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Extending the bandwidth of speckle interferometry
Volume 28, number 1
OPTICS COMMUNICATIONS
January 1979
EXTENDING THE BANDWIDTH OF SPECKLE INTERFEROMETRY
C.G. WYNE Royal Greenwich Observatory, Herstmonceux Castle, Hailsham, Sussex, UK and Imperial College, London, UK
Received 5 October 1978
In speckle interferometry, the scale of the speckle array is wavelength dependent, and this limits the spectral bandwidth that can be used. A simple optical system is described to correct this effect, over a wide spectral range, by introducing a chromatic difference of magnification of opposite sign to the speckle dispersion, and variable in magnitude.
1. Introduction
Speckle interferometry, as was suggested by Labeyde [1], can be used to obtain information from the image formed by a large astronomical telescope, beyond the limits on angular resolution normally imposed by atmospheric turbulence. The method is applicable, for example, to the detection of double stars, the measurement of their separation and magnitude difference, and the measurement of stellar diameters, where these lie within the theoretical limit of resolution set by the telescope diameter, but are masked from direct observation by the seeing. The method has been made possible by the availability of image intensifiers, which enable short exposure star photographs to be taken. With exposures of the order of 0.01 s and an appropriate filter, a star image consists of a random array of speckles, the extent of the array depending on the state of the atmosphere, the speckles being of the order of size of the Airy disk of the telescope. By summing the Fourier transforms of a large number of speckle images, the high resolution information can be extracted. The transforms may be obtained either optically or by computer analysis of digitised data. A speckle interferometer and its use has been described in detail by Dainty and others [2]. It was for this instrument that the device described in this paper was designed. For polychromatic light the speckle pattern of a star image consists of the superposition of monochromatic
speckle patterns, which differ from each other in two ways. There is a difference of scale. Corresponding speckles in different wavelengths are separated from each other in a radial direction, by an amount proportional to their distance from the centre of the pattern and to the wavelength difference. In a monochrome photograph of a polychromatic speckle pattern, individual speckles therefore appear smeared radially, by amounts increasing with their distance from the pattern centre. This dispersion effect is quite large. For the extremes of the visible spectrum, the radial separation between corresponding speckles may be as much as 0.6 for 0.7 of their mean distance from the centre. In addition to the chromatic difference of scale, speckle patterns in different wavelengths are not identical in form. This decorrelation is generally the smaller effect. The chromatic dispersion and decorrelation together produce a blurring of the speckle pattern that necessitates the use of a fairly narrow bandwidth filter (between 8 and 30 nm) and hence limit the application of the speckle interferometer to relatively bright stars. To the extent that the dispersion effect predominates, an optical device correcting the chromatic difference of magnification would allow a wider spectral band width, and the observation of fainter stars. This paper is concerned with the design of such a system. The amount of the blurring of the polychromatic 21
Volume 28, number 1
OPTICS COMMUNICATIONS
speckle image depends on atmospheric conditions; there have been different views as to what was the corresponding variability of the two contributing effects. At the time that the work described in this paper was initiated, it had been suggested that the extent of the dispersion was variable [3] ; this is consistent with the visual appearance of stellar speckle patterns observed at different times. Accordingly, the corrector was designed to have a compensating chromatic difference of magnification continuously adjustable over a wide range. More recent work by Stansberg [4] suggests that the dispersion is in fact invariant, but that this is masked by variable decorrelation. The variable corrector to be described should provide experimental evidence on this question.
