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Design of an all-glass achromatic-fourier-transform lens
Volume 47, number 6
OPTICS COMMUNICATIONS
15 October 1983
DESIGN OF AN ALL-GLASS ACHROMATIC-FOURIER-TRANSFORM LENS *
Chris BROPHY The Institute of Optics, University of Rochester, Rochester, N Y 1462 7, USA Received 12 July 1983
An all-glass F/18 achromatic-Fourier-transform lens is presented. The lens is capable of forming 250 adjacent, achromatized Airy disks across a 7 mm diameter of the optical transform plane. Achromatization is achieved for wavelengths ranging between 450 nm and 650 rim. The final system is approximately 1.2 meters long and is composed of three widely spaced lens groups. Aspects of the lens design are discussed including first-order layout, real ray optimization and practical limitations of all-glass achromatic-Fourier-transform lenses.
1. Introduction
The achromatic-Fourier-transform (AFT) system first proposed by Katyl [1] in 1970 - is an optical device capable of achromatizing two-dimensional Fourier transform patterns. The relationship between scalar fields in the input and output planes of an ideal AFT system [2] is given by
vo¢')= f d2xVoX) exp (-i2rrx.x'/(XoF))
,
(1)
where U is the output field, U0 is the input field, •0 is a design parameter which corresponds to some wavelength in the achromatizable spectral range and F is a parameter which determines the scale of the transform. From eq. (1) we see that U(x') wilt be independent of the illuminating wavelength, ~, if Uo(x ) is independent of k; hence the term, achromatic. The achromatization of optical Fourier transform patterns requires a strongly dispersive optical system. Therefore, almost all previous AFT system designs [ 1 - 1 0 ] have included holographic optical elements (HOE's) which have a V-number or dispersive power an order of magnitude greater than most normal glasses. Unfortunately, these holographic systems can give spurious background patterns in broadband light as a remit of the intrinsic inefficiency of HOE's operating * Research supported in part by Research Corporation. 364
away from the Bragg diffraction condition [ 11 ]. Thus it is worthwhile to consider how well an AFT system can be designed using onty normal glass elements. Wynne [12] has designed an all-glass dispersive optical system for the correction of the wavelength-dependent smearing seen in white light stellar speckle images. This simple but cleverly designed system is capable of operating as an AFT system - achromatizing transform patterns formed in white light (A~, ~ 300 rim). Wynne's system suffers, however, from a relatively low space-bandwidth product. Working at F/170 over a patch in the transform plane 7 m m in diameter, the system is able to form approximately 30 achromatized Airy disks across this diameter. In the present paper an all-glass AFT system design with improved space-bandwidth product is presented. Procedures for first-order layout and optimization are described. Also, limitations of all-glass AFT systems are briefly discussed.
2. First-order design For the purposes of optical design it is useful to consider an AFT system as a type of imaging system. Shown schematically in fig. 1 are the source plane (P1), the diffracting object plane (P2) and the transform plane (P3) - common to all optical Fourier transform systems. In the case of no modulation of the incident
0 030-4018/83/0000-0000/$ 03.00 © 1983 North-Holland
Volume 47, number 6
OPTICS COMMUNICATIONS
AFT
Fig. 1. An AFT system consideredas in imaging system. PI source plane; P2 - diffracting-objectplane; P 3 - transform plane.
