Imaging properties of diffraction gratings

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IMAGING PROPERTIES OF

DIFFRACTION GRATINGS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS DR. IR. C. J. D. M. VERHAGEN, HOOG-LERAAR IN DE AFDELING DER TECHNISCHE NATUUR-KUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN OP WOENSDAG 7 JANUARI 1970 TE 14 UUR

DOOR

WILHELMUS WERNER

natuurkundig ingenieur geboren te Balimbingan (Indonesië)

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DIT PROEFSCHRIFT IS GOEDGEKEURD DOOR DE PROMOTOR PROF. DR. H. DE LANG.

STELLINGEN

I.

Door Namioka is ten onrechte gesteld, dat de uitbreiding van de Rowland cirkel in de derde dimensie een cirkelcilinder oplevert.

Namioka, T. 1959, J. Opt. Soc. Am. 49, 446.

II.

De axiale verplaatsing van het beeld in optische systemen met bewegingscompensatie door middel van roterende polygonale prisma's kan met succes bestreden worden door de vlakken van deze prisma's een negatieve sterkte te geven.

III.

De concentratie van energie in een bepaalde spectrale orde bij tralie-opstellingen met scherend invallende bundels zal groter zijn, indien de bundel invalt nagenoeg even-wijdig aan de groeven in plaats van loodrecht op de groeven zoals gebruikelijk.

IV.

Het verschil van een factor cos fi tussen de uitdrukkingen voor het spectraal scheidend vermogen van een traliespectrometer volgens Mack en medewerkers, en volgens Beutier, wordt door Namioka ten onrechte verklaard met een foutieve afleiding van de karakteristieke vergelijking door Beutier.

Mack, Stehn and Edlen, (1932), J. Opt. Soc. Am. 22, 245. Beutier, H. G. (1945), J. Opt. Soc. Am. 35, 311. Namioka, T. 1959, J. Opt. Soc. Am. 49, 446.

V.

De bepaling van een zogenaamde optimale traliebreedte, waarbij de afmetingen van de geometrisch-optische aberratiefiguur worden gelijk gesteld aan die van de buigings-figuur als gevolg van de pupilbegrenzing, geeft een ongewenste beperking van de be-schrijving van deze aberraties.

VI.

De gebruikelijke bewering, dat twee bij elkaar behorende brandlijnen loodrecht op elkaar staan is in het algemeen onjuist; wel liggen zij in twee elkaar loodrecht snij-dende vlakken.

VII.

Het zou voor het behoud van de oude stadskernen gunstig zijn indien de subsidië-ringsnormen, zoals die voor monumenten-restaurerende instellingen gelden, ook bij restauraties door particulieren worden gehanteerd.

VIIL

Het is wenselijk, dat vertegenwoordigers van verzekeringsmaatschappijen enigermate wiskundig geschoold zijn.

IX.

Het welzijn in Nederland zou bevorderd kunnen worden door de plannen voor de ruimtelijke ordening minder overwegend op de tekentafel op te stellen.

X.

Het is onder bepaalde omstandigheden gewenst de nationale activiteiten in het ruimte-onderzoek voorrang te geven op de activiteiten in internationaal verband.

1

D A N K W O O R D

Gaarne betuig ik mijn erkentelijkheid aan de directie van de Technisch Physische Dienst TNO-TH voor de mij geboden gelegenheid dit werk te voltooien.

Voorts wil ik iedereen bedanken, die aan de tot stand-koming van dit proefschrift heeft medegewerkt; in het bijzonder Ir. G. J. Beernink voor zijn aanmoedigingen, zonder welke dit werk mogelijk niet tot het nu bereikte stadium zou zijn voortgezet.

C O N T E N T S

page Chapter 1. Introduction .

1.1 General remarks 9 1.2 Some current grating mountings 10

1.2.1 Mountings with a plane grating 10 1.2.2 Mountings with a concave grating 11 1.2.2.1 Non-scanning spectrograph making use of the

Rowland circle 11 1.2.2.2 Scanning spectrometers based on the Rowland circle 11

1.2.2.2.1 The Rowland mounting 11 1.2.2.2.2 The Abney mounting 12 1.2.2.2.3 The Paschen-Runge mounting 12

1.2.2.2.4 The Eagle mounting 13 1.2.2.2.5 The "Radius" (Beutler) mounting 14

1.2.2.3 Scanning spectrometers not using the Rowland circle 15

1.2.2.3.1 The Seya mounting 15 1.2.2.3.2 The Johnson-Onaka mounting 15

1.2.2.3.3 The Wadsworth mounting 16 1.3 Considerations for a further study 17 Chapter 2. Theory of the spectral image formation .

2.1 Description of the method 19 2.2 The series expansions of the lightpath function F and the

conditions for image formation 28 2.3 The direction of the principal ray 35 2.4 The general image equation 36 2.5 The first slant angle of the astigmatic line 42

2.6 The length of the astigmatic line 44 2.7 The width of the astigmatic line caused by coma and spherical

aberration 45 2.8 The radius of curvature of the astigmatic line 48

2.9 The second slant angle of the astigmatic line 50

2.10 The orientation of the entrance slit 52 2.11 The radius of curvature of the entrance slit 53

2.12 The choice of the grating surface 55 2.13 Dimensions of grating surface and exit slit 57

2.14 The coefficients C„-I-Có„ 58

2.15 Conclusion 60 Chapter 3. Examples

3.1 Mountings with a plane grating 62 3.2 Mountings with circularly cylindrical gratings 63

3.3 Off-plane mountings with a spherical grating and an object at

infinity 68 3.4 Off-plane mounting with an off-axis paraboloidal grating and

an object at infinity 71 3.5 Formulae of the Rowland circle 73

Appendix 79 Summary 81 Samenvatting 83 References 85

Chapter 1

I N T R O D U C T I O N

1.1 General remarks

For almost a century now diffraction gratings are being used as the dispersing ele-ment in spectroscopic systems. In the greater majority of cases this grating is of the reflecting type, which (among others) has the advantage that the radiation to be analysed need not pass absorbing material as is the case with prisms and transmission gratings. It is therefore possible to study a large part of the electromagnetic spectrum on account of the reflecting properties of the coatings available. As reflection gratings are widely applied, many papers have appeared, describing various aspects of spectral image formation by such gratings.

In this thesis we shall restrict ourselves to the properties of grating systems as seen from the viewpoint of the geometrical aberration theory. We have excluded any questions of physical optics such as, for instance, diffraction due to the finite width of the grating and the intensity distribution over the spectral orders (blaze). Also matters as ruling errors and ghosts will not be considered here.

In literature calculations are usually applied to the description of a certain mount-ing, implying that already in an early stage of the derivations a particular case will be chosen to which the considerations are restricted. In this introduction we shall follow the division usually made. As far as grating arrangements are concerned a differentiation is made between "in-plane" and "off-plane" mountings. The plane referred to here contains the grating normal and is perpendicular to the rulings. Nearly all mountings described so far are "in-plane", which means that the centres of entrance and exit slits are situated in this plane.

One also distinguishes between instruments that image a considerable part of the spectrum, for instance on a photographic plate, and systems containing an entrance slit and an exit slit, thus isolating a specific part of the spectrum.

In the latter instruments, referred to as scanning spectrometers, the spectrum can be scanned by rotation or translation of one or more elements of the system and it is essentially for photo-electric use.

This scanning action should be achieved by the simplest possible mechanism. It is desirable to have the position of the entrance slit and the direction of the entrance beam fixed although a number of known scanning spectrometers do not have these characteristics. We shall now give a brief review of the most important current grating mountings.

1.2 Some current grating mountings

1.2.1 Mountings with a plane grating

In the great majority of cases in which a plane grating is used, this will be in combi-nation with collimation and image forming optics, the latter being almost exclusively of the reflecting type. In these cases the beam incident on the grating is parallel.

