Table Of Contentc. Gomez-Reino
M. V. Perez
C.Bao
Gradient -Index Optics
Springer-Verlag Berlin Heidelberg GmbH
C. Gomez-Reino
M. V. Perez
C.Bao
Gradient-Index Optics
Fundamentals and Applications
t
Springer
Professor Carlos Gomez-Reino
Professor Maria Victoria Perez
Dr. Carmen Bao
University of Santiago de Compostela
Optics Laboratory, Applied Physics Dept.,
Optics and Optometry School and Faculty of Physics
E -15782 Santiago de Compostela
Spain
e-mail: !acgrc@usc.es
ISBN 978-3-642-07568-1
Library of Congress Cataloging-in-Publication-Data
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Gomez-Reino, Carlos:
Gradient index optics: fundamentals and applications / C. Gomez-Reino; M. V. Perez; C. Bao. - Berlin;
Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Tokyo: Springer 2002
ISBN 978-3-642-07568-1 ISBN 978-3-662-04741-5 (eBook)
DOI 10.1007/978-3-662-04741-5
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To Pablo, Clara, Marta, Maria, Javier,
to our families,
and, in memoriam,
toA. Duran
Preface
Currently the terms gradient index or graded index (GRIN) are often used to
describe an inhomogeneous medium in which the refractive index varies from
point to point [P.I-P.2]. There are three basic gradient index types that have been
studied. The first one is the axial gradient. In this case, the refractive index varies
in a continuous way along the optical axis of the inhomogeneous medium. The
isoindicial surfaces (surfaces of constant index) are planes perpendicular to the
optical axis. The second one is the radial GRIN medium in which the index profile
varies continuously from the optical axis to the periphery along the transverse
direction in such a way that isoindicial surfaces are concentric cylinders about the
optical axis. In the special case of the index varying quadratically with the distance
in the transverse direction, the medium is referred to as lens-like or self-focusing
(selfoc). The last type is the spherical gradient. Spherical gradient may be thought
of as a symmetric index around a point so that isoindicial surfaces are concentric
spheres. Equations governing propagation of rays through a spherical gradient are
similar to those of a geodesic lens.
GRIN media occur commonly in nature. Examples are the crystalline lens and
the retinal receptors of the human eye, and the atmosphere of the earth. Shell and
continuous refractive index models are used for the crystalline lens [P.3-P.4], and
a waveguide model is considered for human photoreceptors [P.S]. The atmosphere
of the earth has a refractive index that decreases with height because the density
decreases at higher altitudes. Many unusual atmospheric phenomena, a mirage or
fata morgana being the best-known example, result from the bending of the rays of
light by this gradient [P.6].
In a GRIN medium, the optical rays follow curved trajectories. By an
appropriate choice of the refractive index profile, a GRIN medium can have the
same effect on light rays as a conventional optical component, such as a prism or a
lens. The possibility of using GRIN media in optical systems has been considered
for many years, but the manufacture of materials has been the limiting factor in
implementing GRIN optical elements until the 1970s. In the last 30 years,
however, many different gradient index materials have been manufactured. The
revival of GRIN optics has not been casual; it is connected to a considerable
degree with the enormous development of optical communications systems,
integrated optics, and micro-optics.
Various methods of producing GRIN materials have been developed, but these
processes are limited by the small variation of the index, the small depth of the
gradient region, and the minimal control over the shape of the resultant index
profile. Ion diffusion [P.7] is the most widely used technique for fabrication of
glass GRIN materials. They are manufactured by immersing glass elements in a
molten salt bath for many hours, during which ion diffusion-exchange occurs
through the glass surface and results in the GRIN. Several other methods have
VIII Preface
been reported. The ion-stuffing method offers the possibility of diffusing large
molecules or ions into the glass [P.8], and the sol-gel method involves the
synthesis of a multicomponent alkoxide gel that is shaped by a mold [P.9]. Plastic
GRIN materials have been manufactured by copolymerization and monomer
diffusion [P.IO]. Likewise, of particular interest is use of chemical vapor
deposition techniques to obtain small-diameter GRIN fibers for optical
communications [P.ll]. GRIN glasses with large variation of the index and large
size have been fabricated by fusing together thin layers of glasses of progressively
different refractive indices [P.l2].