2. The corrector: classical theory In the interferometer referred to above, the star image, formed for example at the ]'/14 Cassegrain focus of the Isaac Newton 2.5 metre telescope, is relayed at a 12 times magnification on to the photocathode of an image intensifier whose output is photographed in a series of short exposures by a cine camera. The optical system of the relay consists of two doublets of focal lengths 5 cm and 60 cm separated by a space within which light from the star is collimated. This collimated space accommodates the f'flter, and rotatable prisms to correct the atmospheric dispersion which varies with zenith distance; it also provides a convenient location for the device to correct the chromatic difference of magnification of the speckle pattern star image. This should then be a telescope, of magnification variable up to say I : 1.35 at 400 nm, down to 1 : 0.65 at 700 nm, with a constant unit magnification at some intermediate wavelength, and with negligible aberrations apart from this chromatic effect. This can be achieved in the following way. In a collimated space, a plane-parallel glass block of uniform refractive index is afocal, and free from aberrations. For a single wavelength, a pseudo planeparallel plate can be made by cementing together the curved surfaces of a plane-convex and a plane-concave lens which are made from two glasses having the same refractive index at some mean wavelength ~'0, but differing in dispersion. At the mean wavelength, this 22
January 1979
component will be afocal and aberration free, but will manifest chromatic effects at longer and shorter wavelengths. Similarly, pseudo plane-parallel plates can be compounded from more than two single lens elements. The variable chromatic corrector telescope consists of two such triple components, with an adjustable separation between them. This clearly gives the required invariant unit magnification at the mean wavelength. At other wavelengths, these components will be convergent for shorter and divergent for longer wavelengths, or vice versa, depending on whether the converging constituent lens elements are made from the glasses used for the convergent and divergent elements will simply alter the sign of the optical power k of the complete component at any wavelength. The corrector consists of such a pair of components, whose powers, at any given wavelength, are equal and opposite. The chromatic power of one of the components, at wavelength X, is given by
k(x) where 8n is the index difference at wavelength 8 between the two glasses at a cemented surface of radius r, the summation being over the cemented surfaces of the component. In terms of the conventional first order coefficient of longitudinal chromatic aberration, CI (see for example Welford [5] ), for each component, for the wavelength interval (k - X0), CI =h2k, where h is the incidence height at each surface of a paraxial ray at the mean wavelength ~'0" In our corrector, with components afocal at k0, in a collimated space, h is the same at all surfaces. It follows therefore, since the two components have equal and opposite powers k for all wavelengths, that their CI values are equal and opposite. And since according to conventional theory the CI values of components of a system are additive, the first order longitudinal chromatic aberration of the pair of separated components is zero for all wavelengths, independent of the separation between them. The case is quite different for chromatic difference of magnification. A collimated paraxial ray at k0, incident at height h, has the same height at all surfaces. A corresponding ray at wavelength ~ emerges from the
Volume 28, number 1
OPTICS COMMUNICATIONS
/ - . A . ./
January 1979
• __
~ . . ~ . _
m
k
Fig. 1. Diagramatic: full line, paraxial ray through corrector at wavelength ko; dotted line, paraxial ray at wavelength h.
first component with convergence angle hk(k). Taking the components as thin, with a separation d, the incidence height on the second component is therefore h(1 - dk(k)) (fig. 1). The angular magnification of a telescope is inversely proportional to the ratio of exit and entrance pupil diameters, so for the corrector the angular magnification at wavelength is 1/(1 - dk(k)). For a given value of k(k), a required angular magnification, relative to unity at ~0, may be obtained by adjusting the separation d. The chromatic dispersion in the speckle pattern is such that speckles at longer wavelengths lie further from the common centre. The first component of the corrector should therefore be the one that diverges light at longer wavelengths than the mean.