Such a system was first proposed by Ebert in 1889. It includes one spherical mirror and the entrance slit is fixed. The location of this slit is such, that the beam reflected by the mirror is parallel apart from the influence of the aberrations. The diffracted beams are also parallel, and are again reflected by the same mirror that images the spectrum. In this form the instrument is not used to any great extent. The fact that this system has - in addition to spherical aberration - a lot of coma is the main reason for this.

When in 1952 Fastie described a scanning spectrometer based on this arrangement (see fig. 1.1), it became of more importance. The image quality in this case is very good because by a proper location of the exit slit this mounting is to a first approxima-tion free from coma.

Fig. 1.1

The Ebert-Fastie mounting. For scanning the spectrum the plane grating G is rotated about the central groove. The mirror M and the slits S and S' have fixed posi-tions. The beams incident on and diffracted by the grating are parallel.

Already in 1930 Czerny and Turner described a similar scanning spectrometer, where the collimation and the image formation were performed by two separate spherical mirrors.

In this case as well, the image quality is affected by spherical aberration and some residual coma, because the beam incident on the grating is comatic. It is possible to avoid these aberrations by using two ofF-axis paraboloidal mirrors.

Many variations of the arrangements discussed here, among others those with an off-plane mounted grating, have been described in literature and are used in practice. Furthermore the plane grating is incidentally used in mountings with non-collimated light. The aberration terms of an in-plane mounting of this kind were described by

M. V. R. K. Murty [29]. J. T. Hall did so as well [15]; his results however contain some erroneous statements.

1.2.2 Mountings with a concave grating

The fact that most mountings of the concave grating applied so far are adaptations of the principle that all optical components are located on the Rowland circle, originally stated by Rowland in 1883, we shall briefly discuss here the main optical character-istics of this special arrangement.

When we consider a spherical grating, having a radius of curvature R, and choose the object point on a circle with diameter R tangent to the grating surface, it can easily be proved, that the spectral images are always located on the same circle. This arrangement has the peculiar property, that the spectral image is at all times free from coma, whatever the position of this image and of the object point on the circle will be. Therefore this arrangement is very suitable for the use as a spectrograph, either non-scanning or non-scanning. The quality of the spectral image is mainly determined by spherical aberration. Other aberrations being present are astigmatism, curvature of the astigmatic line and of the paraxial slit image. The spectral image formation on the Rowland circle has been described extensively by H. G. Beutler [1] in 1945.

In spite of several critical remarks which have been made since the appearance of this paper it is still often referred to.

1.2.2.1 Non-scanning spectrograph making use of the Rowland circle

In this arrangement the entrance slit has a fixed position on the Rowland circle and a photographic plate is bent along this circle.

For the purpose of analysing spectra with wavelengths below 50 nm this arrange-ment is used in grazing incidence, in spite of the increasing aberrations, especially astigmatism. The lower wavelength limit of the spectrum depends largely on the angle of incidence. Instruments of this kind have been realized with a glancing angle as small as 10 minutes of arc, giving a lower wavelength limit of 0.05 nm.

1.2.2.2 Scanning spectrometers based on the Rowland circle 1.2.2.2.1 The Rowland mounting

This mounting is shown in fig. 1.2. The entrance slit S has a fixed position in the centre of the rectangular XY coordinate system. The grating G is moved along the Y axis; moreover it is rotated in such a way, that the exit slit S', which is positioned at the grating normal, moves along the X axis. Thus the angle of diffraction is at all times zero. The centre of the Rowland circle moves along a circle having diameter R and centre S.

Fig. 1.2

The Rowland mounting. The en-trance slit S and the enen-trance beam are fixed. The grating G and the exit slit S' are rotated and trans-lated. The diffraction angle tp' is always zero.

1.2.2.2.2 The Abney mounting

This mounting, which is a variation on the Rowland mounting, is shown in fig. 1.3. Here the grating G and the exit slit S' are fixed. The entrance slit is moved along the Rowland circle, together with the light source to be analysed. In many cases this movement appears to be a disadvantage and therefore this mounting is no longer used.

1.2.2.2.3 The Paschen-Runge mounting

For scanning action the exit slit S' must move along the Rowland circle in this

Fig. 1.3

The Abney mounting. The en-trance slit is moved along the Rowland circle and the direction of the entrance beam has to be adjusted. The diffraction angle (p' is always zero.

Fig. L4

The Paschen-Runge mounting. In this mounting is the exit slit S' moved along the Rowland circle. The entrance slit S and the grating G have fixed positions.

mounting, which is shown in fig. 1.4. The grating G and the entrance slit S are fixed. This mounting is frequently used in large photo-electric systems with some tens of fixed slits for a rapid determination of the same number of points of the spectrum.

As a variant this mounting is often used with grazing incidence (see fig. 1.5) which provides a higher reflectivity and dispersion in the vacuum ultraviolet and soft

X ray regions. The aberrations, astigmatism and spherical aberration are large

com-pared with the case of normal incidence, and therefore the aperture of the entrance beam has to be kept small.

Fig. 1.5

The grazing-incidence mounting. The exit slit S' is moved along the Rowland circle. This mounting can be considered as a variant of the Paschen-Runge mounting.

1.2.2.2.4 The Eagle mounting

In this mounting (see fig. 1.6), the angle of incidence is kept almost equal to the angle of diffraction, which minimizes astigmatism. The entrance slit S is fixed. The exit slit S' and the centre C of the Rowland circle rotate around an axis through the entrance slit S. This means, that the grating undergoes a rotation and a translation. The translation is in the direction of the principal ray of the entrance beam, this beam can therefore have a fixed direction.

This mounting has the merit of compactness and is actually used in two forms: the in-plane Eagle mounting and the off-plane Eagle mounting. The former is usually

Fig. 1.6

The Eagle mounting. The entrance slit S has a fixed position. The Rowland circle is rotated about an axis through this slit, which is also the case with the exit slit S'. The grating G is moved in the direction of the entrance beam, which is fixed.

applied to scanning spectrometers and the latter to non-scanning instruments. When the off-plane mounting is applied as a scanning spectrometer we often see the slits positioned symmetrically with respect to the Rowland plane.

It is pointed out, that the equations describing the image formation in the off-plane mounting, as derived in this thesis are at variance with those indicated in the current descriptions of this mounting [16, 31, 32].

1.2.2.2.5 The "Radius" (Beutler) mounting

This mounting is shown in fig. 1.7. Here the entrance slit and the exit slit are both fixed. The grating moves along the Rowland circle to perform a scanning action. This

Fig. 1.7

The "Radius" (Beutler) mounting. In this mounting is the grating G moved along the Rowland circle and have the entrance and exit slits fixed positions. The direction of the entrance beam has to be adjusted.

means that the direction of the principal ray of the entrance beam changes. It is there-fore necessary to adjust the position of the light source to be analysed.

In many cases this will appear to be disadvantageous, in spite of the good image quality and the simple scanning mechanism.

1.2.2.3 Scanning spectrometers not using the Rowland circle 1.2.2.3.1 The Seya mounting

In 1952 Seya described a scanning spectrometer in which a concave grating is rotated about an axis tangential to the grating surface in its centre and parallel to the grooves (see fig. 1.8).

Fig. 1.8

The Seya mounting. Scanning of the spectrum is obtained by a simple rotation of the grating G about the central groove. The slits S and S' are only situated on the Rowland circle in the case the zero-order spectrum is imaged on the exit slit.

The entrance slit, the exit slit as well as the direction of the incident beam are all fixed. It was found that the system was optimized when the angle between the prin-cipal rays of the incident and diffracted beams had the value of 70.25°. The distances of the entrance slit and the exit slit to the grating centre are both equal to R cos (35.13°), in which R is the radius of curvature of the grating surface perpendicular to the grooves.

Namioka discussed this mounting in more detail and considered the use of ellips-oidal gratings in this mounting [35, 36].