field by an object in P2 the transform system will produce an image of the source in the transform plane, P3. For systems requiring spatially-coherent light the source is generally a point (unresolvable object) located directly on axis lying, effectively, an infinite distance away. However, if off-axis points in the source plane are considered it can be shown by simple scalar diffraction theory that an achromatic.Fourier.transform system will image these points with a lateral magnification inversely proportional to wavelength. Therefore, one may speak of an AFT system as an imaging system with the following constraints: 1) The equivalent imaging system must have a magnification inversely proportional to wavelength for all sourceplane points out to some cutoff angle specified in the design. This cutoff angle is related to the maximum spatial frequency passed by the AFT system via the grating equation. 2) There must be no vignetting of rays passing through the aperture stop at angles less than or equal to the cutoff angle. Since the position and size of the aperture stop will be the same as the position and size of the eventual diffracting object this constraint ensures that diffracted light from every point on the diffracting object will reach the transform plane when the system is used in an AFT configuration. The conditions for an imaging system with magnification inversely proportional to wavelength can be expressed as
r'(x)= r0tx/x 01 , u'(x)= u0[x/x0],
(2,3)
where Y'(X) and U'(X) are the axial ray height and axial ray angle, respectively, just after the fmal surface of the lens. Eqs. (2) and (3) ensure the same back focal length for the system in each wavelength while providing an effective focal length that varies inversely with wavelength. The first-order design proceeds by assuming that
15 October 1983
the system can be represented as a series of thin doublets each composed of a crown glass element and a flint glass element. Each doublet is separated from neighboring doublets by an airspace to be determined. The choice of optical materials is limited to normal glasses. In order to maximize the dispersive power of each doublet the greatest possible V-number difference for the flint and crown glasses is desired; thus, SF56 and BK7 were chosen as the flint and crown, respectively. Since optical index and consequently the powers of the various lens elements have a complicated dependence on ~ it is necessary to satisfy eqs. (2) and (3) for a discrete set of wavelengths. Therefore, a set of three wavelengths: XB = 450 nm; ~3 = 550 nm; ~kR = 650 rim, covering the spectral bandpass of the AFT system, is chosen. Eqs. (2) and (3) now represent six equations to be solved algebraically. The unknowns are the curvature differences of the crown and flint elements and the airspaces between doublets. The expressions for Y' and U' in terms of the airspaces and curvature differences are found from propagation of the axial ray in each color incident at the edge of the stop, which in all designs considered is in contrast with the first doublet. The equations used for propagation of the paraxial ray height and angle are the well-known ray transfer and refraction equations from Gaussian optics [13]. Since there are six equations to be solved, it is natural to first choose a system composed of two doublets. In this case there will be six unknowns (i.e. four curvature differences and two airspaces). The sole solution, however, yields negative values for the airspaces and is thus not a physical system. A third crown-flint doublet is necessary in order to give physical AFT systems corrected at three wavelengths. A system of three doublets has nine variables leaving three free parameters with which to characterize the solutions. The selection of these parameters is somewhat arbitrary;however, we chose the heights of the green axial ray on the second and third doublets and the ratio of crown power to flint power (at 550 nm) in the first doublet as the three parameters. As an initial criterion used in selecting some solutions over others, the length of each system is normalized and the systems with the smallest curvature differences are chosen. This procedure leads to several physical solutions which fall into either the Type 1 or Type 2 category as shown in fig. 2. The Type 1 lens has an overall 365
Volume 47, number 6 TYPE
TYPE
1 : (÷)
OPTICS COMMUNICATIONS
(-)
(÷)
(*)
(+~
2 : (÷)
15 October 1983
of lens performance. Constraints, however, must be placed on the relationships between image heights in each zoom position. Shown below is a line of code specifying a constraint of this sort. DEF RGDIFF = Y1 (F2, SI, Z2) - Y1 (F2, SI, Z1)
(4)
Fig. 2. Types of AFT system configurations. The types are characterized by the axial ray height at the lens groups, L2 and L3. The overall powers of the lens groups are shown in parentheses. Also, the path of the green axial ray is shown schematically.
positive lens power for the first doublet, an overall negative power for the second doublet and an overall positive power for the third doublet. The Type 2 lens has overall positive lens powers for all three doublets. The second doublet, L2, in the Type 2 lens acts as a field lens for off-axis rays - turning them back into the system, while L 2 in the Type 1 lens tends to diverge off-axis rays out of the system. Therefore, a particular solution of the Type 2 category is chosen as the starting system.