3. The corrector: further considerations
The discussion in the previous section was in terms of classical first order theory. Monochromatically, at the mean wavelength k 0, the corrector is free from aberrations of all orders. And at other wavelengths, the common forms of higher order aberrations are found to be small. But there is a chromatic effect, of paraxial order, neglected in classical aberration theory, which is of significance for the design of the system. This has recently been analysed by Wynne [6,7]. This is an aberration of the same type as secondary spectrum. It is characteristic of almost all imaging systems of lenses that when achromatised so that paraxial rays of two wavelengths are brought to a common focus, then for intermediate wavelengths the focus lies closer to the lens, and for wavelengths outside the range between the two, it lies further away. This secondary spectrum aberration arises from the dif-
ferent forms of wavelength dependence of index shown by most glasses of high and low dispersions; and it has been generally accepted that it can only be eliminated by the use of anomalous materials. Wynne (loc cit)has shown that an aberration of the same form, but opposite sign, arises when classical achromatism is achieved by the cancellation of terms of opposite sign arising at widely spaced components. This can occur in a high degree in the corrector systems described above. According to classical theory, the chromatic coefficient for a lens system is the surface-by-surface sum of CI terms, discussed above. Wynne [7] has shown that there are additional terms of paraxial order. For the two separated components of our corrector, the true coefficient is EC I + dh2k(k) 2. The second term remains finite when the first sums to zero. From the discussion above, the chromatic difference of magnification of the corrector depends on the value of dk(k). If this be given the required value, in a corrector short enough to be accommodated within the apparatus described in ref. [2], then the chromatic term dh2k(~) 2 becomes undesirably large at the high magnification setting. This can be avoided by an appropriate scaling down of the focal lengths of the 5 cm and 60 cm lenses containing the collimated space. In the final design, given below, at its highest magnification setting, the aberrations nowhere exceed about ~ wavelength over the spectral range 400 to 700 rim.
4. Detailed design of eorrector
Table 1 gives numerical data for the design of a complete relay system with chromatic corrector. A radius of positive sign is convex toward the front, negative concave. 23
Volume 28, number 1
OPTICS COMMUNICATIONS
January 1979
Table 1 Relay optical system with variable chromatic difference of magnification. Magnification at 480 nm, 11.7; dimensions in ram; image diameter 7.0 ram; spectral range 400-700 nm. Radius
Axial separation
Material (Schott glass type)
27.2 to telescope focus
Diameter
air
34.94 Lens 1 focallength 28.6
1.0
SF15
3.0
13.06 2.0
~
K5
-18.09 D1
air
1.0
LaK31
2.0
SF8
1.0
LaK31
16.0 Lens 2 afocalat 480 nm
5.0
-16.0 eo
D2 = 270.0-D 1
air
o0
3.0
LaK31
1.0
SF8
3.0
LaK31
-16.0 Lens 3
12.0
16.0 -242.0 319.0 to image intensifier photocathode
air
1 If nh he the object height from the axis at wavelength hnm and R = (r/7oo - naoo)/](r77oo + n4oo), then to correct for R = 0.70: D 1 = 20, D 2 = 250; to correct for R = 0.49: D 1 = 100, D2 = 170; for R ~. 0:D1 = 270, D 2 = 0.
The focal ratio in the final image space is so large ( a b o u t f[170 at the m e a n wavelength) that a c o r r e c t e d doublet after the collimated space to p r o d u c e this convergence is n o t necessary. In the present design, it is replaced by an appropriate shallow c o n v e x curve on the last surface o f the corrector. The aberrations p r o d u c e d by this surface are negligibly small e x c e p t for a c h r o m a t i c difference o f focus, which is c o r r e c t e d on the first doublet. In use, the separation b e t w e e n lensen 1 and 3 is maintained constant, and the radial c h r o m a t i c dispersion varied by m o v i n g lens 2 to and fro b e t w e e n them. The original system can be further simplified by the elimination o f the rotatable prisms used to correct the transverse c h r o m a t i c dispersion d e p e n d i n g on 24
zenith distance. This correction can be achieved by a small transverse translation o f lens 2. Since neither the change o f refraction index o f a glass, or the change o f refractive index for a pair o f glasses, is linear w i t h wavelength, the correction by a lens system o f the linear dispersion o f the speckle
Table 2 Refractive index from Schott glass catalogue at
LaK31 SF8
400 nm
480 nm
550 nm
700 nm
1.71850 1.73130
1.70601 1.70594
1.69937 1.69361
1.69106 1.67940
Volume 28, number 1
OPTICS COMMUNICATIONS
January 1979
?(mm) 6.0
~
5.0
~;-0
A
B
o
3"0
2-0
, 1"0
I
/.00
i
500
I
600
i
700 ~. (nm)