1.2.2.3.2 The Johnson-Onaka mounting

This mounting, which is shown in fig. 1.9 has in common with the Seya mounting that both slits are fixed and the grating is rotated. The difference is that in the Johnson-Onaka mounting the axis of rotation is chosen in such a way that the de-focusing which occurs in the Seya mounting is minimized. The centre of rotation Q

Fig. 1.9

The Johnson-Onaka mounting. This mounting looks much like the Seya mounting, however, the rota-tion of the grating G is about an axis through point Q in this case.

of the grating is situated on a line through the grating centre and point P, which is midway between the slits on the Rowland circle. The optimum value of OQ is given by:

R sin i((p + (p') 0Q =

l + ^{tan(p — tancp') tan i{(p + (p') (1.1)

1.2.2.3.3 The Wadsworth mounting

This mounting, shown in fig. 1.10, is the only truly stigmatic mounting of the concave spherical grating.

The conditions are, that only the angle of incidence varies for a wavelength varia-tion and that the exit slit S' is always located on the grating normal. Entrance slit S

Fig. 1.10

The Wadsworth mount-ing. For scanning the spectrum the grating G is rotated about the central groove, whereby the exit slit S' is kept at the grating normal. The distance between the grating and the exit slit depends on the angle <p.

and collimating mirror M, necessary to obtain a parallel beam incident on the grating, have fixed positions. However, the distance between the centre of the grating and the spectral image at the grating normal varies with the angle of incidence. This requires an extra mechanism for scanning action.

The concave grating mountings mentioned here usually contain a spherical grating. In the case of grazing incidence and in the Seya mounting, aspherical gratings have been used in order to avoid excessive astigmatism. In these cases the grating surface chosen is a part of an ellipsoid [35] or of a toroid [14].

It will be understood that only a very brief survey of the grating mountings used could be given. For more details and for variants on the mountings described here reference is made to the extensive literature on this subject.

1.3 Considerations for a further study

Upon perusal of the existing literature concerning the spectral image formation, it will be seen that only a few possibilities for the location of the object have been studied extensively. Without overrating the importance of grating mountings with the object neither on the Rowland circle nor at infinity we wish to make it clear, that a descrip-tion of further possibilities for in-plane mountings may be helpful in spectrometer design. A detailed description of these possibilities has so far not been given.

Moreover, it is not necessary to restrict the location of the object point to the plane through the grating normal that is perpendicular to the rulings, i.e. the Rowland plane. Hitherto the off-plane mountings have only been discussed incidentally, whereby the object is chosen at relatively small distances from the Rowland plane, not more then a few degrees. We know descriptions of the off-plane Eagle-mounting, which due to its compactness and its minimum amount of astigmatism will in many cases be chosen as the most suitable one for a high resolution spectrometer for the vacuum ultraviolet. In this paper we shall prove that even in the paraxial approxi-mation these descriptions are inadequate [16, 31, 32].

The inadequacy of these considerations is due to the wrong choice of the pupil coordinates.

Furthermore it will appear, that the orientation of the astigmatic line, to a high degree determining the spectral resolution, deviates from the values in these former descriptions. It is surprising that in some cases the slant angle of the astigmatic line needs a correction of 90 degrees with respect to known results.

Of course, these cases are extreme in so far as the length of the astigmatic line is nearly zero for the small distances from the Rowland plane as taken in these descrip-tions especially in the Eagle mounting, which is known for its small amount of astigmatism. In our opinion this is the reason that this deviation has not been served in the experiments. But for larger off-plane angles the miss-orientation ob-viously cannot be neglected. These large off-plane angles are not only of theoretical

interest. In chapter 3 we shall describe an arrangement where the off-plane angle is 84 degrees (an unconventional case of grazing incidence). This arrangement has been applied successfully. In another new mounting being operational, in which we used an off-axis paraboloidal surface for the grating, the ofiT-plane angle was about 15 degrees.

Proceeding from the facts just mentioned it is evident that also the higher order effects in the image formation (the aberrations) have to be reviewed once more.

It will be found that some essential lightpath defects have been neglected. In the geometrical derivations of the next chapter we have adopted the elegant method, which has also been used by many other authors on this subject who use the principle of Fermat in the expansion of the characteristic function of Hamilton. However, we shall not confine ourselves to the description of image formation in one specific arrangement, as is usually done. Nor will the shape of the grating surface be specified beforehand. Thus, in our calculations we take an arbitrary case concerning the loca-tion of the object as well as the shape of the grating surface. The result is that we are able to consider each particular case of practical interest immediately. Our formulae therefore will also cover the known results of e.g. the Rowland mounting and the plane grating mountings. Several examples of new mountings are given in chapter 3.

There is one aspect, however, for which we wish to deviate from the procedure of most authors. This concerns the dimensions of the aberrations, which will be given as a function of the dimensions of the pupil. The results are comparable with a spot diagram. We shall not define a so-called optimum width of the grating, whereby the dimensions of the geometrical aberrations are equal to the diffraction limit. The designer himself has to verify whether it is the geometrical aberrations or the diffraction pattern which preponderates. This is a well-known procedure in lens design.

Chapter 2

T H E O R Y O F T H E S P E C T R A L I M A G E F O R M A T I O N

2.1 Description of the method

We shall confine ourselves to arrangements with a reflecting grating, which constitutes the only image forming surface in the system. Furthermore, this diffraction grating will act as the pupil of the system.

We shall describe and explain the method using fig. 2.1. We have chosen a right-handed rectangular coordinate system, with its origin 0 on the grating surface and its Z axis normal to the grating surface; the X axis is perpendicular to the grooves on the grating surface. The projections of the grooves on the XY plane are equally spaced parallel straight lines. This is in accordance with the way in which grooves are usually made on a ruling machine. We have designated the coordinates of a point on the grating surface with ^, r] and (,; we shall refer to £, and r\ as the pupil coordinates.

The location of the spectral image of an object point A (x, y, z) and the orientation of the image plane is ascertained by considering intersection B {x', y', z') with an

Fig. 2.1

Coordinate system for a general ray path. Object and image points are given in the X y z coordinate system. The grating surface is given in the I, ?;, f coordinate system; | and »; are referred to as pupil coordinates, u and

V are the pupil coordinates

after the transformation of the f,»; coordinate system into the «, v coordinate system.

arbitrary plane in the image space of a small pencil of light rays, diffracted at a surface element located around a point P (<^, r], Q. In cases of practical interest this surface element can be chosen in such a way, that it is small with respect to the dimensions of the grating, but on the other hand large compared with the grating period, which dimension is in the order of magnitude of the wavelength of light. In the following the small pencil of ligthrays mentioned will be referred to as light ray or ray.

For the optical path F from object point A (JC, y, z) to the intersecting point B (x', y', z') we can write:

F = AP-I-PB =

= [(x-^)^ + ( y - # - f ( z - 0 ' ] * + [ ( x ' - c r + (y-^)^+(z'-C)']* (2.1)

For convenience polar coordinates for the location of the object and its spectral image are introduced. Thus we obtain A {Q, cp, &) and B {Q', cp', 9') with the trans-formations: X y z x' y' z' = Qcos^ sin (p = Qsm9 = Q COS 9 COS (p = Q' COS 9 ' sin (p' = Q' sin 9' = Q' COS 9 ' c o s (p'

The relation between {, r\ and C is given by the shape of the grating surface. This relation can be written for any practical grating surface in a series development as follows:

C = ci,e + ^2in + oi,n' + Si,e + fi2i'ri + ^,^ri' + hvi^ + y,^^+... (2.3)

Note that in general the two planes of the main curvatures of this surface will be oblique with respect to the coordinate system, hence to the grooves.

A study of the run of one particular light ray, gives rise to the question how to find the point at which this ray will be intersected by another ray, coming from the same object point A {Q, (p, 9), but diffracted by the grating surface at a point located at a small distance from point P {^, rj, C).

In other words, what are the conditions for a focus. The conditions used will be based on the principle of Fermat, modified, however, so that it is applicable to diffraction gratings.

Fig. 2.2

The light path differences in the case of diffraction and reflection.

point A (Q, (p, 9), is diffracted at point P ((J, tj, 0 of the grating. This ray will intersect a particular plane in the image space in point B (g', (p', S'). A second ray also coming from A and intersecting the image plane in B, will be diffracted in point Q({-l-A(^, rj, C + AQ.

