01�336098-04$15.00�0© 2001 Optical Society of America
6098 APPLIED OPTICS � Vol. 40, No. 33 � 20 November 2001
moments associated with each transversal field com-ponent � j � s, p� in the form13 of
�x2 � y2 � �r2 �Is
I�r2s �
Ip
I�r2p, (4)
�xu � yv � �r� �Is
I�r�s �
Ip
I�r�p, (5)
�u2 � v2 � ��2 �Is
I��2s �
Ip
I��2p, (6)
where I � Is � Ip. As usual, �r2, ��2, and �r� areclosely connected to the beam size, the beam diver-gence, and the average of the radius of curvature ofthe global beam �the s and p subscripts indicate thatthe beam moments refer to the corresponding fieldcomponents�. In a similar way as for the scalar fieldcase, the radius of curvature, Rayleigh length, andbeam waist plane for each beam component aregiven, respectively, by
Rs,p ��r2s, p
�r�s,p, (7)
zRs,p � ���r2s,p�w
��2s,p�1�2
, (8)
zws,p � �r�s,p
��2s,p, (9)
where subscript w in Eq. �8� indicates that �r2 isevaluated at the waist plane. In terms of the abovemoments, the three-dimensional �3-D� beam qualityparameter, Q3-D,4,13 becomes
Q3-D � �Is
I �2
Q3-Ds� �Ip
I �2
Q3-Dp� �Is Ip
I2 �Q3-Dsp, (10)
where
Q3-D � �r2��2 � �r�2, (11)
Q3-Ds� �r2s��
2s � �r�s2, (12)
Q3-Dp� �r2p��
2p � �r�p2, (13)
Q3-Dsp� �r2s��
2p � �r2p��2s � 2�r�s�r�p. (14)
Note that Q3-Dsand Q3-Dp
represent the beam-qualityparameters associated with each field component.
Our results apply only to stigmatic beams and stig-matic optical systems.17 Q3-D is then related to thebeam-quality factor M2 �Refs. 3 and 6� by the simpleexpression M2 � k�Q3-D�1�2. Q3-D remains invariantfor beams that propagate through such stigmaticABCD systems. This makes Q3-D useful as a beam-quality parameter. Because we deal only with stig-matic beams, we can replace �r2 by �x2, ��2 by �u2,and �r� by �xy everywhere. However, we prefer toretain the general notations introduced in Eqs. �4�–�6� throughout the paper.
3. Beam Quality Changes after Propagation throughMach–Zehnder-Type Systems
We first define what we understand by a MZT sys-tem. The structure of this kind of system is repre-sented in Fig. 1. The configuration resembles that ofa classic Mach–Zehnder interferometer, with the ex-ception of a polarizer in each arm. The transmissionaxes of both polarizers, P1 and P2, were chosen to beorthogonal to avoid interference effects between theemerging beams. After the addition of both polariz-ers, the beam components propagate through differ-ent first-order optical systems. Of course, for theparticular case of having identical ABCD systems, novariation would appear in the spatial and polariza-tion features of the output beam with respect to theinput field.
Since we assume stigmatic nonpolarizing first-order systems, we deal with ABCD matrices com-posed of four submatrices proportional to a 2 � 2identity matrix.17 The second-order moments foreach beam component follow the well-known ABCDrule for scalar fields:
�r2jo � Aj
2�r2ji � 2 AjBj�r�j
i � Bj2��2j
i, (15)
�r�jo � AjCj�r2j
i � � AjDj � BjCj��r�ji
� BjDj��2j
i, (16)
��2jo � Cj
2�r2ji � 2CjDj�r�j
i � Dj2��2j
i. (17)
In Eqs. �15�–�17�, the superscripts i and o denote theinput and output planes and j � 1, 2 refers to the sand p components of the incident beam. To writethe above equations we used A � AI, B � BI, C � CI,and D � DI, where I is the 2 � 2 identity matrix.The explicit dependence of the intensity moments ofthe global beam at the output plane with respect totheir values at the input plane can be obtained atonce by direct substitution of Eqs. �15�–�17� into Eqs.�4�–�6�.
Here we are interested in the beam-qualitychanges generated by MZT systems. For conve-nience, we separately studied two classes of MZTsystems in which the first-order optical systems arefree-propagation sections and magnifiers, and we de-rive the conditions under which the Q3-D parameterof the output beam is optimized.
Fig. 1. Schematic of a MZT system. Planes �i and �o are, re-spectively, the input and the output planes of the system. P1 andP2 are orthogonally oriented linear polarizers.
