Recent Progress In Gradient-Index
`Optics
`
`Susan Houde-Walter
`
`Susan Houde-Walter, "Recent Progress In Gradient-Index Optics," Proc. SPIE
`0935, Gradient-Index Optics and Miniature Optics, (8 April 1988); doi:
`10.1117/12.946906
`Event: 1988 Technical Symposium on Optics, Electro-Optics, and Sensors,
`1988, Orlando, FL, United States
`
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`Recent Progress in Gradient -Index Optics
`Recent Progress
`in Gradient-Index Optics
`
`Susan Houde- Walter
`Susan Houde-Walter
`
`The Institute of Optics, University of Rochester
`The Institute of Optics, University of Rochester
`Rochester, NY 14627
`Rochester, NY 14627
`
`ABSTRACT
`ABSTRACT
`
`A brief summary of gradient -index (GRIN) optical principles
`is given,
`A brief summary of gradient-index
`(GRIN) optical principles
`is given,
`followed by a state -of- the -art review. Much of the important recent work has
`followed by a state-of-the-art review. Much of the important recent work has
`been in the area of GRIN materials, specifically: new glasses for ion exchange,
`been in the area of GRIN materials, specifically: new glasses for ion exchange,
`new ion exchange techniques, GRIN sol -gel glasses and plastic GRIN.
`new
`ion exchange techniques, GRIN sol-gel glasses and plastic GRIN. New
`New
`metrology techniques and fabricated GRIN imaging systems are also discussed.
`metrology techniques and fabricated GRIN imaging systems are also discussed.
`
`1. PRINCIPLES OF GRIN IMAGING OPTICS
`1. PRINCIPLES OF GRIN IMAGING OPTICS
`
`Optical materials which contain a distribution of refractive index and are
`Optical materials which contain a distribution of refractive
`index and are
`(GRIN) materials.
`called
`lens
`design
`used
`are
`gradient -index
`in
`The
`used
`in
`lens design are called gradient-index
`(GRIN) materials.
`The
`parameters that define the index distribution provide valuable new degrees of
`parameters that define
`the index distribution provide valuable new degrees of
`freedom for designers of optical imaging systems1,2.
`freedom for designers of optical imaging systems* »^.
`The index distribution can be represented in its most general form by an
`The
`index distribution can be represented in its most general form by an
`polynomial3,
`index
`index polynomial^,
`
`N(r,z) =
`N(r,z) _
`
`NOO + N0,z + hi,/ + ....
`N00 + N 01z + NO2z2 + ....
`+ r { N10 + N11z + N12z2 + ....
`N,o + V + V + •-
`+ r4 { N20 + N21z + N22z2 + ...
`N +Nz + Nz2 +
`20
`21
`22
`"•
`
`t
`r
`
`+r2pNPc1zp +
`
`(1)
`where z is the coordinate along the optical axis and r
`the radial coordinate.
`where z is the coordinate along the optical axis and r is the radial coordinate.
`is
`Two special cases of the general
`index polynomial have
`expression for
`Two special cases of the general expression for the
`index polynomial have
`the
`proven to be particularly useful in lens design. The first of these, called an
`proven to be particularly useful in lens design.
`The first of these, called an
`axial GRIN,
`represented by equation (1) by eliminating all
`terms which
`is
`axial GRIN,
`is represented by equation
`(1) by eliminating all
`terms which
`include r. As can be seen from the left hand side of Figure 1, the iso- indicial
`include r. As can be seen from the left hand side of Figure 1, the iso-indicial
`surfaces are planes, and the index is a function of the coordinate along the
`surfaces are planes, and
`the
`index
`is a function of the coordinate along the
`An axial GRIN profile in combination with a
`optical
`spherical
`axis only.
`optical axis only.
`An axial GRIN profile
`in combination with a spherical
`is virtually equivalent
`refraction
`surface
`surface, with the
`aspheric
`surface
`is virtually equivalent
`to an aspheric surface, with
`the
`refraction
`to
`an
`varied across the aperture by a local change in refractive index (as opposed to
`varied across the aperture by a local change in refractive index (as opposed to
`local change in curvature).