3. Optimization A starting system is obtained through the procedure outlined above after having replaced the thin doublets with thick cemented triplets. The length of the system is set at 1 meter. This length combined with the conversion of the doublets into triplets gives reasonable values for the curvatures of the lens elements. Care should be taken to preserve the overall power and the crown to flint power ratio in each lens group. Having completed these steps the system is now in a form suitable for real-ray optimization. With the options available on the lens-design program, CODE V [14], there are many possible merit functions which can be constructed for the optimization of an imaging system with 1/~ magnification. One possibility is to consider this system as three separate systems by treating the wavelength and the cutoff field angle as zoomed parameters. The built-in merit function of CODE V, which is roughly the square of the blur circle radius, can then be retained as a measure 366
where RGDIFF is defined to be the difference in ray height at the image surface (SI) for a chief ray (Y1) at the maximum field angle (F2) between zoom position 2 and zoom position 1. If zoom positions 2 and 1 have been specified previously in the optimization sequence to trace only red and green rays, respectively; RGDIFF will be the difference between the red image height and the green image height. In order to avoid the optimization of rays at certain color/field combinations which might be overly constraining, the maximum field angle is scaled in each zoom position to be proportional to wavelength. Thus, if RGDIFF is constrained to be zero, the net effect will be to impose a magnification that is inversely proportional to wavelength at these two wavelengths. A similar constraint is imposed on the difference between the blue and green image heights and constraints are put on the axial ray height differences in each color in order to ensure a common back focus. After approximately 25 optimization cycles the starting lens will be diffraction-limited for a maximum field angle of 0.5 ° at 550 nm which corresponds to an AFT cutoff frequency of 15 mm -1. The term, diffraction-limited, as used here designates a system which for each of the three colors - holds the opd's of all rays to less than k/4 and which - at intermediate wavelengths - holds the deviation of the image height from the 1/k dependence to less than the Airy disk diameter. To achieve diffraction-limited performance at larger field angles it is necessary to reduce the curvatures on the last lens group by adding more elements.
AFT
OIFFRACTING
OBJECT
L3
PLANE
,?-i I ~ eeo - - t
I-loB.* ~ I
Fig. 3. Scale drawing of the final AFT lens. L 1, L2 and L 3 are the three distinct lens groups. All dimensions are given in millimeters.
Volume 47, number 6
OPTICS COMMUNICATIONS
Table 1. Surface data for the final AFT lens shown in fig. 3. All dimensions are in millimeters. THICKNESS 131.0 - 238.0 49.0
85.7 199.2
-33.0 41.1 -27.2 137.0 -44.5 -96.1 92.1
-104.6 -433.8 38.9
45.5 OO
4.0 10.7
SF58 BK7 SF56
12.9
264.3
AIR
4.0 22.9 5.0
8K7 SF56 BK7
880.0
AIR
15.4
BK7 SF56 AIR BK7 AIR
19.4
1.0 20.0 1.0 6.6
!
AIR BK7
25.0
The smaller curvatures resulting from this step - if nothing else - increase the edge diameters of the ele. ments enough to catch the ray bundle at the maximum field angle. An F[18 (at 550 nm) AFT lens is shown in fig. 3. The radius of curvature, axial thickness and glass type for each element are given in table 1. This lens is capable
10"
Ay
SO-
-5. -IO.
,~o
s~o
of diffraction-limited performance for a diffracting object size 10 mm in diameter and for a cutoff spatial frequency of ~27 mm -1 . This gives a space-bandwidth product corresponding to 250 achromatized Airy disks across a diameter of the transform plane. The variation of chief ray height from the AFT condition is shown in fig. 4.
4. Discussion
SF50
18.3
15 October 1983
do X(nm)
Fig. 4. Expectect variation in chief ray height from the AFT condition for the lens of fig. 3. The case shown is for ray angles which correspond to diffraction from a grating with a spatial frequency of 27 lp/mm. The distance between dashed lines is the diameter of the Airy disk for an aperture diameter of 10 mm.