Fig. 2. Curve A shows variation of final image height 77with wavelength, in the absence of a corrector, for dispersion R = 0.70. Curve B shows the residual variation using the corrector system of table I. (If the corrector were used at the f/14 Cassegrain focus of the 2.5 m Isaac Newton Telescope, r/= 3.5 mm corresponds to an angular displacement from the axis of 1.7 arc s). image will not be perfect. However, the residual errors can be reduced to 0.09 of the initial dispersion. To distribute the residuals symmetrically, the index difference between the two glasses of the corrector should be equal and opposite at the extremes of the spectral range. This condition is very approximately met by Schott glass types LaK31 and SF8 at 400 and 700 nm. These two glasses have almost the same refractive index at 480 nm, at which the components are afocal. Table 2 shows their indices. Fig. 2 shows, in curve A, the variation of r/with wavelength for dispersion R = 0.70, in the absence of correction, and curve B shows the residual variation for the appropriate corrector setting. For small dispersions R, the residuals are less.
A relay system with chromatic corrector based on this design has been made and is awaiting trial on a telescope.
References [1] A. Labeyrie, Astr. Astrophys. 6 (1970) 85. [2] D.R. Beddoes, J.C. Dainty, B.L. Morgan and R.J. Scaddon, J. Opt. Soc. Am. 66 (1976) 1247. [3] R.J. Scaddon and J.C. Dainty, Optics Comm. 21 (1977) 51. [4] C.T. Stansberg, Optics Comm. 23 (1977) 303. [5] W.T. Welford, Aberrations of the symmetrical optical system (Academic Press, 1974). [6] C.G. Wynne, Optics Comm. 21 (1977) 419. [7] C.G. Wynne, Opt. Acta 25 (1978) 627.
25
OPTICS COMMUNICATIONS
January 1979
EXTENDING THE BANDWIDTH OF SPECKLE INTERFEROMETRY
C.G. WYNE Royal Greenwich Observatory, Herstmonceux Castle, Hailsham, Sussex, UK and Imperial College, London, UK
Received 5 October 1978
In speckle interferometry, the scale of the speckle array is wavelength dependent, and this limits the spectral bandwidth that can be used. A simple optical system is described to correct this effect, over a wide spectral range, by introducing a chromatic difference of magnification of opposite sign to the speckle dispersion, and variable in magnitude.
1. Introduction
Speckle interferometry, as was suggested by Labeyde [1], can be used to obtain information from the image formed by a large astronomical telescope, beyond the limits on angular resolution normally imposed by atmospheric turbulence. The method is applicable, for example, to the detection of double stars, the measurement of their separation and magnitude difference, and the measurement of stellar diameters, where these lie within the theoretical limit of resolution set by the telescope diameter, but are masked from direct observation by the seeing. The method has been made possible by the availability of image intensifiers, which enable short exposure star photographs to be taken. With exposures of the order of 0.01 s and an appropriate filter, a star image consists of a random array of speckles, the extent of the array depending on the state of the atmosphere, the speckles being of the order of size of the Airy disk of the telescope. By summing the Fourier transforms of a large number of speckle images, the high resolution information can be extracted. The transforms may be obtained either optically or by computer analysis of digitised data. A speckle interferometer and its use has been described in detail by Dainty and others [2]. It was for this instrument that the device described in this paper was designed. For polychromatic light the speckle pattern of a star image consists of the superposition of monochromatic
speckle patterns, which differ from each other in two ways. There is a difference of scale. Corresponding speckles in different wavelengths are separated from each other in a radial direction, by an amount proportional to their distance from the centre of the pattern and to the wavelength difference. In a monochrome photograph of a polychromatic speckle pattern, individual speckles therefore appear smeared radially, by amounts increasing with their distance from the pattern centre. This dispersion effect is quite large. For the extremes of the visible spectrum, the radial separation between corresponding speckles may be as much as 0.6 for 0.7 of their mean distance from the centre. In addition to the chromatic difference of scale, speckle patterns in different wavelengths are not identical in form. This decorrelation is generally the smaller effect. The chromatic dispersion and decorrelation together produce a blurring of the speckle pattern that necessitates the use of a fairly narrow bandwidth filter (between 8 and 30 nm) and hence limit the application of the speckle interferometer to relatively bright stars. To the extent that the dispersion effect predominates, an optical device correcting the chromatic difference of magnification would allow a wider spectral band width, and the observation of fainter stars. This paper is concerned with the design of such a system. The amount of the blurring of the polychromatic 21