`Figure 2 schematically illustrates the correction
`a local change
`in curvature).
`Figure 2 schematically illustrates the correction
`a
`of spherical aberration using an axial GRIN profile. Axial GRIN profiles have
`of spherical aberration using an axial GRIN profile. Axial GRIN profiles have
`
`(1)
`
`2
`2
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`/ SPIE Vol. 935 Gradient-Index Optics and Miniature Optics (1988)
`/ SPIE Vol 935 Gradient -Index Optics and Miniature Optics (1988)
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`collimators4,5,
`binocular
`of
`correct
`aberration
`spherical
`used
`spherical aberration of collimators 4 *^, binocular
`been
`to
`to correct
`been used
`objectives6 and distortion in eyepieces7.
`objectives^ and distortion in eyepieces7 .
`
`AXIAL GRIN
`
`N(z) = N00 + N01 z
`
`Z->
`
`RADIAL GRIN
`
`N(z) = N00
`
`r2 = x2 +
`
`1. Axial GRIN profile geometry (left) and radial GRIN
`Figure
`Figure 1. Axial GRIN profile geometry (left) and radial GRIN
`profile geometry (Right).
`(Right).
`profile geometry
`
`AXIAL GRIN
`AXIAL GRIN
`SINGLET
`SINGLET
`
`HOMOGENEOUS
`HOMOGENEOUS
`SINGLET
`SINGLET
`
`<
`
`>
`
`UNDERCORRECTED
`UNDERCORRECTED
`SPHERICAL ABERRATION
`SPHERICAL ABERRATION
`
`singlet with
`a homogeneous
`Figure
`in a homogeneous singlet with
`Spherical
`aberration in
`Spherical aberration
`Figure 2.
`2.
`spherical aberration using
`Correction of
`Correction of spherical aberration using
`spherical
`surfaces
`(left).
`spherical surfaces (left).
`an axial GRIN profile (right).
`an axial GRIN profile (right).
`
`3
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`The second type is the radial GRIN profile, shown on the right hand side of
`The second type is the radial GRIN profile, shown on the right hand side of
`terms in equation
`This representation is obtained by eliminating all
`is obtained by eliminating all terms in equation
`This representation
`Figure 1.
`Figure
`1.
`(1) which include z.
`Its representation simplifies to the following expression:
`Its representation simplifies to the following expression:
`(1) which include z.
`N(r) = Noo + N10r2 + N20r4 + ...
`( 2 )
`The isoindicial surfaces of the radial GRIN are concentric cylinders about the
`The isoindicial surfaces of the radial GRIN are concentric cylinders about the
`optical axis. Radial GRIN lenses that have plano surfaces are known as Wood
`optical axis. Radial GRIN lenses that have piano surfaces are known as Wood
`lenses, after R.W. Wood who invented them in 19058.
`lenses, after R.W. Wood who invented them in 1905**.
`
`interesting features of radial GRIN design can be associated with
`interesting features of radial GRIN design can be associated with
`Several
`Several
`individual coefficients
`in equation (2).:
`in equation (2).:
`individual coefficients
`
`1.) The index of the "base" glass is given by the constant term, N00; it is the
`1.) The index of the "base" glass is given by the constant term, NQO» it is the
`index that one specifies in a homogenous design.
`index that one specifies in a homogenous design.
`2.) N10 is the coefficient of the quadratic term. This term has the most
`is the coefficient of the quadratic term. This term has the most
`2.) NIO
`important consequences for GRIN optics. The power of a thin Wood lens is
`is
`The power of a thin Wood lens
`important consequences for GRIN optics.
`given by:
`given by:
`°grad - -2N10í
`t
`= -2N
`6
`(3)
`1 °
`grad
`(3 )
`grad is the power due to the GRIN profile and t is the thickness of the
`is the power due to the GRIN profile and t is the thickness of the
`where <&grad
`where
`This power gives the designer two new degrees of freedom in
`in
`two new degrees of freedom
`the designer
`This power gives
`element.
`element.
`controlling first -order properties: Ni 0 and element thickness,
`controlling first-order properties: NIO and element thickness, t.
`t.