In cases of practical interest A | can be considered to be small with respect to the dimensions of the grating. Comparing ray APB with ray AQB, there will in general be a difference in the light path, amounting to CP + PD = AF. This light path difference corresponds with a phase difference In-AFjk, where I represents the wave-length of the light. In the direction of the diffracted beams, which are the main maxima of the interference pattern, formed by all the grooves together, the phase difference between the two rays works out at K-2n.

We write K = —m-n, where m is an integer, which denotes the order of diffraction in question. If the distance is noted as A^ = n-d, where d represents the grating period, we obtain:

A F _ _ A(J

(2.4) Because A^ will be very small compared with the dimensions of the grating and d will be in the order of magnitude of the wavelength A, we may write:

dF mX

T

Next we shall consider a ray being diffracted in point S{^,ri + Ari, C + AQ shifted with respect to point P {^, rj, C) parallel to the grooves. Because by the excursion A/j on the grating surface there are no phase leaps as come across in the «J direction, no phase difference arises between the two rays APB and ASB and the difference in the light path will be zero. According to the principle of Fermat the second condition for a focus therefore will be:

For the study of image formation we have to make a proper choice of pupil coor-dinates. This choice, which generally does not get so much notice, is, however, an important one. The directions of the pupil coordinates for which the conditions for reflection and diffraction have to be fulfilled must define the collections of light rays all lying in the same plane instead of crossing each other. With these collections only it is possible to find the location of the paraxial images. In the usual case of systems with rotation symmetry it is evident that for reasons of symmetry one pupil coordinate is taken in the plane, which contains the optical axis of the system and the principal rays of object beam and image beam. This plane is usually referred to as the meridional plane. The second pupil coordinate will always be chosen normal to this meridional plane.

However, in the case of spectral image formation by means of a diffraction grating there are no such simple considerations of symmetry. Because generally the principal rays of the incident and diffracted beams and the grating normal are not situated in the same plane, we cannot define a meridional plane in the usual way. Thus it is by no means clear for what directions in the pupil plane the partial derivatives of the light path function F must be taken in order to find the equations describing image for-mation.

When the object is situated in the XZ plane (see fig. 2.1) and the grating surface is symmetrical with respect to this plane, the XZ plane may be looked upon as a meri-dional plane. In that case the location of the merimeri-dional and the sagittal image for all wavelengths can immediately be found from conditions (2.5) and (2.6) respectively. Considering the zero order image formed by an aspherical grating (or mirror) the sym-metry with respect to the meridional plane may be disturbed by an oblique orientation of the main curvatures of the grating surface and the choice of the pupil coordinates is therefore not obvious in this case. If we consider the zero order image in an off-plane mounting (object out of the XZ off-plane) with a spherical grating the choice of the pupil coordinates normal and parallel to the grooves will obviously not be suitable for the purpose, because the meridional plane, in this case containing one of the pupil coordinates, is neither normal nor parallel to the grooves. Nevertheless, in literature on this subject we often find instances of such a wrong choice of the pupil coordinates normal and parallel to the grooves.

It is appropriate therefore to introduce new pupil coordinates u and v, found by a transformation of ^ and r]. The question arises, whether this coordinate system is a rectangular or an oblique one. It may in general be said that the pupil coordinate system to be introduced has to be an oblique-angled system, which will be proven in the following chapters. The new coordinate system is found (see fig. 2.3) by a rotation of the £, axis round the Z axis through a variable angle i/',, which gives coordinate u, and a rotation of the ri axis round the Z axis through a variable angle i/'2, which gives coordinate v.

1

Fig. 2.3

The coordinate systems in the pupil plane.

The transformation equations we have to introduce, are: ^ = Mcosi/'j — usini/'j

ri = usini/'i + t;cosi/'2

(2.7)

With these equations together with (2.5) and (2.6) we can derive the conditions to be used for the description of the geometric optical image formation with the diffraction grating. We obtain: dF du dF di 8F dri mX (2.8) dF dv dF di dF drj mX . , di dv dt] dv d (•2.9)

The characteristic function F a s written in equation (2.1) is developed in a series expan-sion expressed in variables u and v. Applying (2.8) and (2.9) to gratings with extended apertures we can show that these conditions can only be met rigorously for the first and second order terms of function F.

In order to keep the path length defects arising from the higher order terms small enough, we have to reduce the aperture of the grating. However, in practice we cannot keep the aperture below this limit, we need larger apertures. We have therefore to describe the image defects, usually called aberrations. In general the path length defects as well as the number of contributing terms will grow rapidly when increasing the dimensions of the grating. In literature the image defects are as a rule described with the aid of the equations:

SF mX . ,,

^ = - ^ + AL (2.10)

- ^ = AM' (2.11)

dr]

in which AL' and AM' are the deviations in the direction cosines of the final ray from the ideal focussing direction.

Obviously the deviations AL' and AM' are functions of ^ and rj. They can also be written in the form of a series expansion. Usually one finds the following elaboration of equation (2.10): the distance As between the intersections of the main ray and an arbitrary ray, diffracted at point P, with the image plane normal to the main ray is given by:

As = <PB>-AL' (2.12)

Herein one takes: <PB> x Q', SO that (2.10) is written as:

ÊL^-'^ + ^ (2.13)

di p Q'

Therefore all higher order terms in dFjdi, have been equalised to ASIQ', which is the angular deviation A(p' from cp'.

Apart from the fact that <PB> has been multiplied with a deviation in a direction cosine instead of with an angular deviation in order to find the aberration in image

As, which differences in a factor cos (p', the description used is only a first order

approach. Because we have to discuss terms in ^^ and £,* in F, the approximation <PB> K, Q' is not justified. Furthermore, the deviation AL' has to be taken for the direction cosine of the ray diffracted in P instead of for the direction cosine of the main ray of the entire beam. A more precise analysis for in plane mountings (see fig. 2.4) yields

A. = - < P ^ > ^ (2.14) cos {(p' — a)

Fig. 2.4

- Cross aberrations in the

(f'—a. t^o dimensional case.

By substituting (2.14) and the relation AL' = cosa-Aa in (2.10) we find:

dF mX , , ^ As mX , , , e' . , ,^ .^. jr= --j- + coscccos{(p - a ) - p ^ = - - ^ - I - c o s a cos((p - a ) ^ p g y A ( p (2.15)

which has to be developed in a series expansion.

This has also to be done with equation (2.11). For the three-dimensional case this relation becomes more complex, but we expect it will give us the solution of our problem.

However, this procedure was found to be less suitable and we have tackled the problem in a slightly different way, whereby the off-plane mountings have also been included.

There are two effects, which we have to take into account. The first one relates to the way the light path function has to be defined. The deviation of the rays from the ideal focusing direction for larger apertures is explained by posing the intersecting point with the image plane of a ray diffracted at point P (^, rj, C) deviates from the intersecting point of a ray diffracted at 0. This will influence light path function F. We have to compute the terms of function F responsible for this deviation. This will be done by the introduction of the new variables A^', A(p' and A9'. For the ray APB the intersecting point has the coordinates

Q' = QÓ+Ae'

cp' = <pó + A(p' (2.16)

9' = 9o + A3'

in which (^0, (P'Q, 9'Q) are the coordinates of the intersecting point with the image plane of the ray, diffracted in the centre 0 of the grating surface. Point BQ {QO, (Po, ^o) will

be called paraxial image point. T h e displacements from this paraxial image, the geometric optical aberrations, are given by A^', Q'Q COS 9'o'A(p' and Q'QAS'.