A third lens group was necessary in order to achieve the 1/?, image magnification at three wavelengths. For the system of fig. 3 the maximum deviation from the 1/?, image height at intermediate wavelengths was just less than the Airy disk diameter (see fig. 4). If the spectral bandpass were extended from 200 nm to 300 nm (i.e. the width of the visible spectrum) the deviation would have exceeded the Airy disk size. One way to accommodate a larger bandwidth without reducing the aperture size might be to require 1/), magnification at more than three wavelengths. Correction at four wavelengths with the three-lens-group system was attempted but the system failed to optimize as if it were overconstrained. Thus, a fourth lens group may be necessary for good AFT performance over the entire visible range when large diffracting~bject diameters are to be used. In addition to the limitation on spectral bandwidth there is also a limit to the maximum spatial frequency carried by an all-glass AFT system which is ultimately determined by the system's length. The cutoff spatial frequency of the design in fig. 3 corresponds to a half field of view in the equivalent imaging system of only 1° at 650 nm. On the other hand a long system is necessary for good AFT correction. This can be explained in terms of a relatively recent addition to the theory of chromatic aberration [15]. The classical theory of paraxial chromatic aberration is inadequate in that it assumes the same ray height in each wavelength for the calculation of surface contributions. In fact, for widely spaced elements the ray heights in certain colors can become so different at some surfaces that additional contributions of first order can result. It is these extra contributions which must be managed in the design of an AFT system made out of normal glass. Thus the axial ray behavior, which determines the ?,-dependent magnification and which requires large ray height differences in widely separated wavelengths, is aided by 367
Volume 47, number 6
OPTICS COMMUNICATIONS
a long system length while the required off-axis ray behavior, which determines the cutoff spatial frequency of the AFT system, is restricted by a long system length. This tradeoff is inevitable ; however, improved performance with shorter system lengths might be obtained by moving the diffracting-object plane inside the physical system. The placement of the stop position inside an imaging system generally leads to accommodation of larger field angles. Future all-glass AFT systems must concentrate on increasing the aperture diameter, spectral bandwidth and cutoff spatial frequency. Realistically, one cannot expect these systems to be much shorter than 1 meter; for applications where a small physical size is important, folding of the optical path will be necessary. For comparison, holographic AFT systems with spatial frequency cutoffs on the order of 50 m m -1 have been made which are about a half a meter long. From the experience of designing the system of fig. 3 a cutoff frequency of 50 m m -1 is probably near - if not over - the upper limit for all-glass systems. To accommodate even higher spatial frequencies HOE's will almost certainly be required and the use of off-axis HOE configurations [7] may provide a means of overcoming problems with diffraction efficiency. Nevertheless, all-glass AFT systems are fairly easily handled by computer-aided lens design techniques and can provide adequate performance for several applications [5,9,16]. The lens of fig. 3 and table 1 is currently being constructed and will be used in polychromatic speckle and image processing experiments.
368
15 October 1983
Acknowledgements The author wishes to thank Professors G.M. Morris and R.E. Hopkins for their helpful advice on this subject.
References [1] R.H. Katyl, Appl. Optics 11 (1972) 1241. [2] G.M. Morris, Appl. Optics 20 (1981) 2017. [3] R. Ferriere, J.P. Goedgebuer and J.C. Vienot, Optics Comm. 31 (1979) 285. [4] E.N. Leith and J.A. Roth, Appl. Optics 18 (1979) 2803. [5] G.D. Collins, Appl. Optics 20 (1981) 3109. [6] G.M. Morris, Optics Comm. 39 (1981) 143. [7] J. Upatnieks, J.G. Duthie and P.R. Ashley, Tech. Rep. RR-82-5, U.S. Army Missile Command, Redstone Arsenal (1982). [8] R.E. Hopkins and G.M. Morris, J. Opt. Soc. Am. 72 (1982) 1150A. [9] N. George and G.M. Morris, in: Advances in opti~l information processing (SPIE, Vol. 388, 1983). [10] R. Ferriere and J.P. Goedgebuer, Appl. Optics 22 (1983) 1540. [11] T. Stone and N. George, Optics Lett. 7 (1982) 445. [12] C.G. Wynne, Optics Comm. 28 (1979) 21. [13] R. Kingslake, Lens design fundamentals (Academic Press, New York, 1978). [ 14 ] CODE V is a registered trademark of Optical Research Associates. [15] C.G. Wynne, Optica Acta 25 (1978) 627. [16] C. Brophy and G.M. Morris, J. Opt. Soc. Am. 73 (1983) 87.