`
`Because the GRIN profile has
`its own optical power,
`is effective
`is effective at
`it
`its own optical power,
`the GRIN profile has
`it
`Because
`at
`that depend on optical power.
`correcting aberrations
`Petzval
`field curvature
`Petzval field curvature
`that depend on optical power.
`correcting aberrations
`field curvature due to the
`The contribution to Petzval
`case in point.
`the
`to
`to Petzval field curvature due
`The contribution
`in point.
`is a case
`is
`a
`is given by
`presence of the gradient
`presence of the gradient is given by
`
`649rad
`
`grad
`2
`
`-2N10í
`_
`<NO/ (N00)2
`(N00
`(N00)
`(4)
`curvatures
`surface
`contribution
`the
`analogy with
`the
`in
`in
`surface curvatures
`due
`in
`the
`to
`to
`the contribution due
`analogy with
`in
`homogeneous design. Note that
`inversely proportional
`this contribution is
`to
`inversely proportional
`is
`homogeneous design. Note that this contribution
`to
`By using the GRIN profile to increase the power of an
`the square of N0 0 .
`By using the GRIN profile to increase the power of an
`the square of NQO-
`field curvature than by using
`the designer incorporates less Petzval
`than by using
`element,
`less Petzval field curvature
`the designer incorporates
`element,
`equivalent
`surface
`curvatures.
`the
`the equivalent surface curvatures.
`
`2
`
`It has no effect on power or Petzval
`3.) N2O is the fourth power coefficient.
`It has no effect on power or Petzval
`3.) N20 is the fourth power coefficient.
`It can be used to correct other aberrations without changing
`field curvature.
`to correct other aberrations without changing
`It can be used
`field curvature.
`field curvature or first -order properties.
`previously corrected Petzval
`In this
`this
`In
`previously corrected Petzval field curvature or first-order properties.
`way spherical aberration can be decoupled from Petzval
`field curvature, etc..
`way spherical aberration can be decoupled from Petzval field curvature, etc..
`The conditions for zero paraxial axial color (PAC) in a GRIN system are
`in a GRIN system are
`The conditions for zero paraxial axial color (PAC)
`derived in a paper by McLaughlin, et al9. The derivation is reproduced below.
`derived in a paper by McLaughlin, et al^. The derivation is reproduced below.
`
`The total paraxial axial color of a lens composed of thin lens elements in air
`The total paraxial axial color of a lens composed of thin lens elements in air
`
`is
`is
`
`4
`4
`
`/ SPIE Vol. 935 Gradient -Index Optics and Miniature Optics (1988)
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`all elements
`all elements
`ai A + Asurf
`1
`total == -V 7 Y2 (A* ^ + Ad) J
`PACtotal
`'
`grad
`Ygrad 7
`>^ ' J a\ v Ysurf
`,2
`,2
`«
`U ak
`a
`
`A
`
`surf
`
`surf
`V
`
`(5)
`(5)
`where ya is the height of the axial ray, and u'ak is its final angle. A (I) surf is
`is its final angle. AO surf is
`is the height of the axial ray, and u' a k
`where y a
`the difference in power of the surfaces due to the wavelength dependence of
`to the wavelength dependence of
`in power of the surfaces due
`the difference
`the base index. This can be shown to be
`the base index. This can be shown to be
`-
`
`(6)
`(6)
`
`where V is the Abbe number of dispersion:
`where V is the Abbe number of dispersion:
`
`Nood-1
`
`V
`
`N
`
`OOF" OOC
`- N
`ooc
`ooF
`
`(7)
`(7)
`Here Nood, NOOF and NOOC represent the base indices at yellow, blue and red
`Here NQOd» NO OF and NQOC represent the base indices at yellow, blue and red
`light respectively. O(1)grad is the difference in power of the GRIN profile due
`is the difference in power of the GRIN profile due
`light respectively. AOg ra d
`In the case
`to the wavelength dependence of the coefficients in equation (8).
`In the case
`to the wavelength dependence of the coefficients in equation (8).