Historically, Maxwell [P.13] was among the first to consider inhomogeneous
media in optics, when, in 1854, he described a GRIN lens, known as Maxwell's
fish eye, of spherical symmetry with the property that points on the surface, and
within the lens are sharply focused at conjugate points. In 1905, Wood [P.14]
constructed a cylindrical lens by a dipping technique whereby a cylinder of gelatin
is produced with refractive index axial symmetry. Thin transverse sections with
flat end-faces act like converging or diverging lenses, depending on whether the
index is a decreasing or increasing function of the radial distance.
Nearly 40 years later, Luneburg [P.I] analyzed ray propagation through
inhomogeneous media. He described a variable index, spherically symmetric
refracting structure performing perfect geometric imaging between two given
concentric spheres. The refractive index profile of such a structure has been
expressed by an integral equation. Luneburg solved it explicitly for the case where
one of the spheres is of infinite radius, and the second is coincident with the edge
of the lens. In the classical Luneburg lens, any parallel bundle of rays passing
through the lens converges at a point located on the surface of the lens.
Generalizations of the classical Luneburg lens were introduced later. In 1955,
Stettler [P.15] considered the cases when the rays completely encircle the center of
the lens, and in 1958 Morgan [P.16] allowed refractive index discontinuities. More
recently, interesting contributions of the generalized Luneburg lens have been
reported by Sochacki [P.17-P.18]. The Luneburg lens is of importance for
applications in optics and, for instance, it may be used in integrated optics
[P.19-P.24].
Luneburg also analyzed light propagation through a GRIN medium with
hyperbolic secant refractive index profile by conformal mapping of a sphere onto a
plane. In 1951 and 1954, Mikaelian [P.25] and Hetcher et al. [P.26] describe a
radial index profile in terms of a hyperbolic secant function in a cylindrical rod to
provide focusing. For this profile, all meridional rays are sharply imaged
periodically within the rod. More recently, light propagation in optical fibers
[P.27], planar waveguides [P.28], and aspherical laser resonators [P.29] with
hyperbolic profiles has been studied as an analogy with the eigenstates of the
stationary Schrodinger equation in a hyperbolic potential, which is a special case
of the Poschl-Teller potential [P.30].
In the field of image formation by GRIN lenses, considerable effort has been
directed toward geometrical and wave optics. Several methods were used to
Preface IX
compute ray trajectories in inhomogeneous media. A convenient way of solving
the ray equation was suggested by Montagnino [P.31]. Kapron [P.32] analyzed
paraxial ray tracing to derive imaging properties by a GRIN lens. Sands [P.33-
P.35] and Moore [P.36-P.37] have used Buchdahl theory for third-order
aberrations in rotationally symmetric systems. Buchdahl theory has been extended
to fifth-order aberrations by Gupta et al. [P.39]. A study of aberrations of selfoc
fibers was made by Brushon [P.40]. Equations for the limiting rays in GRIN lenses
were derived by Harrigan [Po 41]. Works on these and another geometrical optics
topics have been summarized in the books of Marchand [P.2], Sodha and Ghatak
[P.42], Kravtsov and Orlov [P.43], and Greisukh et al. [P.44].
Geometrical optics provide enough knowledge of position, size, and aberrations
of the image to allow the design of a GRIN element with reasonably good
performance. However, it is also necessary to analyze the performance of the
GRIN element by physical optics and guided-wave theory. In this way, Yariv
[P.45] studied modal propagation in quadratic index fibers and applied it to the
problem of image transmission. Iga et al. [P.46] described imaging and
transforming properties in distributed-index lenses and planar waveguides.
Gomez-Reino et al. [P.47-P.49] have analyzed paraxial imaging, Fourier
transforming transmission, and modal propagation in a GRIN rod, and have also
shown that the rod can be represented by a transmittance function equivalent to the
conventional lens transmittance function.
Many applications have been found for GRIN materials in science and
technology. Linear arrays of radial GRIN rod lenses can be used in the
photocopying industry [P.50-P.53]. GRIN lenses can be incorporated within an
image intensifier as an image--enhancing system for low light-level applications
[P.54] and are used in medical endoscopes with high numerical aperture and low
f-number [P.55-P.57]. Light pulses are focused by GRIN lenses onto optical
memory and compact disk systems for writing and reading information [P.58-
P.61]. Within the communication area, most optical devices for manipulating and
processing signals in optical fiber transmission systems include GRIN lenses for
carrying out typical functions such as on-axis imaging, collimation, focusing, and
off-axis imaging [P.62-P.63]. Taking advantage of these inherent functions of the
GRIN lenses, a wide variety of devices have been designed and fabricated. An
optical fiber connector is the simplest application of the GRIN lenses for
numerical aperture conversion. GRIN attenuators are mainly used to equalize
optical signals. Directional couplers and wavelength-division de/multiplexers
using GRIN lenses have been developed by inserting beam splitters and
interference filters or gratings. One of the key devices in optical transmission
systems is the switch and the most typical type is the device where a GRIN lens is
moved to switch an optical path. An isolator using GRIN lenses is important to
prevent reflected beams from reaching laser sources. An optical bus
interconnection system consisting of cascade arrays of GRIN lenses is used as a
free-space three-dimensional optical interconnect [P.64]. Finally, GRIN lenses are
also used for optical sensing. Intensity-modulated fiber-optic sensors employ
X Preface
GRIN lenses to improve coupling efficiency and sensing characteristics [P.65-
P.67].