OPTICS COMMUNICATIONS
15 October 1983
DESIGN OF AN ALL-GLASS ACHROMATIC-FOURIER-TRANSFORM LENS *
Chris BROPHY The Institute of Optics, University of Rochester, Rochester, N Y 1462 7, USA Received 12 July 1983
An all-glass F/18 achromatic-Fourier-transform lens is presented. The lens is capable of forming 250 adjacent, achromatized Airy disks across a 7 mm diameter of the optical transform plane. Achromatization is achieved for wavelengths ranging between 450 nm and 650 rim. The final system is approximately 1.2 meters long and is composed of three widely spaced lens groups. Aspects of the lens design are discussed including first-order layout, real ray optimization and practical limitations of all-glass achromatic-Fourier-transform lenses.
1. Introduction
The achromatic-Fourier-transform (AFT) system first proposed by Katyl [1] in 1970 - is an optical device capable of achromatizing two-dimensional Fourier transform patterns. The relationship between scalar fields in the input and output planes of an ideal AFT system [2] is given by
vo¢')= f d2xVoX) exp (-i2rrx.x'/(XoF))
,
(1)
where U is the output field, U0 is the input field, •0 is a design parameter which corresponds to some wavelength in the achromatizable spectral range and F is a parameter which determines the scale of the transform. From eq. (1) we see that U(x') wilt be independent of the illuminating wavelength, ~, if Uo(x ) is independent of k; hence the term, achromatic. The achromatization of optical Fourier transform patterns requires a strongly dispersive optical system. Therefore, almost all previous AFT system designs [ 1 - 1 0 ] have included holographic optical elements (HOE's) which have a V-number or dispersive power an order of magnitude greater than most normal glasses. Unfortunately, these holographic systems can give spurious background patterns in broadband light as a remit of the intrinsic inefficiency of HOE's operating * Research supported in part by Research Corporation. 364
away from the Bragg diffraction condition [ 11 ]. Thus it is worthwhile to consider how well an AFT system can be designed using onty normal glass elements. Wynne [12] has designed an all-glass dispersive optical system for the correction of the wavelength-dependent smearing seen in white light stellar speckle images. This simple but cleverly designed system is capable of operating as an AFT system - achromatizing transform patterns formed in white light (A~, ~ 300 rim). Wynne's system suffers, however, from a relatively low space-bandwidth product. Working at F/170 over a patch in the transform plane 7 m m in diameter, the system is able to form approximately 30 achromatized Airy disks across this diameter. In the present paper an all-glass AFT system design with improved space-bandwidth product is presented. Procedures for first-order layout and optimization are described. Also, limitations of all-glass AFT systems are briefly discussed.
2. First-order design For the purposes of optical design it is useful to consider an AFT system as a type of imaging system. Shown schematically in fig. 1 are the source plane (P1), the diffracting object plane (P2) and the transform plane (P3) - common to all optical Fourier transform systems. In the case of no modulation of the incident
0 030-4018/83/0000-0000/$ 03.00 © 1983 North-Holland
Volume 47, number 6
OPTICS COMMUNICATIONS
AFT
Fig. 1. An AFT system consideredas in imaging system. PI source plane; P2 - diffracting-objectplane; P 3 - transform plane.
