various aspects of the invention will evolve from the following detailed description of the preferred embodiments thereof which should be taken in conjunction with the hereinbefore described drawings . the optical layout of the invention as a whole is shown in fig1 by referenced character 10 . optical system 10 includes as its key element a diffraction grating 12 , as shown in fig1 and 2 . the grating has a surface 14 which is reflective to the electromagnetic radiation of interest which is emitted by an extended source 16 and passes through an entrance slit 18 along a principal axis 20 . pre - optics 22 may be inserted between a compact source 24 and the entrance slit in order to efficiently direct the passage of such light . the reflective surface of the grating is composed of a set of minute grooves 26 which provides an interference pattern in the reflected ( i . e . diffracted ) beam exiting the grating along a principal axis 28 . an exit slit 30 may be employed to allow passage of an exceedingly narrow band of wavelengths δλ to a target or other detection means 32 . in this manner , optical system 10 effectively transmits only a desired wavelength emitted by the source , and hence performs the function of a monochromator . to provide focusing in the direction of the grating grooves ( i . e . normal to dispersion ), a mirror 46 may be interposed anywhere between the light source 16 and the target 32 . if the mirror is cylindrical and oriented orthogonal to the grating surface , this arrangement can greatly concentrate or collimate the image intensity at the target ( or any intermediate position ) without sacrifice of spectral resolution . however , an actual mirror will possess slope errors ( ripples , twisting , etc .) and other surface irregularities which will distort the wavefront in the dispersion direction of the grating . this will degrade the spectral resolution unless the mirror is placed on the side of either slit opposite the grating . if such a mirror is placed between the exit slit and the target , then the focusing of light through the entrance slit by the pre - mirror will also not degrade . in addition , a post - mirror ( not shown ), similar in function to the pre - mirror , can be used to further concentrate the image brightness by refocusing or collimating the exit slit onto the target in the dispersion direction . to provide selection of the desired transmitted wavelength , the grating is rotated about a fixed axis 34 so as to scan a continuous band of wavelengths past the exit slit , as depicted in fig1 . to maintain a fixed beam direction for principal light rays 20 and 28 which intersect at the grating center ( or pole ) w = 0 , the rotation axis is preferred to pass through this pole . given a constant included angle , 2θ , between the incident principal ray 20 and diffracted principal ray 28 , the required angle of incidence , α , and angle of diffraction , β , relative to the grating surface normal are : where σ o is the groove spacing at the pole , and m is the spectral order . entrance slit 18 and exit slit 30 are ideally provided with transverse positioning means 42 and 44 , respectively . this enables adjustment of the angle 2θ and thereby using equation ( 1 ) provides an accurate absolute wavelength calibration . such slit assemblies may be model vsa - 300 available commercially from hettrick scientific , inc . such alignment , and also the rotational alignment of the slit length along the direction of the grating grooves , may be facilitated by use of a visible alignment source 48 incident normally on the grating surface and diffracted through the slits . to minimize the optical aberrations , the spacings between the grating grooves 26 are allowed to vary as a function of their position , w , across the grating surface . such a function may be expressed in various ways , but for purposes of illustrating the degree of aberration correction provided , it is written here as a polynomial : where n i are constant coefficients which determine the local spacings between the centers of each groove , where i = 2 , 3 , 4 , etc . given straight grooves , the error δl in path length between entrance and exit slit , and the resulting optical aberration δλ in wavelength for a non - principal ray 36 which strikes aperture coordinate w ( fig1 ) is given to a good approximation by : δλ =( σ . sub . o / m )( f . sub . 2 w + 3 / 2f . sub . 3 w . sup . 2 + 2f . sub . 4 w . sup . 3 + . . . ) ( 4 ) and where r is the radius of curvature of the grating , r is the length of principal ray 20 connecting the entrance slit and the grating pole , and r &# 39 ; is the length of diffracted principal ray 28 connecting the grating pole and the exit slit . the image at the transmitted wavelength is considered to be &# 34 ; in focus &# 34 ; if f 2 = 0 , and the image profile is to a first order of approximation symmetrical if there is no coma ( f 3 = 0 ). given the constraint of fixed entrance and exit slits , these two conditions may both be honored at two wavelengths of choice , λ 1 and λ 2 , if the grating has a finite radius and is simply rotated about its pole to select wavelength . from equations 1 - 8 , this solution leads to the following choice of parameters : ## equ2 ## where subscripts 1 and 2 refer to the quantity as determined using α and β derived from equation ( 1 ) at the chosen wavelengths λ 1 and λ 2 , and where for conciseness of equations ( 9 ) and ( 10 ) the following dimensionless parameters are defined : a = γ sin α . sub . 