Volume 28, number 1
OPTICS COMMUNICATIONS
speckle image depends on atmospheric conditions; there have been different views as to what was the corresponding variability of the two contributing effects. At the time that the work described in this paper was initiated, it had been suggested that the extent of the dispersion was variable [3] ; this is consistent with the visual appearance of stellar speckle patterns observed at different times. Accordingly, the corrector was designed to have a compensating chromatic difference of magnification continuously adjustable over a wide range. More recent work by Stansberg [4] suggests that the dispersion is in fact invariant, but that this is masked by variable decorrelation. The variable corrector to be described should provide experimental evidence on this question.
2. The corrector: classical theory In the interferometer referred to above, the star image, formed for example at the ]'/14 Cassegrain focus of the Isaac Newton 2.5 metre telescope, is relayed at a 12 times magnification on to the photocathode of an image intensifier whose output is photographed in a series of short exposures by a cine camera. The optical system of the relay consists of two doublets of focal lengths 5 cm and 60 cm separated by a space within which light from the star is collimated. This collimated space accommodates the f'flter, and rotatable prisms to correct the atmospheric dispersion which varies with zenith distance; it also provides a convenient location for the device to correct the chromatic difference of magnification of the speckle pattern star image. This should then be a telescope, of magnification variable up to say I : 1.35 at 400 nm, down to 1 : 0.65 at 700 nm, with a constant unit magnification at some intermediate wavelength, and with negligible aberrations apart from this chromatic effect. This can be achieved in the following way. In a collimated space, a plane-parallel glass block of uniform refractive index is afocal, and free from aberrations. For a single wavelength, a pseudo planeparallel plate can be made by cementing together the curved surfaces of a plane-convex and a plane-concave lens which are made from two glasses having the same refractive index at some mean wavelength ~'0, but differing in dispersion. At the mean wavelength, this 22
January 1979
component will be afocal and aberration free, but will manifest chromatic effects at longer and shorter wavelengths. Similarly, pseudo plane-parallel plates can be compounded from more than two single lens elements. The variable chromatic corrector telescope consists of two such triple components, with an adjustable separation between them. This clearly gives the required invariant unit magnification at the mean wavelength. At other wavelengths, these components will be convergent for shorter and divergent for longer wavelengths, or vice versa, depending on whether the converging constituent lens elements are made from the glasses used for the convergent and divergent elements will simply alter the sign of the optical power k of the complete component at any wavelength. The corrector consists of such a pair of components, whose powers, at any given wavelength, are equal and opposite. The chromatic power of one of the components, at wavelength X, is given by
k(x) where 8n is the index difference at wavelength 8 between the two glasses at a cemented surface of radius r, the summation being over the cemented surfaces of the component. In terms of the conventional first order coefficient of longitudinal chromatic aberration, CI (see for example Welford [5] ), for each component, for the wavelength interval (k - X0), CI =h2k, where h is the incidence height at each surface of a paraxial ray at the mean wavelength ~'0" In our corrector, with components afocal at k0, in a collimated space, h is the same at all surfaces. It follows therefore, since the two components have equal and opposite powers k for all wavelengths, that their CI values are equal and opposite. And since according to conventional theory the CI values of components of a system are additive, the first order longitudinal chromatic aberration of the pair of separated components is zero for all wavelengths, independent of the separation between them. The case is quite different for chromatic difference of magnification. A collimated paraxial ray at k0, incident at height h, has the same height at all surfaces. A corresponding ray at wavelength ~ emerges from the
Volume 28, number 1
OPTICS COMMUNICATIONS
/ - . A . ./
January 1979
• __
~ . . ~ . _
m
k
Fig. 1. Diagramatic: full line, paraxial ray through corrector at wavelength ko; dotted line, paraxial ray at wavelength h.