field by an object in P2 the transform system will produce an image of the source in the transform plane, P3. For systems requiring spatially-coherent light the source is generally a point (unresolvable object) located directly on axis lying, effectively, an infinite distance away. However, if off-axis points in the source plane are considered it can be shown by simple scalar diffraction theory that an achromatic.Fourier.transform system will image these points with a lateral magnification inversely proportional to wavelength. Therefore, one may speak of an AFT system as an imaging system with the following constraints: 1) The equivalent imaging system must have a magnification inversely proportional to wavelength for all sourceplane points out to some cutoff angle specified in the design. This cutoff angle is related to the maximum spatial frequency passed by the AFT system via the grating equation. 2) There must be no vignetting of rays passing through the aperture stop at angles less than or equal to the cutoff angle. Since the position and size of the aperture stop will be the same as the position and size of the eventual diffracting object this constraint ensures that diffracted light from every point on the diffracting object will reach the transform plane when the system is used in an AFT configuration. The conditions for an imaging system with magnification inversely proportional to wavelength can be expressed as
r'(x)= r0tx/x 01 , u'(x)= u0[x/x0],
(2,3)
where Y'(X) and U'(X) are the axial ray height and axial ray angle, respectively, just after the fmal surface of the lens. Eqs. (2) and (3) ensure the same back focal length for the system in each wavelength while providing an effective focal length that varies inversely with wavelength. The first-order design proceeds by assuming that
15 October 1983
the system can be represented as a series of thin doublets each composed of a crown glass element and a flint glass element. Each doublet is separated from neighboring doublets by an airspace to be determined. The choice of optical materials is limited to normal glasses. In order to maximize the dispersive power of each doublet the greatest possible V-number difference for the flint and crown glasses is desired; thus, SF56 and BK7 were chosen as the flint and crown, respectively. Since optical index and consequently the powers of the various lens elements have a complicated dependence on ~ it is necessary to satisfy eqs. (2) and (3) for a discrete set of wavelengths. Therefore, a set of three wavelengths: XB = 450 nm; ~3 = 550 nm; ~kR = 650 rim, covering the spectral bandpass of the AFT system, is chosen. Eqs. (2) and (3) now represent six equations to be solved algebraically. The unknowns are the curvature differences of the crown and flint elements and the airspaces between doublets. The expressions for Y' and U' in terms of the airspaces and curvature differences are found from propagation of the axial ray in each color incident at the edge of the stop, which in all designs considered is in contrast with the first doublet. The equations used for propagation of the paraxial ray height and angle are the well-known ray transfer and refraction equations from Gaussian optics [13]. Since there are six equations to be solved, it is natural to first choose a system composed of two doublets. In this case there will be six unknowns (i.e. four curvature differences and two airspaces). The sole solution, however, yields negative values for the airspaces and is thus not a physical system. A third crown-flint doublet is necessary in order to give physical AFT systems corrected at three wavelengths. A system of three doublets has nine variables leaving three free parameters with which to characterize the solutions. The selection of these parameters is somewhat arbitrary;however, we chose the heights of the green axial ray on the second and third doublets and the ratio of crown power to flint power (at 550 nm) in the first doublet as the three parameters. As an initial criterion used in selecting some solutions over others, the length of each system is normalized and the systems with the smallest curvature differences are chosen. This procedure leads to several physical solutions which fall into either the Type 1 or Type 2 category as shown in fig. 2. The Type 1 lens has an overall 365
Volume 47, number 6 TYPE
TYPE
1 : (÷)
OPTICS COMMUNICATIONS
(-)
(÷)
(*)
(+~
2 : (÷)
15 October 1983
of lens performance. Constraints, however, must be placed on the relationships between image heights in each zoom position. Shown below is a line of code specifying a constraint of this sort. DEF RGDIFF = Y1 (F2, SI, Z2) - Y1 (F2, SI, Z1)
(4)
Fig. 2. Types of AFT system configurations. The types are characterized by the axial ray height at the lens groups, L2 and L3. The overall powers of the lens groups are shown in parentheses. Also, the path of the green axial ray is shown schematically.
positive lens power for the first doublet, an overall negative power for the second doublet and an overall positive power for the third doublet. The Type 2 lens has overall positive lens powers for all three doublets. The second doublet, L2, in the Type 2 lens acts as a field lens for off-axis rays - turning them back into the system, while L 2 in the Type 1 lens tends to diverge off-axis rays out of the system. Therefore, a particular solution of the Type 2 category is chosen as the starting system.