1 cos . sup . 2 α . sub . 1 - sin α . sub . 2 cos . sup . 2 α . sub . 2 - γδ . sup . 2 sin β . sub . 1 cos . sup . 2 β . sub . 1 + δ . sup . 2 sin β . sub . 2 cos . sup . 2 β . sub . 2 ( 13 ) b =- γ sin α . sub . 1 cos α . sub . 1 + sin α . sub . 2 cos α . sub . 2 + 2γδε sin β . sub . 1 cos . sup . 2 β . sub . 1 - 2δε sin β . sub . 2 cos . sup . 2 β . sub . 2 + γδ sin β . sub . 1 cos β . sub . 1 - δ sin β . sub . 2 cos β . sub . 2 ( 14 ) c =- γε . sup . 2 sin β . sub . 1 cos . sup . 2 β . sub . 1 + ε . sup . 2 sin β . sub . 2 cos . sup . 2 β . sub . 2 - γε sin β . sub . 1 cos β . sub . 1 + ε sin β . sub . 2 cos β . sub . 2 ( 15 ) δ =( γ cos . sup . 2 α . sub . 1 - cos . sup . 2 α . sub . 2 )/( cos . sup . 2 β . sub . 2 - γ cos . sup . 2 β . sub . 1 ) ( 16 ) ε =[ γ ( cos α . sub . 1 + cos β . sub . 1 )-( cos α . sub . 2 + cos β . sub . 2 )]/( cos . sup . 2 β . sub . 2 - γ cos . sup . 2 β . sub . 1 ) ( 17 ) from equations 9 - 18 , the resulting design parameters are determined to be : using these parameters , curves 200 , 202 and 204 of fig4 are the individual optical aberrations calculated from equations 1 through 8 as functions of the selected wavelength . as constrained above , the resolution is extremely high in the immediate vicinity of the two chosen correction wavelengths ( 100 å and 200 å ). however , because equation 5 provides for the same optimum value of n 2 only for these discrete wavelengths , the image sharpness degrades rapidly elsewhere . the result is a spectral resolution which is generally as poor as the prior art design ( hettrick et al , 1986 ) employing an equally - spaced spherical grating ( curves 100 , 102 and 104 ) with the same groove density , angular deviation and system length ( r + r &# 39 ;). to overcome this limitation , a trick is employed which takes full advantage of the fact that the grating grooves are unequally spaced . at each wavelength other than the two chosen ones , the grating is translated along its surface tangent . this effectively changes the grating parameters as continuous functions of the amount of translation δw . due to the large radius of curvature r , the movement of the grating surface away from its rotation axis is small [ r ( 1 - cos φ )] if the grating is translated along the straight chord 40 fixed relative to the grating surface ( fig1 ). thus , in practice the grating translation need not be constrained strictly along the curved grating surface , greatly simplifying the mechanical design ( discussed below ). from the geometry defined in fig3 it follows that equations 1 - 8 remain valid with the following substitutions : and where terms in the groove space variation ( equation 2 ) higher than n 4 have not been retained . note that φ = 0 for a plane grating . using the same numerical parameters previous given ( and setting n 4 = 0 ), curves 300 , 302 and 304 of fig4 are the calculated results of optimizing δw to eliminate defocusing at each wavelength , by numerical iteration of equations ( 20 ) and ( 21 ) with φ chosen such that f 2 = 0 from equation ( 5 ). all wavelengths are now sharply in focus , the new limit to the optical resolution being spherical aberration , which decreases as the third power of the numerical aperture . it therefore has a resolvable half - energy width approximately a factor of four smaller than the extremum aberration plotted in fig4 c . the resulting spectral resolution has thus improved approximately two orders of magnitude , from 0 . 5 å ( curve 200 ) to 0 . 003 å ( curve 304 divided by four ). further correction is available by use of non - zero values for n 4 . from equation ( 22 ) it is clear that this term will significantly change the substituted value of n 3 * as the grating is translated ( δw ≠ 0 ). through iteration of the above numerical example , it was determined that a value of n 4 =- 6 . 99 × 10 - 7 mm - 4 eliminates coma near the center of the spectrum ( curve 402 ), with the coma becoming in practice negligible elsewhere compared to the spherical aberration ( which itself has been reduced in the process , as seen in curve 404 ). the resulting symmetrical image allows the use of accurate modeling techniques to further enhance the spectral resolution of a recorded spectrum . as plotted in fig4 d , only modest amounts of translation are required , thereby allowing efficient use of the grating aperture while maintaining a fixed beam direction . furthermore , because this motion functions to remove only a modest amount of residual aberration the required accuracy of translation is trivial by contemporary standards . due to the grating translation , an unbaffled grating will give rise to an angular aperture whose center ( or effective principal axis ) shifts by an equal amount . if this needs to be eliminated , then an entrance baffle 50 ( fig1 ) may be inserted prior to the grating to ensure underillumination . a translation of δw combined with a full grating ruled width of w , leads to a baffled grating width of w - δw . in the above numerical example , the assumed baffled width of 50 mm and the translation of 25 mm requires a ruled grating width of 75 mm . thus , only a modest amount of the grating is unused . the small required grating size , modest translational travel , undemanding translation accuracy and ability to use simple rectilinear motion , leads to a simple and compact mechanical design ( fig5 ), using commercially available components . grating 12 is contained in a holder 52 by use of set screws 54 which are threadingly engaged in the holder . to prevent distortion of the grating surface , set screws 54 may be spring - loaded ball plungers , such as part no . 