`of a radial GRIN profile,
`is given by differentiating equation (3):
`of a radial GRIN profile, it is given by differentiating equation (3):
`it
`
`-N Jt
`. = -2(NI
`A(|>
`64grad = -2 (N10F- NIX)
`10C'
`v 10F
`Ygrad
`
`By defining a measure of dispersion for each coefficient
`By defining a measure of dispersion for each coefficient
`analogy to equation (7),
`to equation (7),
`analogy
`
`(8)
`(8)
`(2)
`in equation
`in equation (2)
`
`in
`in
`
`and Lcb grad can be written
`and A<I>g ra(j can be written
`
`Nod
`jOd
`. .
`V = ———————
`)
`- N
`(N
`jo
`0
`(NjoF -
`Njoc)
`
`4>
`°g rad
`
`V10
`
`AOgrad
`
`The total PAC for a radial GRIN thin lens singlet will be eliminated if
`The total PAC for a radial GRIN thin lens singlet will be eliminated if
`
`(9)
`(9)
`
`(10)
`( 1 0)
`
`»,urf _ '^grad
`°grad
`°surf
`V10
`V " V 10
`V
`(11)
`(11)
`the same when more than one separate element
`The procedure is
`than one separate element is present,
`the same when more
`is
`The procedure
`is present,
`but the change in axial ray height must be included in the calculations.
`but the change in axial ray height must be included in the calculations.
`Radial GRIN lenses have been used in design studies for color and Petzval
`Radial GRIN lenses have been used in design studies for color and Petzval
`camera objectives10.
`curvature
`correction
`field
`and camera objectives^. The
`binocular9
`in binocular^
`field curvature correction
`The
`and
`in
`
`5
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`inclusion of radial GRIN in an optical imaging system generally allows
`the
`imaging system generally allows
`the
`in an optical
`inclusion of radial GRIN
`to reduce of the number of elements by a
`factor of three while
`designer
`to reduce of the number of elements by a factor of three while
`designer
`Figures 3 and 4 illustrate this point.
`maintaining system performance.
`Figure
`Figure
`Figures 3 and 4 illustrate this point.
`maintaining system performance.
`is a layout of the new double Gauss camera objective reported by Mandler in
`is a layout of the new double Gauss camera objective reported by Mandler in
`3
`3
`is a high -quality camera objective, compatible with 35 -mm film
`is a high-quality camera objective, compatible with 35-mm film
`This
`This
`1980.
`1980.
`format, operating at f/2 with a focal length of 50 mm and a half -field of view
`format, operating at f/2 with a focal length of 50 mm and a half-field of view
`(HFOV) of 21.8 °. This objective was redesigned using two elements, each with a
`(HFOV) of 21.8°. This objective was redesigned using two elements, each with a
`The layout of the GRIN camera
`common, nearly parabolic index profile.
`the GRIN camera
`layout of
`The
`index profile.
`common, nearly parabolic
`objective and a schematic of the index profile are shown in Figure 4.
`The GRIN
`objective and a schematic of the index profile are shown in Figure 4. The GRIN
`essentially equivalent performance to the homogeneous design by
`design has
`the homogeneous design by
`to
`design has essentially equivalent performance
`Mandler.
`Mandler.
`
`Figure 3. New Double Gauss camera objective. Focal length = 50
`Focal length = 50
`Figure 3. New Double Gauss camera objective,
`mm, f /2, hfov = 210. From reference 10.
`mm, f/2, hfov = 21°. From reference 10.
`
`N(r)
`
`Figure 4. Equivalent radial GRIN camera objective.
`Figure 4. Equivalent radial GRIN camera objective.
`From reference 10.
`50 mm, f /2, hfov = 21 °.
`50 mm, f/2, hfov = 21°. From reference 10.
`
`Focal length =
`Focal length =
`
`Radial GRIN lenses are also used as focusing elements in small geometries.
`Radial GRIN lenses are also used as focusing elements in small geometries.
`Selfoc(R) rods have been
`These are usually called GRIN or Selfoc(R) rods.
`Selfoc( R ) rods have been
`These are usually called GRIN or Selfoc(R) rods.