Substituting the coordinates (2.16) in the light path function F , we find the final expansion of F with the aid of the Taylor series

FiQ',(p',9') = F(Q'O,(P'O,9'O) + 'd^F dF . , dF . , dF ^., AQ + A(p' + AS -I-_dQ' dcp' d9'

+ i

d(p'd9' J ff'o. "P'o. ^'o- (2.17)

The second eflfect of importance for o u r calculations is related to the conditions for image formation. These conditions have to be adjusted in order to be applicable t o rays

part of the

e'o^ö'^sin 0, cosfl,

'cos !)'QA(P'

A(e, (p, 9)

incident much off-axis on the grating surface, so that the angle between these rays and the principal ray of the entire beam is not negligibly small. In fig. 2.5 we have indicated two light rays, one diffracted in point P (u, v, C) of the grating and the other at a distance Aw in point Q (u + Au, v, C + AQ. In the image plane chosen normal to the principal ray the aberrations for the two light rays from P and Q will be g'o 0'„ and

g'o i0'„ + ö9'„) respectively. These rays include and angle 8^, with the plane defined by

the u axis and the principal ray. The projections of these rays in this plane include an angle (?, with the principal ray itself. By the choice of the image plane normal to the principal ray we introduce an extra light path difference AF between the two rays diffracted in P and Q, equal to —g'o sin fi cos d^öd'^.

In this case as well we take the distance AM small compared with the dimensions of the grating surface, which is allowed because the wave front of the diffracted beam will be a continuous plane. We may therefore write instead of condition (2.8) the following more precise condition for image formation by a diffraction grating:

SF mX / • T a 8^'y /T io\

^r—= ^cositf, — öosinéicosöi-r— (2.18)

du d du

Likewise (2.9) can also be written more precisely, which gives:

SF mX . , , . T ;v 5Öi' .- . „ .

^ — = -t-—r-sin 1/^2 - 00 sin 02 cos 02 ^ - (2.19)

dv d d v

In this Ö2 is the angle included by the ray diffracted in P and the plane defined by the principal ray of the diffracted beam and the v axis. The angle between the projection of ray PB in this plane and the principal ray is

^2-In the following chapters we shall use the conditions (2.18) and (2.19) for the description of image formation. The procedure to be followed is to select the terms of a particular order in u and i; from the equation obtained. For these terms it is required that the conditions are closely met. This is possible by introducing the new variables Ag', A(p' and A9', which are as a matter of course expansions in u and t; similar to the light path function F.

When more precise conditions for image formation and the Taylor expansion (2.17) of function F are applied, a series expansion containing a number of novel third and fourth order terms in u and v is obtained. Furthermore it will be found that upon the application of the new pupil coordinate system (u, v) it is possible to avoid errors in the paraxial approximation caused by a less appropriate choice of the pupil coordinates as found in literature.

Finally it is pointed out, that in first instance the image plane will be chosen normal to the principal ray in the paraxial image point and that one of the results of our calculations will be the reorientation of this plane to obtain coincidence with the

2.2 The series expansions of the light path function F and the conditions for image formation

In order to obtain a general series expansion of light path function F, we have to find expansions for the parts AP and PB of the light path (see fig. 2.1).

From fig. 2.1 it follows immediately:

(APy^(x-0' + (y-r,)' + iz-0'

= x^ + y^ + z^ + ^' + r]^ + C-2x^-2yri-2zC (2.20)

Applying the transformations (2.2) and (2.7) we obtain:

(APy = g^-j-u^-fi)^ —2MI;COSI/'I sin 1/^2 + 2Müsini/'iCosi/'2 +

— 2ecos9sin(p(ucosi/'] — t> sin 1/^2) +

— 2e sin 9 (u sin i/* 1 + v cos i/'2) — 2gl^ cos 9 cos (p + C^- (2.21) Here we have to substitute an expression for C that can be derived from (2.3) by

sub-stitution of the transformations (2.7). We obtain: C =

-|-u^(aiCOs^i/'i+a2sini/'iCosi/'i-|-a3Sin^i/'i)-l--I- Mi;[ —2aiCOs t/'iSin i/'2 +a2(cos i/'jcos i/'2 — sin i/^jsin i/'2) + 2a3sin i/^^cos ij/2 +

+ v^((Xisin^{j/2 — a2sin i/'2Cos i/'2 + a3Cos'^i/'2) +

-|-u^(j8iC0S^i/'i-l-jS2sin lAiCos^i/'i H-^jsin^i/^iCOS l/'t

^-^4sin^^/'l)-l--|-u^(;[—3;SiCos^i/'isin i/'2+^2(^2 sin i/'jcos i/'isin i/'2+cos^i/',cos t/'2)+

+ 1^3(2 sin i/^icos i/'icos 1J/2 — sin^i/'isin 1/^2) + 3j34sin^i/'iCOS 1/^2] +

-t- M(;^[3^iCos i/' iSin^i/'2 + Pzi^^^ i/'isin^i/'2 - 2 cos ^|/^sin i/^2COS 1/^2) + -I- ^3(cos i/',cos^i/'2 — 2 sin i/' jsin (/'2COS 1/^2) + 3j64sin i/']Cos^i/^2] + -I- v^{ — Pisin^il/2 + j82sin^i/'2COS ij/1 — jSjsin (/'2COS^i/'2 + PA!^O%^4'2) +

+ u*(yiCOS*\pi + )'2sin ipicos^ij/i + yssin^i/^icos^i/^i -f y^sin^i/'icos i/', -I- yssin^i/^ j) 4-..

(2.22) A final expression for AP is found after substituting this expression for ( in (2.21) and taking the series expansion of the square root. A similar expression will be found for

PB upon replacing g, cp and 9 by g', q)' and 9' respectively. For the total light path AP + PB = F we find:

F = g + g'-uiCi+C'i)-v{C2-C'2) + iu\C3 + C'3) + uviC^+C'^) +

+ ^v\C, + C's) + lu\C^ + C'e) + WviC^ + C^) + i«i''(C8 + Cg) +

+ i(;^(C9 + C ^ ) - f V ( C , o + C;o)+... (2.23) where the coefficients C„ and C^ are contributed by AP and PB respectively.

In order to obtain the coefficients representing the geometric optical aberrations, we have to substitute the coordinates introduced in (2.16). In other words we have to develop expression (2.23) with the aid of the Taylor series (2.17). The result of this development up to and including the terms of the third order in u and v and the terms in u'*^ is

F = e4-eó + T i + T 2 - l - T 3 + ...-l-T,9-l-T2o + Ae' + ... (2.24)

in which: Ti T2 T3 T4 T5 Te T7 Ts T , T.o T u T12 Ti3 = — u(cos9sin(pcosi/' = —1)( — cos 9 sin (jo sin = +iu\C3 + C'o,). = +uviC^ + C^J. = +^v\C, + CÓ,). = +^u\Ce + C^,). = +iu^v(C, + C^;). = +iuv\Cs + Ci.). = +^v\C, + C^,). = +iu\C,o + CiJ. = -u[cos9ócoS(i9Ócos

1 -1- sinS sin i/'i -1- cos Sósin cp'ocosif/i + sin Sósin i/'t). i/'2 + sin9cosi/'2" - cos 9'osm (pósin 1/^2 + sin SQCOS I/'2, •

i/' 1A9' — (sin 9'osm (JOQCOS ij/^ — cosSósin i/',)AS'].

= - ü[ — COS SQCOS cp'osin 1/^2 Acjo' -1- (sin Sósin (p'osin 1/^2 + cos SQCOS 1/^2) AS'].

= -l-u[icos9ósin<pócosi/ri(A<p')^ + sinSÓ cos (^QCOS ilfiAcp'A9' + + (^cos Sosin (PoCos i/'i -j- ^sin Sósin i/'i) (AS')^].

Ti4 = -|-y[ —icos9ósin<pósini/'2(A<p')^ —sin9ócos(/)ósinl/'2A(p'A^'-|-— (icos Sósin cp'oSin 1/^2 —sin9ócos(/)ósinl/'2A(p'A^'-|-— isin SQCOS 1/^2) (A9')^] •

T i 5 = +iM^^ (cos 9ósin cpócos I L Qo

+ 2cos9ósin<pó(aiCOS^i/', -|-a2'

-I-i/' 1 -j- sin Sósin -I-i/'i) (cos Sócos <pócos -I-i/'i) +

a2sini/'iCosi/'i-f-a3sin^i/'i) A<p'-|-2

(cos 9'osia cp'ocos i/» i -I- sin Sósin i/'i) (— sin Sósin (pócos i/'i -I- cos Sósin i/'i) +

Qo

+ 2 sin9'oCOS(p'o{(XiCOS^lpi+a2&irnljiCOS ij/1+cc^sin^il/i) \A9'>.