10001p manufactured by northwestern tools , inc . of dayton , ohio . the grating surface 14 is registered against pins 53 attached to the holder . by means of screws 56 , holder 52 is rigidly bolted to slide plate 58 which translates relative to end plates 60 and 82 by means of interposed inner ways 62 bolted to the slide plate by screws 63 , outer ways 64 bolted to the end plates by screws 65 , and ball or roller bearings 66 . thus , grating 12 is allowed to translate along the direction 40 ( normal to the plane of fig5 ) as illustrated in fig1 . bearing assemblies consisting of hardened ways and bearings are commercially available from numerous sources , and need not be discussed further . the translation is driven by a rotating cam 68 which is in surface contact with a spherical ball 70 press fit into a bar 72 which is bolted to the slide plate 58 by means of screws 76 . this surface contact can be assured by extension spring means 74 connecting the cam 68 and the bar 72 by use of pins 76 attached to both members . the grating rotation is also driven by a rotating cam 78 maintained in surface contact with a ball 80 press - fit into a radius bar 82 , by spring and pin means 81 and 83 , respectively , as detailed in fig6 . radius bar 82 also functions as one of the end plates as discussed above . both end plates 60 and 82 are supported by pivots 84 mounted by means of clamps 87 bolted to the stationary end members 86 using screws 89 . similarly , clamps 61 provide mounting of the opposite ends of pivots 84 onto end plates 60 and 82 by use of screws 67 . hence , as bar 82 is pushed by cam 78 , the end plates 60 and 82 co - rotate about pivots 84 together with the attached grating 12 . if pivot 84 is a flexural pivot , then spring 81 and pins 83 are not required . support members 86 are affixed to a base plate 88 by means of bolts 90 . correlation of rotation between the cams 68 and 78 is accomplished by mounting them to a common cam shaft 91 going through their centers of rotation . the shaft is free to rotate about is central axis by means of radial bearings 92 . an anti - backlash worm gear 93 is also rigidly attached ( e . g . press - fit or clamped ) to shaft 91 and is driven by a worm 94 connected to a drive shaft 95 which is free to rotate within radial / thrust bearings 96 fit into base plate 88 . translation of worm 94 is prevented by external retaining rings 97 , one of which is supported against leaf spring 98 . if external rotation means 99 is applied to drive shaft 96 , both the wavelength drive cam 78 and the grating translation cam 68 will rotate synchronously . this provides for the automatic determination of the amount of translation required at each wavelength . to obtain the highest intrinsic mechanical accuracy , the wavelength cam may have a circular surface and be rotated off of its geometrical center , as illustrated in fig6 . alternatively , the cam 78 may be machined to a non - circular shape in accordance with providing a linear relationship between the angle of cam rotation and the selected wavelength . due to the functional relationship shown in fig4 d , as well as the correction necessary to remove the translation of bar 72 due to rotation only of bar 82 , the translation cam is theoretically required to have a non - circular shape . however only modest accuracy is required and therefore it may be easily manufactured . the required accuracy may be easily derived from fig4 a and 4d , revealing a defocus of approximately 0 . 4 å per 25 mm of travel . thus , even a 0 . 5 mm error in translation would result in less than 0 . 01 å of residual aberration . due to the translation of the varied space grating , the spacing at pole 34 ( fig1 ) will change , and thus a conventional sine - bar mechanism will not result in linearity between the angle of rotation of a lead screw member ( engaged to a normal pusher block ) and the resulting wavelength selected . however , as the dominant change to the pole spacing is due to the n 2 term [ equations ( 2 ) and ( 20 )], such linearity can be regained to a high degree of accuracy without resort to a cam , by use of an inclined plane as the pusher block . fig7 shows such an optional wavelength drive , in which a lead screw 77 ( which may be attached directly or indirectly to drive shaft 95 of fig5 ) is threadingly engaged into a block 79 whose surface is inclined at an angle to the rotation of the lead screw . in this configuration , a different bar 85 is rigidly attached to slide plate 58 of fig5 and is oriented along the direction 40 of the grating translation . as the translation cam moves this slide plate along the direction of its ways 62 ( resulting in a change in the distance from ball 80 to the fixed grating pole 34 ), the ball 80 attached to radius bar 85 will migrate over the inclined plane pusher . this results in a built - in mechanical correction to the grating angle in linear relationship to the amount of grating translation , as required by the n 2 term and the variation in incident angle given by equation ( 24 ). for example , given the above numerical example , with n 2 =- 1 . 63766 mm - 2 , and a radius bar length of 556 mm , the pusher block inclination angle would be 1 . 66 °. the residual non - linearity is calculated to be approximately 0 . 