3. Optimization A starting system is obtained through the procedure outlined above after having replaced the thin doublets with thick cemented triplets. The length of the system is set at 1 meter. This length combined with the conversion of the doublets into triplets gives reasonable values for the curvatures of the lens elements. Care should be taken to preserve the overall power and the crown to flint power ratio in each lens group. Having completed these steps the system is now in a form suitable for real-ray optimization. With the options available on the lens-design program, CODE V [14], there are many possible merit functions which can be constructed for the optimization of an imaging system with 1/~ magnification. One possibility is to consider this system as three separate systems by treating the wavelength and the cutoff field angle as zoomed parameters. The built-in merit function of CODE V, which is roughly the square of the blur circle radius, can then be retained as a measure 366
where RGDIFF is defined to be the difference in ray height at the image surface (SI) for a chief ray (Y1) at the maximum field angle (F2) between zoom position 2 and zoom position 1. If zoom positions 2 and 1 have been specified previously in the optimization sequence to trace only red and green rays, respectively; RGDIFF will be the difference between the red image height and the green image height. In order to avoid the optimization of rays at certain color/field combinations which might be overly constraining, the maximum field angle is scaled in each zoom position to be proportional to wavelength. Thus, if RGDIFF is constrained to be zero, the net effect will be to impose a magnification that is inversely proportional to wavelength at these two wavelengths. A similar constraint is imposed on the difference between the blue and green image heights and constraints are put on the axial ray height differences in each color in order to ensure a common back focus. After approximately 25 optimization cycles the starting lens will be diffraction-limited for a maximum field angle of 0.5 ° at 550 nm which corresponds to an AFT cutoff frequency of 15 mm -1. The term, diffraction-limited, as used here designates a system which for each of the three colors - holds the opd's of all rays to less than k/4 and which - at intermediate wavelengths - holds the deviation of the image height from the 1/k dependence to less than the Airy disk diameter. To achieve diffraction-limited performance at larger field angles it is necessary to reduce the curvatures on the last lens group by adding more elements.
AFT
OIFFRACTING
OBJECT
L3
PLANE
,?-i I ~ eeo - - t
I-loB.* ~ I
Fig. 3. Scale drawing of the final AFT lens. L 1, L2 and L 3 are the three distinct lens groups. All dimensions are given in millimeters.
Volume 47, number 6
OPTICS COMMUNICATIONS
Table 1. Surface data for the final AFT lens shown in fig. 3. All dimensions are in millimeters. THICKNESS 131.0 - 238.0 49.0
85.7 199.2
-33.0 41.1 -27.2 137.0 -44.5 -96.1 92.1
-104.6 -433.8 38.9
45.5 OO
4.0 10.7
SF58 BK7 SF56
12.9
264.3
AIR
4.0 22.9 5.0
8K7 SF56 BK7
880.0
AIR
15.4
BK7 SF56 AIR BK7 AIR
19.4
1.0 20.0 1.0 6.6
!
AIR BK7
25.0
The smaller curvatures resulting from this step - if nothing else - increase the edge diameters of the ele. ments enough to catch the ray bundle at the maximum field angle. An F[18 (at 550 nm) AFT lens is shown in fig. 3. The radius of curvature, axial thickness and glass type for each element are given in table 1. This lens is capable
10"
Ay
SO-
-5. -IO.
,~o
s~o
of diffraction-limited performance for a diffracting object size 10 mm in diameter and for a cutoff spatial frequency of ~27 mm -1 . This gives a space-bandwidth product corresponding to 250 achromatized Airy disks across a diameter of the transform plane. The variation of chief ray height from the AFT condition is shown in fig. 4.
4. Discussion
SF50
18.3
15 October 1983
do X(nm)
Fig. 4. Expectect variation in chief ray height from the AFT condition for the lens of fig. 3. The case shown is for ray angles which correspond to diffraction from a grating with a spatial frequency of 27 lp/mm. The distance between dashed lines is the diameter of the Airy disk for an aperture diameter of 10 mm.