(cos SQCOS cp'oCOS i/' 1) (— cos Sósin ^ósin i/'2 + sin Sócos i/'2) + I L Qo

(cos Sósin ^ócos i/^ i -f sin Sgsin i^ O ( — cos SQCOS ^ósin il/2) +

Qo

T16 = +UV

Qo

'2 + a2C0s(/'icos\J/2 — a2sini/'isinij/2 + 2a3sini/'icos 1P2)

Acp'-I--t-cos öósin ^^ÓC — 2a 1 cos 1/'1 sin 1/'

•ósin i/'2 + sin SQCOS i/'i) +

(— sin Sósin ^JQCOS ij/1 -I- cos 9ósin lA 1) (— cos 9ósin cp'osin i/'2 + sin 9'oC

- Qo

(cos 9'ositi (p'ocos lA 1 -I- sin Sósin i/'i) (sin Sósin «pósin i/'2 -f- cos SQCOS ij/2) +

Qo

+ sin9'ocos<p'o{ — 2aiCos(/'isin 1/^2 + a2COSi^icos^^2 — a2sini^iSini/'i + 2a3sini/'icosi/'i) A9' >.

Ti7 = -l-it^^<^ ( —cos9ósin(?'ósini/'2 + sin9óc IL Qo

+ 2cos9ósin(pó(aisin^i/'2 —a2sini/'2COsi/'2 + a3COS^i/'2) Acp' +

1 Sósin «pósin i/'2 + cos 9QCOS I/'2) + 1 SQCOS 1^2) (— cos SQCOS (pósin I/TJ) +

2

— (— cos öósin (p'o^in 1/^2 + sin SQCOS ij/2) (-1- sin ^

Qo

+ - — ( - c o s Ö ^ i L Qo

+ 2sin9ócos(pó(aisin^i/'2 —a2sin Qo

i^jcos 1^2-f a3COS^i/'2) A9']

Tig = —iu^ -[1—(cosSósin^ócosi/'i-l-sinSósini/'i)^].

(eó)

T j 9 = -1- Mf (cos Sósin (pócos i/'i -I- sin Sósin i/'i) ( — cos 9ósin (pósin i/'i + sin SQCOS t/'2).

(eó)

T20 = —iu^ - [ I - ( - c o s 9 ó s i n 9 ) ó s i n i / ' 2 + sin9ócosi/'2)^].

(eó)

Equation (2.24) is the general series expansion of the light path function F, which is the basis of the following calculations. When the present expansion with those found in literature is compared we find that apart from the terms T6-T12, a number of novel terms Ti 3-T20 appear, having the same order in u and v. It will be found that in many practical cases a considerable number of these new terms are fully comparable in magnitude with those considered hitherto. The coefficients C„-I-CÓ„ are given in full in Chapter 2, section 14.

The terms of the partial derivatives of expansion (2.24) to variables u and v satis-fying the conditions (2.18) and (2.19) describe the image formation. In the following sections we shall show that from the first order terms in u and t; in F we obtain the angular coordinates cp'o and 9'o of the paraxial image.

Furthermore, only complying with (2.18), we find from the second order terms in F two values for g'o, each value with a certain combination of the angles i/'i and if/2- The two angles i/'i and i/'i will also have two solutions. As a result of the striking sym-metry with respect to u and v in the series expansion of F, we find while complying with (2.19) the same two values for g'o. However, each of them with another combi-nation of I/'I and i/'2. It will therefore be found that for both astigmatic images among other things for the determination of the position and the width of the image, we can confine ourselves to condition (2.18). The two combinations of eó. i/'i ^"d '/'2. as found with condition (2.18), do not comply with condition (2.19). Here we have a defect in image formation, usually called astigmatism. It also appears that for the description of the length of the image we only need condition (2.19).

Considering one of the combinations g'o, i/'i and i/'2 complying with (2.18) we now wish to give the procedure for a further description of the image in question. This procedure is correct for the two combinations found by us. In order to study the dimensions of the cross-section of the beam with a plane containing the paraxial image point, we select two collections of light rays. One collection includes the rays that are diffracted at the u axis in the pupil plane and the other originates from the v axis.

The collection of rays diffracted at the v axis may be considered by application of condition (2.18) as well as condition (2.19). In the case of (2.18), in which the partial derivative of F is taken to the variable u, for case M = 0 we find an expression for the curve formed by the intersections of the collection of rays in question with the image plane. The part of this curve defined by the chosen dimension of the pupil in the

v direction, so the total amount of aberration in question, will be obtained from

condition (2.19). Herein the partial derivative of F i s taken to variable v. Considering the collection of rays diffracted at the u axis we find the expression for the inter-secting curve with the aid of condition (2.19). The extend of the aberration as a func-tion of the dimension of the pupil will in this case be obtained from (2.18).

To execute our procedure, we must first define the intersections of the rays with the image plane Ag' = 0, which is normal to the principal ray in point (g'o, q>'o, So).

An example of such a cross-section of the beam is given in fig. 2.6. From the col-lection of rays diffracted at the u axis we find the width of the spectral image.

The aberration caused by the terms in u^ of the equation found with (2.18) will be referred to as coma. The terms in «^ of this equation determine the spherical aber-ration. With the collection of rays diffracted at the v axis we obtain the length / of the astigmatic line and its radius of curvature R^. From both collections we obtain an expression for the angle between the astigmatic line and the normal to the direction of the dispersion. We shall call this angle the first slant angle of the astigmatic line, which will be represented by the symbol cp.

Finally we have to calculate the angle Q, giving the reorientation of the image plane in order to find optimum sharpness of the total astigmatic image. This angle Q will be referred to as second slant angle. Deviations ei and £2, as indicated in fig. 2.6, are mainly caused by this incorrect orientation of the image plane.

For a more exact description of the deviations such as ei and 62 we have also to consider third order terms such as u^v etc. in the earlier mentioned equation found by application of (2.18). However, because in cases of practical interest it may be expected that they will not disturb the description of the image seriously, we shall not study these terms. Furthermore we expect, that the description of these terms will only be significant, upon choosing curved pupil coordinate axes. The coeflScients of these terms are moreover very complex.

In this study the terms in u^ of the partial derivative of F to u, describing spherical aberration, are the only third order terms considered. In the equation found by the

Fig. 2.6

Example of a spot diagram. O Intersecting points of

rays emerging from the «-coordinate

D Intersecting points of rays emerging from the u-coordinate

A Intersecting point of ray diffracted at («max. «max) in the pupil plane

application of (2.19) we shall not consider the second order terms, the term in v^ in-cluded.

We shall now classify the meanings of the subsequent terms of expansion (2.24). When condition (2.18) is used we obtain from:

Ti = term independent on « and u: direction of the principal ray. T2 = zero.

T3 = term u | the azimuths i/'i and i/'2 of the pupil coordinates u and v and the T = term v I location of the paraxial images.

T5 = zero.

Tg = termw^: coma and, combined with term uAcp' of T15, spherical aberration. T7 = term uv: second slant angle of the astigmatic line.

Tg = termy^: radius of curvature of the astigmatic line projected in the image plane Ag' = 0.

Tg = zero.

Tio = term M^: spherical aberration.

Ti 1 = term Acp' \ first slant angle of the astigmatic line. term AS' f angular measure of the aberrations.

these two terms are equal to two terms of the expansion of

— g'oSm4>i cos 8lidO'Jdu) in condition (2.18) and

there-fore disappear. term u{dAcp'/du)

term u(dA9'/du)

T , , = term v(dAcp du) , , , , ^ , • • ,• , , . „ , ' , ^ second slant angle of the astigmatic line. term v{dA9 du) '

radius of curvature of the astigmatic line projected in the plane Ag' = 0

T13 = term {Acp')^ term Acp'A9' term (AS')^

further terms with derivatives of Acp' and AS' are not going to be considered because they are of the third order in u and v (except for u^).