`from the Nippon Sheet Glass
`commercially for
`years
`available
`several
`the Nippon Sheet Glass
`from
`for several years
`available commercially
`Company (NSG) in Japan, and are now distributed in the United States through
`Company (NSG) in Japan, and are now distributed in the United States through
`
`6
`6
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`/ SPIE Vol. 935 Gradient -Index Optics and Miniature Optics (1988)
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`a number of vendors.
`These have been used for coupling and multiplexing in
`a number of vendors. These have been used for coupling and multiplexing in
`referred to paper 935 -05 of this
`telecommunications applications (the reader is
`telecommunications applications (the reader is referred
`to paper 935-05 of this
`session by W.J. Tomlinson) and as relay lenses in endoscopes (the reader
`session by WJ. Tomlinson) and as relay lenses
`in endoscopes (the reader is
`is
`referred to paper 935 -03 of this session by D.C. Leiner).
`referred to paper 935-03 of this session by D.C. Leiner).
`
`Radial GRIN focusing rods are used in strip arrays for commercial desktop
`Radial GRIN focusing rods are used in strip arrays for commercial desktop
`(the reader is referred to paper 935 -02 of this session by J.D. Rees).
`copiers (the reader is referred
`to paper 935-02 of this session by J.D. Rees).
`copiers
`The idea here is
`that each lens has very short conjugates so that the entire
`The idea here
`is that each lens has very short conjugates so
`that the entire
`system length is on the order of a centimeter.
`Each lens images a
`system
`length
`is on
`the order of a centimeter.
`Each
`lens
`images a small
`small
`fraction of the field. The image from each lens overlaps with those of adjacent
`fraction of the field. The image from each lens overlaps with those of adjacent
`Although the total
`superposition image.
`lenses
`lenses
`in a superposition
`image. Although
`the
`total scanned field
`is
`large,
`scanned field is
`in a
`large,
`each lens element has a small aperture and conjugate distances, and needs to
`each lens element has a small aperture and conjugate distances, and needs
`to
`be corrected for a small field and aperture only.
`be corrected for a small field and aperture only.
`
`A different
`index representation than equation (2)
`is commonly used for
`is commonly used for
`index representation
`than equation (2)
`A different
`GRIN rods, although either are equally valid. The GRIN rod index profile is
`GRIN rods, although either are equally valid. The GRIN rod index profile is
`written in terms of the refractive index squared:
`written in terms of the refractive index squared:
`
`n2(r) = n* { 1 - (gr)2 + h4(gr)4 + h6(gr)6 + ...}
`n2(r) = no { 1 - (gr)2 + h4(gr)4 + h6(gr)6 + ...}
`
`(12)
`(12)
`where g is
`the focusing factor, and the h's
`referred to as
`the h's are
`the focusing factor, and
`where g
`is a constant, referred
`to as
`a constant,
`are
`additional constants used to described the index distribution. C.B. Wooley gives
`additional constants used to described the index distribution. C.B. Wooley gives
`conversion the index polynomial
`coefficients
`an approximate conversion
`the
`index polynomial coefficients and
`the
`index
`an approximate
`index
`and the
`squared polynomial assuming eighth order expansions as detailed belowi 1:
`squared polynomial assuming eighth order expansions as detailed below 11 :
`Noo • %
`= n
`N
`
`g2
`no
`N ..?£
`2
`10
`Nio
`2
`4
`n2 h
`0 4g -
`2n0
`h6 g6 - 2Nio N2o
`h6gs - 2N10 N20
`2 n0
`2 no
`h8 g8 - 2Ni0 N3o- N2o
`h8 g8 - 2 Nio N30 - N220
`2n
`2 no
`0
`
`N20
`
`N
`Nao -
`so
`N
`N4o -
`40
`
`2
`Nio
`
`(13)
`(13)
`
`isoindical
`profiles with
`Spherical GRIN profiles
`spherical
`index
`spherical
`index profiles with
`Spherical GRIN profiles are
`isoindical
`are
`The planar
`They are sometimes used in micro -optics applications.
`surfaces.
`They are sometimes used
`in micro-optics applications.
`The planar
`surfaces.