first component with convergence angle hk(k). Taking the components as thin, with a separation d, the incidence height on the second component is therefore h(1 - dk(k)) (fig. 1). The angular magnification of a telescope is inversely proportional to the ratio of exit and entrance pupil diameters, so for the corrector the angular magnification at wavelength is 1/(1 - dk(k)). For a given value of k(k), a required angular magnification, relative to unity at ~0, may be obtained by adjusting the separation d. The chromatic dispersion in the speckle pattern is such that speckles at longer wavelengths lie further from the common centre. The first component of the corrector should therefore be the one that diverges light at longer wavelengths than the mean.
3. The corrector: further considerations
The discussion in the previous section was in terms of classical first order theory. Monochromatically, at the mean wavelength k 0, the corrector is free from aberrations of all orders. And at other wavelengths, the common forms of higher order aberrations are found to be small. But there is a chromatic effect, of paraxial order, neglected in classical aberration theory, which is of significance for the design of the system. This has recently been analysed by Wynne [6,7]. This is an aberration of the same type as secondary spectrum. It is characteristic of almost all imaging systems of lenses that when achromatised so that paraxial rays of two wavelengths are brought to a common focus, then for intermediate wavelengths the focus lies closer to the lens, and for wavelengths outside the range between the two, it lies further away. This secondary spectrum aberration arises from the dif-
ferent forms of wavelength dependence of index shown by most glasses of high and low dispersions; and it has been generally accepted that it can only be eliminated by the use of anomalous materials. Wynne (loc cit)has shown that an aberration of the same form, but opposite sign, arises when classical achromatism is achieved by the cancellation of terms of opposite sign arising at widely spaced components. This can occur in a high degree in the corrector systems described above. According to classical theory, the chromatic coefficient for a lens system is the surface-by-surface sum of CI terms, discussed above. Wynne [7] has shown that there are additional terms of paraxial order. For the two separated components of our corrector, the true coefficient is EC I + dh2k(k) 2. The second term remains finite when the first sums to zero. From the discussion above, the chromatic difference of magnification of the corrector depends on the value of dk(k). If this be given the required value, in a corrector short enough to be accommodated within the apparatus described in ref. [2], then the chromatic term dh2k(~) 2 becomes undesirably large at the high magnification setting. This can be avoided by an appropriate scaling down of the focal lengths of the 5 cm and 60 cm lenses containing the collimated space. In the final design, given below, at its highest magnification setting, the aberrations nowhere exceed about ~ wavelength over the spectral range 400 to 700 rim.
4. Detailed design of eorrector
Table 1 gives numerical data for the design of a complete relay system with chromatic corrector. A radius of positive sign is convex toward the front, negative concave. 23
Volume 28, number 1
OPTICS COMMUNICATIONS
January 1979
Table 1 Relay optical system with variable chromatic difference of magnification. Magnification at 480 nm, 11.7; dimensions in ram; image diameter 7.0 ram; spectral range 400-700 nm. Radius
Axial separation
Material (Schott glass type)
27.2 to telescope focus
Diameter
air
34.94 Lens 1 focallength 28.6
1.0
SF15
3.0
13.06 2.0
~
K5
-18.09 D1
air
1.0
LaK31
2.0
SF8
1.0
LaK31
16.0 Lens 2 afocalat 480 nm
5.0
-16.0 eo
D2 = 270.0-D 1
air
o0
3.0
LaK31
1.0
SF8
3.0
LaK31
-16.0 Lens 3
12.0
16.0 -242.0 319.0 to image intensifier photocathode
air
1 If nh he the object height from the axis at wavelength hnm and R = (r/7oo - naoo)/](r77oo + n4oo), then to correct for R = 0.70: D 1 = 20, D 2 = 250; to correct for R = 0.49: D 1 = 100, D2 = 170; for R ~. 0:D1 = 270, D 2 = 0.