Ti4 = not to be considered.

T , , = term uAcp' I , . , ,

\ spherical aberration.

term «AS j term u^(dA(p'/du) term u^(dA9'/du)

these two terms are equal to two terms of the expansion of —g'o sin ^f cos 8^{d9'Jdu) in condition (2.18) and therefore disappear.

Tjg = term rA^j' I radius of curvature of the astigmatic line projected in the term i;AS' J plane Ag' = 0.

Ti7 = two terms, which we shall not consider.

Ti8 = term uAg': second slant angle of the astigmatic line. term u^{dAg'jdu): negligible.

Ti9 = term vAg': radius of curvature of the astigmatic line when projected in the

(Ag'—g'oAB') plane, which will not be considered. term u- v{dAg'/8u): negligible.

T20 = term ti^(Mö'/5M): negligible.

Ag' = term 5AÖ'/5M: negligible.

When combinations g'o, i/'i and 1^2 complying with (2.18) are used, we have together with condition (2.19) the following interpretation of the terms.

Ti = zero.

T2 = term independent on u and v: direction of the principal ray. T3 = zero.

T4 = term u: coefficient is zero.

T5 = term v. length of the astigmatic line.

Of all further terms we select only those, which are of interest in the approximation put forward in this treatment.

T,, = term u(dAcp'ldv) I „ . . /';!AQ'/a \ > coefficient IS zero. term u{dA9 jdv) J

T12 = tcvm Acp' 1 first slant angle of the astigmatic line, angular measure of the term AS' ( aberrations.

these two terms are equal to two terms of the expansion of

-g'o sin^2^os82(d9',/dv) in condition (2.19) and

there-fore disappear. term v{dA(p'ldv)

term vidA9'/dv)

After the expansion of the light path function F we obviously have to develop the conditions for image formation. The term in (2.18) being dependent on u and v, which in this connection has to be considered, is —g'o sin fi cos 8i{dö'Jdu).

In (2.19) this will be the term —g'o sin ^2 cos 82id0'i/dv). Because we do not consider terms of the third order in u and v in the equation found by the application of (2.18), except for the terms M', it is permitted to write cos 8^x1. For comparable reasons we may write cos 82 ~ i. We have therefore to develop sin ^i and sin f 2 in a series expansion. It is easy to see that (?i can be considered as the angle between the two vectors (flip ai^, «i^) and («2,, ^2^, ^23) in which:

«1, = eócosSósincp; «12 = eósinSÓ «13 = eócosSócostpó

, n' • ' / (2.25)

''21 = öo cos SQ sin «Po — M cos 1/'1 «22 ^ eósinSÓ — usini/'i

«23 = eócosSócostpó - C Using the well-known relation:

cos ,?i = «.••«2. + «i2-«22 + «i3-«23 (2.26)

\{a,\ + a,\ + a,\)W{a2] + a2\ + a2\f\

taking the square root of 1 — cos^c^i and developing this in a series in u we find:

sinc^i = ( — )[1—(cosSgsincpócosi/'j

-|-sinSósini/'i)^]*-|-\QO/

u^'

-!-( — ) (cos Sosin <pócos ^^ 1 -f- sin Sósin i/* 1) [ 1 — (cos Sósin cp'oCOS 1/' j -I- sin Sósin 1/' 1 )^]*

\Q'OJ

X [1—(cos Sósin (pocos I/'I-I-sin Sósin i/'i)^ — eócos

SócostpóC'^iCOS^i/'1-1--I- a2sin(/'iCOSi/'i-|-a3sin^i/'i)]+ ... (2.27) This expression, together with cos 81 w 1 has to be substituted in (2.18).

The term sin ^2 in condition (2.19) may be derived similarly with the result:

sin 02 = I — )[1—(cos Sósin (pósini/'2 —sin Sócosi/'2)^]*+... (2.28)

This expansion can be broken off" after the first term, because terms v^ in (2.19) that give a higher order approach of the length of the astigmatic line, are not considered in this study. The reason for this is, that in the vast majority of cases of practical interest a better approach is not required.

Besides this, in cases when this better approach could be significant, we also have to take into account the correct form of the pupil. We have then already reached the stage of ray-tracing the system.

2.3 The direction of the principal ray

expansion (2.24) of the light path function F. This will be done stepwise. In each section we shall select terms representing a certain aspect of image formation. Applying (2.18) and considering only the terms that are independent on u and v we find:

-^— = — (cos S sin <j!) +cos Sósin (pó) cos I/'I—(sin S +sin So) sin I/'I = -7-cos i/'i (2.29)

With condition (2.19) we obtain:

—-= -|-(cosSsin(j9-l-cosSósin(pó)sini/'2 —(sinS-|-sinSÓ)cosi/'2 = + —7-sin\ii2 (2.30)

The solutions of the equations (2.29) and (2.30) are:

sinS + sinSÓ = 0 (2.31)

We recognize here the well-known grating equations, giving the direction of the principal ray of the diffracted beam. As expected we find that angles i/'i and ^2 don 't appear in these equations, which means that there is no influence of the orientation of the pupil coordinates u and v on the direction of the principal ray of the diffracted beam.

2.4 The general image equation

After determining the direction of the principal ray of the diffracted beam, giving the directional coordinates cp'o and So of the spectral image for a certain wavelength, we have to find an expression for the coordinate g'o, representing the image distance. This expression follows from the first order terms in u and 1; of the equations found by the application of (2.18) and (2.19) to the expansion (2.24) of the light path func-tion F. Applying (2.18) we obtain:

- | ^ = + u(C3 + CÓ3) + K Q + Co,) = 0 (2.33)

This must be valid for all values of u and v and the solutions are therefore C3 -I- CÓ3 = 0 and C4-I-CÓ, = 0; written in fuU:

-i [1—(cosS sin ^ cos I/'I -l- sinSsini/'j)^]

— 2cosScos(i()(aiCOS^i/'i-l-a2sini/'iCosi/'i +a3sin^i/'i) , [1—(cosSósintpócosi/'i-f-sinSósini/'i)^]

-I-Qo

— 2cosSócos<i9Ó(aiCOS^i/'i-l-a2sini/'iCOsi/'i+a3sin^i/'i) = O (2.34) and:

[(cosSsin <pcos I/'I -f-sinSsini/'i)( —cosSsin(psini/'2-l-sinScosi/'2) -I-+ cos I/'I sin i/'2 — sin i/'icos i/'j] — cos Scos cp [ — 2aiCos i/'iSin i/'2 -I-+

+ a2(cosi/'iCosi/'2 —sini/'iSini/'2)-l-2a3sini/'iCosi/'2] +

[(cos Sósin ipócos I/'I -l- sin Sósin i/'i) (— cos Sósin <pósin ij/2 + sin Sócos 1^2) + Qo

+ cos i/'isin i/'2 — sin i/'icos i/'2] — cos S^cos cpó[^ — 2aiCos i/'iSin i/'2

-t--f a2(cosi/'iCOsi/'2 —sini/'isini/'2)-l-2a3sini/'iCosi/'2] = 0 (2.35) We shall refer to equation (2.34) as the general image equation of a diffraction grating,

because when the "in-plane" mounting with a grating symmetrical with respect to the XZ plane is considered, equation (2.35) disappears and equation (2.34) reduces to the equation which is normally used in literature as the general or paraxial image equation. If the shape of the grating surface, the value of mX/d, and the position of the object point are taken into consideration as given the only three variables in (2.34) and (2.35) after the substitution of (2.31) and (2.32) are g'o, i/'i and i/'2. It will be observed that it is possible to eliminate variable g'o by substitution of the expression for ^0 from (2.34) in (2.35). One equation is then obtained with two variables, ipi and i/'i- When this equation is evaluated in a more comprehensible form we find:

(tani/'itani/'2-i-l)(Xtan^i/'i-|-Btani/'i-l-C) = 0 (2.36) in which the coefficients A, B and C will be given below. If the first factor in (2.36) is

zero we find solutions cot i/'2 = —tan i/'i. However, these solutions have no practical significance as they represent cases with coinciding pupil coordinates. Therefore (2.36) reduces to:

^ t a n V i + B t a n i / ' i - l - C = 0 (2.37) The coefficients A, B and C are indicated by the functions:

A = (-|-2a3sinScosSsin(pó —a2Cos^S)(cos(p-f-cos(pó) +

sinS cos^S(sin cp + sin cp'o).