`the Tokyo Institute of Technology,
`microlens array, developed at
`is an example
`microlens array, developed at the Tokyo Institute of Technology,
`is an example
`These are made as hemispherical
`of spherical GRIN profiles
`lenslets
`of spherical GRIN profiles 12
`These are made as hemispherical
`lenslets
`12
`embedded in a substrate glass (see discussion of field -assisted ion exchange in
`embedded in a substrate glass (see discussion of field-assisted ion exchange in
`They are used singly or sandwiched together to form a spherical
`section 3.2).
`section 3.2). They are used singly or sandwiched together to form a spherical
`lens (see Figure 5). Multiple stacks of these lens arrays in combination with
`lens (see Figure 5). Multiple stacks of these lens arrays
`in combination with
`
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`
`filters, modulators,
`fiber optic
`for fiber optic coupling and
`for
`etc. have been proposed
`coupling and
`filters, modulators, etc. have been proposed
`gives a review of
`integrated optics applications. Chapter 10 of reference 13
`gives a review of
`integrated optics applications. Chapter 10 of reference 13
`the proposed technology. Figure 6 is a schematic of one such stacked array.
`the proposed technology. Figure 6 is a schematic of one such stacked array.
`r n(r)
`
`(a)
`
`(b)
`
`r n(r)
`
`HEMISPHERICAL LENS
`HEMISPHERICAL LENS
`
`n2
`
`f
`
`STACKED SPHERICAL LENS
`STACKED SPHERICAL LENS
`Planar microlenses: hemispherical
`(top) and stacked
`(top)
`and stacked
`Figure 5.
`Planar microlenses: hemispherical
`Figure 5.
`From reference 12.
`spherical (bottom).
`From reference 12.
`spherical (bottom).
`
`2-D ARRAYED
`2 -D ARRAYED
`OPTICAL COMPONENTS
`OPTICAL COMPONENTS
`
`FIBER
`
`FIBER
`
`PLANAR MICROLENS
`PLANAR MICROLENS
`
`Figure 6. Assembly of stacked planar optics using two microlens
`Figure 6. Assembly of stacked planar optics using two microlens
`arrays for fiber coupling.
`From reference 13.
`From reference 13.
`arrays for fiber coupling.
`
`2. REQUIRED GRIN PARAMETERS
`2. REQUIRED GRIN PARAMETERS
`
`to be controlled in the manufacture of
`the manufacture of
`There
`in
`are three basic parameters
`to be controlled
`three basic parameters
`There are
`the maximum attainable refractive
`GRIN lens blanks.
`index
`the maximum attainable refractive
`first
`index
`The
`is
`first
`The
`lens blanks.
`is
`GRIN
`length in any radial GRIN lens
`(An). The An determines the focal
`in any radial GRIN lens
`difference
`length
`(An). The An determines the focal
`difference
`to make
`that contains a non -zero quadratic coefficient.
`Therefore,
`in order
`to make
`in order
`Therefore,
`that contains a non-zero quadratic coefficient.
`For positive lenses, which are the most
`large An's are required.
`For positive lenses, which are the most
`lenses,
`fast
`large An's are required.
`fast lenses,
`common type,
`in the center
`and decreases
`the center and decreases
`refractive
`index is highest
`in
`is highest
`the
`index
`the refractive
`type,
`common
`towards the edge.
`towards the edge.
`
`negative An has been
`largest
`reported in
`fabricated
`the patent
`the patent
`in
`The
`reported
`fabricated negative An has been
`largest
`The
`literature in a patent assigned to Nippon Sheet Glass (U.S. Patent #4,462,663)
`14
`literature in a patent assigned to Nippon Sheet Glass (U.S. Patent #4,462,663) 14 .
`
`8
`8
`
`/ SPIE Vol. 935 Gradient-Index Optics and Miniature Optics (1988)
`/ SPIE Vol. 935 Gradient -Index Optics and Miniature Optics (1988)
`
`8
`
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`
`The inventors reported a fabricated An of -0.13 by ion exchange in glass.
`The inventors reported a fabricated An of -0.13 by ion exchange in glass. Such
`Such
`large negative An's are possible by melting a very high index homogeneous
`index homogeneous
`large negative An's are possible by melting a very high
`thallium cation can
`containing thallium.