20 November 2001 � Vol. 40, No. 33 � APPLIED OPTICS 6099
A. Free Propagation
In this case the beam freely propagates along the twoarms of the MZT system, but the respective propaga-tion distances differ. As a result, the ABCD matrixelements are
A1 � 1, B1 � L1, C1 � 0, D1 � 1, (18)
A2 � 1, B2 � L2, C2 � 0, D2 � 1. (19)
We could achieve the implementation of differentfree-space propagation distances in the two arms ofthe optical device, for example, by properly shiftingthe totally reflecting mirrors in arm 1 of the inter-ferometer. This would enable us to control the dif-ference L2 L1, which, as will be shown later, is theparameter we must optimize. By using Eqs. �10�–�17� we get
�Q3-D � Q3-Do � Q3-D
i
�Is Ip
I2 ��L2 � L1�2��2s��
2p � 2�L2 � L1�
� ��r�p��2s � �r�s��
2p��, (20)
where the averages are evaluated at the input plane.From Eq. �20� we can show the following necessaryand sufficient condition for improving the quality pa-rameter ��Q3-D � 0�:
�Q3-D � 0 N �L2 � L1� � 2� Hs � Hp�, (21)
provided that
sign�L2 � L1� � sign�Hs � Hp�, (22)
where
Hs,p ��r�s,p
��2s,p. (23)
We then conclude that the quality parameter can beimproved by means of the above MZT system forthose beams that fulfill conditions �21� and �22�.Therefore, no improvement can be achieved for fieldsthat exhibit similar behavior along the s and p axes.This improvement would be optimum when
L2 � L1 � Hs � Hp. (24)
In such a case
��Q3-D�opt � Is Ip
I2 ��2s��2p�Hs � Hp�
2. (25)
To gain further insight into the physical meaning ofEq. �25� let us consider the distance L between thewaist planes associated with each component of theinput beam. In terms of L, Eq. �20� can be rewrittenas follows:
�Q3-D �Is Ip
I2 ��2s��2p��L2 � L1�
2 � 2L�L2 � L1��.
(26)
The optimum quality improvement is then attainedwhen
L2 � L1 � L (27)
and ��Q3-D�opt reads
��Q3-D�opt � Is Ip
I2 ��2s��2p L2. (28)
From Eqs. �27� and �28� it can be concluded that theoptimum quality improvement is reached when thedifference between the free-propagation sections ofthe MZT arrangement exactly compensates for thedistance between the waist planes of the input beamcomponents. In this sense, note that Hs and Hp arethe distances between the input and the waist planesassociated with the s and p components �compareEqs. �9� and �23��. In particular, for distances muchlonger than the Rayleigh length �i.e., in the far field�,Hs and Hp could be understood as the respective radiiof curvature.
B. Magnifiers
We now consider MZT systems whose ABCD compo-nents are magnifiers. A magnifier is a first-orderoptical system that multiplies the divergence by afactor m 1, increasing the minimum beam size by afactor m. This effect leads to a modification of theRayleigh length of the beam �see Eq. �8��. The ele-ments of the corresponding ABCD matrices are
A1 � m1, B1 � 0, C1 � 0, D1 � m1 1, (29)
A2 � m2, B2 � 0, C2 � 0, D2 � m2 1. (30)
Following a similar procedure to that carried out forfree propagation, we obtain
�Q3-D �Is Ip
I2 ��m2 � 1��r2s��2p
� �m 2 � 1��r2p��2p��
2s, (31)
where
m � m1 m2 1, (32)
with m1 and m2 being the lateral magnification of therespective first-order systems. Consequently, thoseMZT systems whose magnifiers give rise to the samem factor will produce identical variations of the Q3-Dparameter.
From Eq. �31� it is easy to see that �Q3-D � 0 onlyif a � �r2s��
2p � �r2p��2s � b. �As a result, thisprocedure cannot be applied to beams with the samebehavior along the s and p axes.� When a differsfrom b, one can improve the beam quality by choosingtwo magnifiers that satisfy
min�1, b�a� � m2 � max�1, b�a�. (33)
For a physical interpretation of this condition we con-sider, for the sake of simplicity, that the waists ofboth components are placed at the input plane of theMZT system. In such a case, the moments �r2s and
6100 APPLIED OPTICS � Vol. 40, No. 33 � 20 November 2001
�r2p in Eq. �31� are evaluated at the waist plane. Byreferring to Eq. �8�, it then follows that the qualityparameter will be optimized provided that
m2 � zRp�zRs
. (34)
We then finally get
��Q3-D�opt � Is Ip
I2 ��2s��2p� zRs
� zRp�2. (35)
Equation �35� shows that, to optimize the qualityparameter, the MZT system should compensate forthe differences in Rayleigh lengths associated witheach incident beam component.
4. Concluding Remarks
The results reported in Section 3 suggest that a gen-eral scheme to improve the beam-quality parameterof a partially polarized beam might consist of a MZTsystem that contains free-propagation sections tocompensate for the different waist positions of theircomponents followed by a second MZT with magnifi-ers to compensate for the corresponding Rayleighlengths.