A third lens group was necessary in order to achieve the 1/?, image magnification at three wavelengths. For the system of fig. 3 the maximum deviation from the 1/?, image height at intermediate wavelengths was just less than the Airy disk diameter (see fig. 4). If the spectral bandpass were extended from 200 nm to 300 nm (i.e. the width of the visible spectrum) the deviation would have exceeded the Airy disk size. One way to accommodate a larger bandwidth without reducing the aperture size might be to require 1/), magnification at more than three wavelengths. Correction at four wavelengths with the three-lens-group system was attempted but the system failed to optimize as if it were overconstrained. Thus, a fourth lens group may be necessary for good AFT performance over the entire visible range when large diffracting~bject diameters are to be used. In addition to the limitation on spectral bandwidth there is also a limit to the maximum spatial frequency carried by an all-glass AFT system which is ultimately determined by the system's length. The cutoff spatial frequency of the design in fig. 3 corresponds to a half field of view in the equivalent imaging system of only 1° at 650 nm. On the other hand a long system is necessary for good AFT correction. This can be explained in terms of a relatively recent addition to the theory of chromatic aberration [15]. The classical theory of paraxial chromatic aberration is inadequate in that it assumes the same ray height in each wavelength for the calculation of surface contributions. In fact, for widely spaced elements the ray heights in certain colors can become so different at some surfaces that additional contributions of first order can result. It is these extra contributions which must be managed in the design of an AFT system made out of normal glass. Thus the axial ray behavior, which determines the ?,-dependent magnification and which requires large ray height differences in widely separated wavelengths, is aided by 367
Volume 47, number 6
OPTICS COMMUNICATIONS
a long system length while the required off-axis ray behavior, which determines the cutoff spatial frequency of the AFT system, is restricted by a long system length. This tradeoff is inevitable ; however, improved performance with shorter system lengths might be obtained by moving the diffracting-object plane inside the physical system. The placement of the stop position inside an imaging system generally leads to accommodation of larger field angles. Future all-glass AFT systems must concentrate on increasing the aperture diameter, spectral bandwidth and cutoff spatial frequency. Realistically, one cannot expect these systems to be much shorter than 1 meter; for applications where a small physical size is important, folding of the optical path will be necessary. For comparison, holographic AFT systems with spatial frequency cutoffs on the order of 50 m m -1 have been made which are about a half a meter long. From the experience of designing the system of fig. 3 a cutoff frequency of 50 m m -1 is probably near - if not over - the upper limit for all-glass systems. To accommodate even higher spatial frequencies HOE's will almost certainly be required and the use of off-axis HOE configurations [7] may provide a means of overcoming problems with diffraction efficiency. Nevertheless, all-glass AFT systems are fairly easily handled by computer-aided lens design techniques and can provide adequate performance for several applications [5,9,16]. The lens of fig. 3 and table 1 is currently being constructed and will be used in polychromatic speckle and image processing experiments.
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15 October 1983
Acknowledgements The author wishes to thank Professors G.M. Morris and R.E. Hopkins for their helpful advice on this subject.
References [1] R.H. Katyl, Appl. Optics 11 (1972) 1241. [2] G.M. Morris, Appl. Optics 20 (1981) 2017. [3] R. Ferriere, J.P. Goedgebuer and J.C. Vienot, Optics Comm. 31 (1979) 285. [4] E.N. Leith and J.A. Roth, Appl. Optics 18 (1979) 2803. [5] G.D. Collins, Appl. Optics 20 (1981) 3109. [6] G.M. Morris, Optics Comm. 39 (1981) 143. [7] J. Upatnieks, J.G. Duthie and P.R. Ashley, Tech. Rep. RR-82-5, U.S. Army Missile Command, Redstone Arsenal (1982). [8] R.E. Hopkins and G.M. Morris, J. Opt. Soc. Am. 72 (1982) 1150A. [9] N. George and G.M. Morris, in: Advances in opti~l information processing (SPIE, Vol. 388, 1983). [10] R. Ferriere and J.P. Goedgebuer, Appl. Optics 22 (1983) 1540. [11] T. Stone and N. George, Optics Lett. 7 (1982) 445. [12] C.G. Wynne, Optics Comm. 28 (1979) 21. [13] R. Kingslake, Lens design fundamentals (Academic Press, New York, 1978). [ 14 ] CODE V is a registered trademark of Optical Research Associates. [15] C.G. Wynne, Optica Acta 25 (1978) 627. [16] C. Brophy and G.M. Morris, J. Opt. Soc. Am. 73 (1983) 87.