`univalent
`glass
`then
`The
`then be
`thallium cation can
`The univalent
`be
`thallium.
`glass containing
`at an elevated
`exchanged with another univalent cation,
`such as potassium,
`exchanged with another univalent cation, such as potassium, at an elevated
`The large difference in polarizabilities of the exchanging ions
`ions
`in polarizabilities of the exchanging
`temperature.
`large difference
`The
`temperature.
`rise to a corresponding change in refractive index.
`Unless
`gives
`ion
`the
`Unless
`ion
`index.
`the
`in refractive
`to a corresponding change
`rise
`gives
`larger concentration of thallium will
`taken to equilibrium,
`thallium will
`larger concentration of
`is
`exchange
`to equilibrium, a
`taken
`is
`exchange
`a
`remain in the lens center, while most of the incoming cation will be found
`lens center, while most of the incoming cation will be found
`the
`in
`remain
`near the glass
`result,
`illustrates
`this point.
`Figure 7
`surface.
`the
`As a result,
`the
`this point.
`illustrates
`Figure 7
`the glass surface.
`near
`As
`a
`refractive index will be lowered at the glass surface.
`refractive index will be lowered at the glass surface.
`
`0.5mm
`0.5mm
`Concentration profiles of index modifier elements in a
`in a
`Figure 7. Concentration profiles of index modifier elements
`Figure 7.
`radial GRIN rod.
`From reference 14.
`radial GRIN rod. From reference 14.
`
`Positive An's are also desirable for GRIN imaging systems. Misawa, et al,
`Positive An's are also desirable for GRIN imaging systems. Misawa, et al,
`reported the largest positive An to date.
`This was accomplished by diffusing
`This was accomplished by diffusing
`to date.
`reported the largest positive An
`thallium into Schott TiF6 optical glass. The resulting An was 0.2712. Although
`thallium into Schott TiF6 optical glass. The resulting An was 0.27 12 . Although
`the resulting glass is somewhat brittle,
`it has excellent optical transparency.
`it has excellent optical transparency.
`the resulting glass is somewhat brittle,
`
`axial GRIN imaging systems designs
`radial
`Large
`are
`imaging systems designs are commonly
`commonly
`radial and axial GRIN
`Large
`and
`constrained to An's no greater than 0.10. Bigger An's may be useful, but they
`constrained to An's no greater than 0.10. Bigger An's may be useful, but they
`is thought to be the largest value that
`are usually held to this limit because it
`are usually held to this limit because it is thought to be the largest value that
`can be routinely fabricated in large geometries.
`large geometries.
`in
`can be routinely fabricated
`
`Many GRIN lens designs also require
`The profile
`large profile depths.
`The profile
`large profile depths.
`lens designs also require
`Many GRIN
`depth is equal to the radius of a radial GRIN lens.
`In order to make radial GRIN
`In order to make radial GRIN
`depth is equal to the radius of a radial GRIN lens.
`reach
`profile
`of
`objective
`sometimes
`designs,
`lens
`tens
`depths must
`tens of
`reach
`sometimes
`lens designs, profile depths must
`objective
`millimeters9,10.
`to find applications for which
`An alternative approach is
`to find applications for which
`is
`An alternative approach
`millimeters^* 10 .
`small radial GRIN elements are useful. The Selfoc(R) rod lens is one example of
`small radial GRIN elements are useful. The Selfoc(R) rod lens is one example of
`this philosophy.
`this philosophy.
`
`the profile depth determines the maximum allowable
`In axial GRIN lenses,
`the maximum allowable
`In axial GRIN lenses, the profile depth determines
`sag of the lens surface.
`In this case the required depths are smaller, generally
`In this case the required depths are smaller, generally
`sag of the lens surface.
`on the order of a few millimeters.
`on the order of a few millimeters.
`
`9
`SPIE Vol. 935 Gradient -Index Optics and Miniature Optics (1988) /
`SPIE Vol. 935 Gradient-Index Optics and Miniature Optics (1988) / 9
`
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`
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`
`index profile is also important to the GRIN lens designers.