The focal ratio in the final image space is so large ( a b o u t f[170 at the m e a n wavelength) that a c o r r e c t e d doublet after the collimated space to p r o d u c e this convergence is n o t necessary. In the present design, it is replaced by an appropriate shallow c o n v e x curve on the last surface o f the corrector. The aberrations p r o d u c e d by this surface are negligibly small e x c e p t for a c h r o m a t i c difference o f focus, which is c o r r e c t e d on the first doublet. In use, the separation b e t w e e n lensen 1 and 3 is maintained constant, and the radial c h r o m a t i c dispersion varied by m o v i n g lens 2 to and fro b e t w e e n them. The original system can be further simplified by the elimination o f the rotatable prisms used to correct the transverse c h r o m a t i c dispersion d e p e n d i n g on 24
zenith distance. This correction can be achieved by a small transverse translation o f lens 2. Since neither the change o f refraction index o f a glass, or the change o f refractive index for a pair o f glasses, is linear w i t h wavelength, the correction by a lens system o f the linear dispersion o f the speckle
Table 2 Refractive index from Schott glass catalogue at
LaK31 SF8
400 nm
480 nm
550 nm
700 nm
1.71850 1.73130
1.70601 1.70594
1.69937 1.69361
1.69106 1.67940
Volume 28, number 1
OPTICS COMMUNICATIONS
January 1979
?(mm) 6.0
~
5.0
~;-0
A
B
o
3"0
2-0
, 1"0
I
/.00
i
500
I
600
i
700 ~. (nm)
Fig. 2. Curve A shows variation of final image height 77with wavelength, in the absence of a corrector, for dispersion R = 0.70. Curve B shows the residual variation using the corrector system of table I. (If the corrector were used at the f/14 Cassegrain focus of the 2.5 m Isaac Newton Telescope, r/= 3.5 mm corresponds to an angular displacement from the axis of 1.7 arc s). image will not be perfect. However, the residual errors can be reduced to 0.09 of the initial dispersion. To distribute the residuals symmetrically, the index difference between the two glasses of the corrector should be equal and opposite at the extremes of the spectral range. This condition is very approximately met by Schott glass types LaK31 and SF8 at 400 and 700 nm. These two glasses have almost the same refractive index at 480 nm, at which the components are afocal. Table 2 shows their indices. Fig. 2 shows, in curve A, the variation of r/with wavelength for dispersion R = 0.70, in the absence of correction, and curve B shows the residual variation for the appropriate corrector setting. For small dispersions R, the residuals are less.
A relay system with chromatic corrector based on this design has been made and is awaiting trial on a telescope.
References [1] A. Labeyrie, Astr. Astrophys. 6 (1970) 85. [2] D.R. Beddoes, J.C. Dainty, B.L. Morgan and R.J. Scaddon, J. Opt. Soc. Am. 66 (1976) 1247. [3] R.J. Scaddon and J.C. Dainty, Optics Comm. 21 (1977) 51. [4] C.T. Stansberg, Optics Comm. 23 (1977) 303. [5] W.T. Welford, Aberrations of the symmetrical optical system (Academic Press, 1974). [6] C.G. Wynne, Optics Comm. 21 (1977) 419. [7] C.G. Wynne, Opt. Acta 25 (1978) 627.
25