3 å . in practice , external means 99 comprise various conventional gears , bearings , cranks , motors , counters , etc . as convenient for the particular application , and to enable accurate monitoring of the grating position and orientation . in addition , as such a mechanism may be used in combination with soft x - ray radiation , it would be situated inside a vacuum , requiring use of conventional rotary motion feedthroughs which are commercially available . the mechanical couplings between the wavelength selection and grating translation may of course be replaced by separate external drives correlated by a computer . the use of varied spaced grooves combined with suitable motions of the grating surface as a function of wavelength , permits the virtual elimination of significant optical aberrations in the monochromatic image . thus , linear dispersion becomes the limiting factor in determining the attainable spectral resolution of such a system . in the above numerical example , the spherical aberration limit of 0 . 005 å will be reached only if the slits are approximately 5 microns wide . this motivates the use of two ( or more ) such systems in series , wherein the individual dispersions add to increase the slit - limited resolution . in such configurations , the exit slit for the first system is placed at the same position as the entrance slit for the second system , and so forth . the use of this intermediate slit is , however , optional , as the additive dispersion will continue to disperse the various wavelengths in a one - to - one mapping onto the final exit slit plane . alternatively , the path length for all rays ( principal and extrema ) may be made constant by use of two ( or more ) such systems arranged such that the dispersions cancel . in this case , an intermediate slit must be used to eliminate the background of overlapping wavelength images . such a ( common path length ) monochromator preserves the time resolution of the incident beam , and is therefore of interest in the context of pulsed sources . while in the foregoing , a preferred embodiment of the present invention has been set forth in detail for the purposes of making a complete disclosure of the invention , it may be apparent to those of skill in the art that numerous changes may be made in such detail without departing from the spirit and principles of the invention . in particular , optimizations other than equations 9 - 18 may be devised to minimize the required grating translation over desired wavelength regions , or to eliminate aberrations at other wavelengths . a plane grating may also be substituted for the concave grating illustrated above , in the case where both object and image are real . in this case , there are insufficient degrees of freedom to provide δw = 0 at two distinct wavelengths . the required amount of translation necessary to provide in - focus imaging is therefore larger than for the preferred embodiment . however one finds , through use of equations 1 - 8 and 19 - 22 with an exceedingly large value for r , that the required translation is minimized if a maximum infinite de - magnification is chosen ( i . e . the object is a plane wave , and the image is diffracted into a negative spectral order ). unfortunately , assuming the grating is baffled to maintain a fixed direction for the principal ray , the larger translation requires a correspondingly larger grating . as another example , the same technique may used to dramatically improve the properties of a plane grating having a virtual image and real object [ e . g . the erect field optical system of hettrick et al ( u . s . pat . no . 4 , 776 , 696 )]. as with the concave grating embodiment discussed above , the introduction of grating translation in the latter design will also result in perfectly focused images at all scanned wavelengths , thereby increasing the spectral resolution . in this case , the amount of required translation is very small , due to the fact that the erect field imaging is nearly in focus prior to such translation . however , the improvement is significant for high resolution applications . as a further example of the versatility of the present invention , the grating surface may be aspherical , which can be used to self - focus in the direction parallel to the grooves without the introduction of an additional mirror reflection . while in the foregoing , embodiments of the present invention have been set forth in considerable detail for the purpose of making a complete disclosure of the invention , it may be apparent to those of skill in the art that numerous changes may be made in such detail without departing from the spirit and principles of the invention . in a fundamentally different application , the proposed translation of a varied - space grating may be used by itself ( i . e . without any rotation required ) to optimize the focusing of any varied - space grating . for example , this method may be used to optimize the resolution of a spectrum recorded by the prior art gratings of hettrick et al ( 1983 ), hettrick et al ( u . s . pat . no . 4 , 776 , 696 ), harada et al ( u . s . pat . no . 4 , 312 , 569 ), harada et al ( 1984 ), or meekins et al ( u . s . pat . no . 4 , 578 , 804 ). finally , it should be recognized that while the invention has been applied in detail to electromagnetic radiation , it may be utilized in principle with wavelike radiation of any nature , including acoustic waves .