`The shape of the
`The shape of the index profile is also important to the GRIN lens designer5 .
`have been shown to be useful in axial GRIN collimators4
`Linear index profiles
`Linear index profiles have been shown to be useful in axial GRIN collimators4
`e used in focusing rods 14. More general index profile
`Parabolic profiles ar
`index profile
`in focusing rods 14 . More general
`Parabolic profiles are used
`are useful
`well- corrected multi -element designs with conventional
`types
`in well-corrected multi-element designs with conventional
`types are useful
`in
`10.
`specifications
`specifications ^.
`
`The above specifications mandate new materials in order to fabricate GRIN
`in order to fabricate GRIN
`The above specifications mandate new materials
`lens designs. As recently as three years ago, the number of GRIN lens designs
`lens designs. As recently as three years ago, the number of GRIN lens designs
`the axial GRIN collimator4 the
`that had been fabricated was
`limited to
`the
`the axial GRIN collimator4
`to
`limited
`fabricated was
`that had been
`arrays12 and those designs which incorporated the Selfoc(R)
`microlens
`the Selfoc(^) rods.
`those designs which incorporated
`microlens arrays 12 and
`rods.
`fabrication and testing technologies have advanced to the
`Lately, both the
`the
`to
`technologies have advanced
`testing
`the fabrication and
`Lately, both
`The remainder of this paper
`point where new designs are being fabricated.
`The remainder of this paper
`point where new designs are being fabricated.
`recent advances.
`reviews
`these
`these recent advances.
`reviews
`
`3.1 Ion Exchange in Glass
`3.1 Ion Exchange in Glass
`
`3. GRIN MATERIALS
`3. GRIN MATERIALS
`
`The most common method for
`fabricating GRIN materials
`cationic
`is
`is cationic
`fabricating GRIN materials
`for
`The most common method
`diffusion from a molten salt
`The base
`into a batch -melted (or "base ") glass.
`The base
`diffusion from a molten salt into a batch-melted (or "base") glass.
`glass can be thought of as containing three categories of oxide components.
`thought of as containing three categories of oxide components.
`glass can be
`the intermediates, and the glass modifiers.
`The formers
`The glass formers,
`The formers
`the glass modifiers.
`intermediates, and
`the
`The glass formers,
`determine the basic molecular structure of the glass and can form glasses by
`determine the basic molecular structure of the glass and can form glasses by
`themselves. Examples of glass formers are silica (Si02), phosphorous pentoxide
`themselves. Examples of glass formers are silica (SiO2), phosphorous pentoxide
`(P205), germanium dioxide (Ge02) and boron oxide (B203).
`The intermediates
`(P2O5), germanium dioxide (GeO2) and boron oxide (6203). The intermediates
`also contribute to the structure of the glass, but cannot be made into glasses by
`also contribute to the structure of the glass, but cannot be made into glasses by
`themselves. Alumina (A1203), lead oxide (PbO) and zincite (ZnO) are examples
`themselves. Alumina (A12O3), lead oxide (PbO) and zincite (ZnO) are examples
`The network modifiers are weakly bound to the other
`of glass intermediates.
`The network modifiers are weakly bound to the other
`of glass intermediates.
`glass components by non -bridging oxygens and so have little effect on the
`the
`little effect on
`glass components by non-bridging oxygens and so have
`glass structure. These include the alkali oxides, e.g. soda (Na20), potash (K20 )
`glass structure. These include the alkali oxides, e.g. soda (Na2O), potash (K2O)
`and lithia (Li20), and many other oxides such as lime (CaO), barium oxide (BaO),
`and lithia (Li2O), and many other oxides such as lime (CaO), barium oxide (BaO),
`cesium oxide (Cs20) and rubidium monoxide (Rb20), etc. A complete table of
`cesium oxide (Cs2O) and rubidium monoxide (Rb2O), etc. A complete table of
`glass components and their classifications can be found in reference15.
`in reference 15 .
`glass components and their classifications can be found
`
`The basic picture of the structure of oxide glasses is
`the random network
`is the random network
`The basic picture of the structure of oxide glasses
`In this model the glass formers and intermediates are arranged in a
`in a
`interm