B = [ — 2aiCos^S-i-2a3(l—cos^Ssin^(pó)] (cos (/)-!-cos (?)ó) +

cos^S(sin^(a —

sin^taó)-Q

C = [ —2a,sinScosSsin(pó + a2(l~cos^Ssin^<))ó)](cos<p + cos(pó) +

l ,

+ -sinS(l —cos Ssin(psin(/)o)(sin(/) + sin(pó).

For I/'I we generally find two solutions, i/'i, and \pi^, yielding with the aid of (2.34) in two values go, and 0Ó2 foi" Qo- In general the two distances go, and eÓ2 are not equal. It is to be expected that application of (2.19) will yield the same distances. In doing so and only considering the first order terms in u and v, we have:

^ = + M(C4 + Co J + v{C, + Co,) = 0 (2.38)

which will be complied with if C4-I-CÓ, = 0 and Cj-l-CÓ, = 0.

Writing these equations in full and eliminating g'o in the same way as done with (2.34) and (2.35), we obtain:

(tani/'itani/'2-l-l)(Ctan^i/'2-Btani/'2 + /l) = 0 (2.39) in which the coefficients A, B and C are the same as in (2.37). Here as \vell the solutions

cot i/'2 = —tan I/'I are of no practical interest. Therefore (2.39) reduces to:

C tan V 2 -Bt-dnil/2 + A = 0 (2.40)

with solutions i/'2, and i/'22.

Combining the solutions for 1/', and t/'^ we find four solutions for the pupil coor-dinate system, defined by a rotation of the <^ axis and the i] axis respectively by the angles:

1. I/'I, and i/'2,

2. I/'I, and i/'2, (2.41) 3. lAi, and i/'a,

The values of these angles as found from (2.37) and (2.40) are: ^ , - B + ( e ' - 4 / l C ) * , - . , , tan I/'I, = ^-^j '- (2.42)

eo„„ = ±l±(^=id£)! (2.44,

for the pupil coordinate system, that are applicable, are: 1. I/'I, combined with i/'2,

4. I/'I J combined with i/'22

The appearance of two solutions fully meets expectations, because we have in optical image formation of one point in general two line shaped images, usually referred to as astigmatic lines. These are in accordance with the two main curvatures of a general-ly shaped wave front of the beam in the image space.

Using the equations (2.43) and (2.44) we can write:

coti/'2, = cot( ipi^-^ + Kn] (2.46)

in which K will be an integer. From (2.42) and (2.45) we obtain:

cot i/'i, = cot( I/'I, - ^ -f- KTI ) (2.47)

The final solutions of the pupil coordinates used in the following sections, are indi-cated by a rotation of the £, axis and the rj axis respectively over the angles:

«Al = I/'I,

(2.48)

with which one astigmatic line is described, and ^1 = "Au

1A2 = "At. " 2 + ^ "

for a description of the second astigmatic line (see fig. 2.7). It is pointed out here that I/'I, and i/'i^ are the solutions of the quadratic equation (2.37).

We shall choose that value of K with which the u, v, z coordinate system is a right handed one, hence [I/'I—1/'21 < -l-7r/2.

It will be seen that the same distances g'o^ and g'o^ are obtained from (2.19). Note that I/'I and i/'2 are dependent on wavelength X. In the description of the image formation by a diffraction grating another pupil coordinate system is required for each wavelength.

The orientations of the pupil coordinates are given by the solutions of equation (2.37) in the form as pointed out in (2.48) and (2.49). In general the solutions of (2.37) are such, that the coordinate system will be an oblique one.

The advantage of the use of the two different coordinate systems as indicated in (2.48) and (2.49) will be, that for both astigmatic images it is among other things possible to find the position of the paraxial image and the width of the spectral image in the direction of the dispersion with the aid of condition (2.18). It is also possible to determine the length of the astigmatic images with one formula derived from (2.19). We must therefore only derive one set of formulae for the description of both astig-matic images.

What exactly happens with the rotation of the pupil coordinates is indicated by the following consideration.

We shall only consider the rays diffracted at a line coinciding with the u coordinate in the pupil plane, which is exactly the same as the solution cot i/'2 = —tan i/'i. This means that in the calculations we have to substitute v = 0 and that the angle i/'2 is of no interest . Applying (2.18) we now have only one equation for g'o and i/'i; C3 + CÓ3 = 0. This may be written as: g'o = g'oi^i). An infinite number of distances

g'o is found dependent on the orientation of the u coordinate. We have proved, that

the equation g'o = g'oi^i) together with the condition:

^ ^ = 0 (2.50)

# 1

will result immediately in equation (2.37) if we eliminate variable ^0- It niay therefore be concluded that one astigmatic line will appear there where eó(i/'i) will have a minimum and the other at the maximum. These are the only two collections of rays, diffracted at a line in the pupil plane containing 0, having an intersecting point in the

paraxial approach. In all other cases we find, that the collections of rays, when complying with Fermat, will cross each other, at the narrowest distance.

The same result is obtained from g'o = eó(i/'2). derived from C5 -I- Co, = 0, com-bined with dg'oiip 2)1^^2 = 0.

the solutions of equation (2.37)

^ 1 = ^ , , 1^2= lAj =1/', - ; r / 2 + K '''2 = ^2='!'^ -nl2+Kn combination 3 >1 •/-I = %, combination 4 'J'1 = K

i>j = i/(j = i|/, —;t/2+K7r 1^2 = ^2i = (''i2-'r/24- Kn

2.5 The first slant angle of the astigmatic line

Having defined the location of the astigmatic line, endeavours will now be made to find its orientation and its dimensions as brought about by the geometrical aberra-tions.

The astigmatic image is described in a rectangular coordinate system with its origin in the paraxial image point with coordinates g'o, cp'o and So. The coordinates themsel-ves are Ag', g'o cos 9'oAcp' and öóAS'. The coordinate A^' coincides with the principal ray, g'o cos 9'oAcp' is in the direction of the dispersion and g'oA9' is normal to this direction.

For simplicity a description will first be given of image formation in a plane normal to the principal ray at Ag' = 0. Later we shall determine the angle Q through which this image plane has to be rotated round the g'o cos 9'oAcp' axis in order to obtain optimum imaging for the entire astigmatic line.

First we shall try to find an expression for the angle cp between the astigmatic line, when projected in the image plane Ag' = 0, and the normal to the direction of the dispersion. The angles (/> and Q will be referred to as the first and the second slant angle of the astigmatic line respectively.

An expression for the angle </> will be derived from equation dFjdu = 0 with M = 0, defining the curve where the rays going through the v axis in the pupil plane inter-sect plane Ag' = 0. Taking all other values of u, we obtain a collection of curves forming together the blurred image for one wavelength.

The tangent to the curve dF/du = 0 with M = 0 is obtained considering term T u in equation (2.24). This yields

I-^—I = —cosSócos(i()ócosi/'iA(i()'-l-(sinSósin(pócosi/'i—cosSósini/'i)AS'=0 (2.51) Equation (2.51) represents the q' axis of the coordinate system (p', q') in the image plane A^' = 0, which is oriented at an angle cp with respect to the coordinate system

{Q'Q COS 9'O Acp', g'o AS') (see fig. 2.8).

We can easily see that

sine/) = ( — sin Sósin ^ócos I/'I + cos Sósin I/'I) (2.52)

C\Q'O

cos cp = (cos (PÓCOS I/'I) (2.53)

CiQo

The coefficient c^ is found by taking sin^cp + cos^cp = 1; we obtain

Ci = — [ ( —sinSósin(pócosi/'i-l-cosSósini/'i)^-l-(cos<pócosi/'i)^]* (2.54) Qo