In this sense it should be noted that a third illus-trative example of the MZT system, namely, the so-called lenslike birefringent media,18 can be used toimprove the beam-quality parameter. This kind ofelement can be considered either as a thin lens madeof an anisotropic spatially homogeneous material or aplane-parallel plate made of a birefringent uniaxialmedium whose �inhomogeneous� principal indices ex-hibit a �transversal� quadratic dependence on the dis-tance to the propagation axis. This type of mediumcan be understood as a MZT system with a thin lensin each arm. As was reported in Ref. 18, this systemimproves the beam quality by compensating for thedifferent average radius of curvature of each fieldcomponent of the input beam. It thus seems that, toimprove the Q3-D parameter, the ABCD systems atthe arms of the MZT arrangement should be chosenand driven to compensate for the differences betweenthe second-order intensity moments of the beam com-ponents.
This research was supported by the Ministry ofEducation and Culture of Spain under project PB97-0295. We also thank an anonymous referee for thevaluable suggestions to improve the manuscript.
References and Notes1. S. Lavi, R. Prochaska, and E. Keren, “Generalized beam pa-
rameters and transformation law for partially coherent light,”Appl. Opt. 27, 3696–3703 �1988�.
2. M. J. Bastiaans, “Propagation laws for the second-order mo-
ments of the Wigner distribution function in first-order opticalsystems,” Optik 82, 173–181 �1989�.
3. A. E. Siegman, “New developments in laser resonators,” inOptical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2–14�1990�.
4. J. Serna, R. Martınez-Herrero, and P. M. Mejıas, “Parametriccharacterization of general partially coherent beams propagat-ing through ABCD optical systems,” J. Opt. Soc. Am. A 8,1094–1098 �1991�.
5. H. Weber, “Propagation of higher-order intensity moments inquadratic-index media,” Opt. Quantum Electron. 24, 1027–1049 �1992�.
6. ISO standard 11146, Lasers and Laser Related Equipment.Test methods for laser beam parameters: beam widths, di-vergence angle, and beam propagation factor �InternationalOrganization for Standardization, Geneva, Switzerland, 1999�.
7. R. Martınez-Herrero, P. M. Mejıas, and G. Piquero, “Qualityimprovement of partially coherent symmetric-intensity beamscaused by quartic phase distortions,” Opt. Lett. 17, 1650–1651�1992�.
8. A. E. Siegman and J. Ruff, “Effects of spherical aberration onlaser beam quality,” in Laser Energy Distribution Profiles,J. M. Darchuk, ed., Proc. SPIE 1834, 130–139 �1992�.
9. J. Serna, G. Piquero, P. M. Mejıas, and R. Martınez-Herrero,“Parametric characterization of Hermite–Gauss mode beamsand Gauss–Schell model fields propagating through super-Gaussian apertures,” Opt. Quantum Electron. 28, 1039–1048�1996�.
10. S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generatingradially polarized beams interferometrically,” Appl. Opt. 29,2334–2339 �1990�.
11. N. Kugler, S. Dong, Q. Lu, and H. Weber, “Investigation of themisalignment sensitivity of a birefringence compensated two-rod Nd:YAG laser system,” Appl. Opt. 36, 9359–9366 �1997�.
12. T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson,and M. J. Rooks, “Circularly symmetric operation of aconcentric-circle-grating, surface-emitting, Al�Ga�GaAsquantum-well semiconductor laser,” Appl. Phys. Lett. 60,1921–1923 �1992�.
13. Q. Lu, S. Dong, and H. Weber, “Analysis of TEM00 laser beamquality degradation caused by a birefringent Nd:YAG rod,”Opt. Quantum Electron. 27, 777–783 �1995�.
14. R. Martınez-Herrero, P. M. Mejıas, and J. M. Movilla, “Spatialcharacterization of general partially polarized beams,” Opt.Lett. 22, 206–208 �1997�.
15. J. M. Movilla, G. Piquero, P. M. Mejıas, and R. Martınez-Herrero, “Parametric characterization of nonuniformly polar-ized beams,” Opt. Commun. 149, 230–234 �1998�.
16. F. Gori, “Matrix treatment for partially polarized, partiallycoherent beams,” Opt. Lett. 23, 241–243 �1998�.
17. G. Nemes, “Measuring and handling general astigmaticbeams,” in Laser Beam Characterization, P. M. Mejıas, H.Weber, R. Martınez-Herrero, and A. Gonzalez-Urena, eds. �So-ciedad Espanola de Optica, Madrid, 1993�, pp. 325–358.
18. G. Piquero, J. M. Movilla, P. M. Mejıas, and R. Martınez-Herrero, “Beam quality of partially polarized beams propagat-ing through lenslike birefringent elements,” J. Opt. Soc. Am. A16, 2666–2668 �1999�.
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