This book provides an in-depth, self-contained treatment of GRIN optics and
describes the light propagation through inhomogeneous media, which covers basic
as well as specialized results. This book should be useful to students, professors,
research engineers, physicists, and optometrists in the field of lightwave
communications, imaging and transforming systems, and vision sciences as a
textbook or a reference book. Chapter 1 deals with the fundamentals of light
propagation in inhomogeneous media in the framework of the geometrical optics
limit for very large wavenumbers. Chapter 2 provides a general description of
light propagation in GRIN media by means of a linear integral transform. Chapters
3 and 4 study the laws of transformation of uniform and non-uniform beams
through and by GRIN lenses. Effects of gain and losses in GRIN materials on light
propagation are treated in Chap. 5. Chapter 6 concerns GRIN media with
hyperbolic secant refractive index profiles. Chapter 7 generalizes the self-imaging
phenomenon to GRIN media. Light propagation in the crystalline lens of the
human eye by GRIN optics is considered in Chap. 8; Chap. 9 shows some devices
that can be made by GRIN lenses.
The authors wish to thank those persons who have contributed to making this
book a reality. In particular, the authors are grateful to Prof. L. Gamer and Dr.
C.R. Fernandez-Pousa for helpful discussions and valuable criticism. The authors
also express grateful thanks to M.T. Flores-Arias, M.A. Rama Varela and 1.M.
Rivas-Moscoso for constructive suggestions.
The writing of this book constitutes an activity in the Education Program of the
University of Santiago de Compostela (USC). A short course based on some
chapters of this book was offered to graduate students at USC and was well
received. The financial support of the Xunta de Galicia and Spanish Government,
under PGIDT and TIC plans, for the original work by the authors reported in this
book is gratefully acknowledged.
Contents
1 Light Propagation in GRIN Media ......................................... .
1.1 Introduction ................................................................... .
1.2 Vector Wave Equations...... ........ ....... ......... .......... ...... ......... 1
1.3 Scalar Wave Equation......................................................... 4
1.4 Parabolic Wave Equation..................................................... 6
1.5 Ray Optics: Axial and Field Rays ............................................. 9
2 Imaging and Transforming Transmission Through GRIN Media ...2 5
2.1 Introduction.................................................................... 25
2.2 The Kernel Function.......................................................... 25
2.3 Imaging and Fourier Transforming Through GRIN Media ............... 30
2.4 Fractional Fourier Transforming in GRIN Media. . . . ..... . . . . . ... . . . ...... 33
2.5 Modal Representation of the Kernel........................................ 37
3 GRIN Lenses for Uniform Illumination ..................................... 43
3.1 Introduction................................................................. ... 43
3.2 Transmittance Function of a GRIN Lens for Uniform Illumination ..... .43
3.3 GRIN Lens Law: Imaging and Fourier Transforming by GRIN Lens.. 50
3.4 Geometrical Optics of GRIN Lenses...................................... 56
3.5 Effective Radius, Numerical Aperture, Aperture Stop, and Pupils..... 63
3.6 Diffraction-Limited Propagation of Light in a GRIN lens ................. 71
3.7 Effect of the Aperture on Image and Fourier Transform
Formation.................................................................... ... 79
4 GRIN Lenses for Gaussian Illumination ................................... 87
4.1 Introduction................................................................. ... 87
4.2 Propagation of Gaussian Beams in a GRIN Lens.......................... 87
4.3 GRIN Lens Law: Image and Focal Shifts.................................. 97
4.4 Effective Aperture ............................................................. 104
5 GRIN Media with Loss or Gain ................................................ l 09
5.1 Introduction................................................................... 109
5.2 Active GRIN Materials: Complex Refractive Index. .......... ...... .... 109
