## Elasticity Constants of Isotropic Materials

The elasticity theory of isotropic media involves two elasticity constants characterizing the elastic properties of the material which thus is sometimes called a Fig. 1.48 Torsion of a beam of square cross-section in the large deformation theory (from La Torsion des Prismes by Saint-Venant 91 ) Fig. 1.48 Torsion of a beam of square cross-section in the large deformation theory (from La Torsion des Prismes by Saint-Venant 91 ) bi-constant theory. However, four different constants are usually...

## Info

15 Sophie Germain 68 established a biharmonic equation in memoirs to the French Academy, the last in 1815, as attempts to explain the vibration modes of an elastic plate. This problem was brought to her attention by the fascinating experiments by Chladni 28 showing various patterns formed by the sand which moved on a vibrating plate, when excited by acoustic strings of different tones, until reaching the two-dimensional positions of the nodes. After seeing Chladni's patterns -or nodal curves -...

## G V gJ m

This angle is zero at the poles and maximum near t n 4. In practice, liquid mirror bearings are levelled using a precise bubble level that is sensitive to the vertical direction defined by the acceleration vector g ac where ac is the centripetal acceleration. The deviation of g ac from the true vertical is corresponding to the sin2t term of (7.52). In this case, the required adjusting angle is just the cos t term this term defines the direction of vector m + to coalign with g ac. Under this...

## E

The choice of n 1 is preferable for making small corrective plates. For medium to large plates, a uniform loading at q2 0.85bar with n 4 i.e. q1 0.2125bar allows reducing the amount of glass outside the clear aperture. The Sphe 3 adjustment is achieved by slight under or over pressure during the end of the polishing while maintaining the constancy of the n -ratio. Vacuum-pressure controllers allows maintaining the right pressures q1, q2 and the constancy of their ratio. For aspherizing plates...

## M

Fig. 9.5 Anastigmatic monocentric magnifier (A) corrected in all orders for the three first aberrations. Lens components (B) individually satisfying the Abbe sine condition. In a high power microscope, these are classically used for reducing the beam divergence in the front part of the objective (C) where the final imaging convergence must be achieved by cemented lenses Fig. 9.5 Anastigmatic monocentric magnifier (A) corrected in all orders for the three first aberrations. Lens components (B)...

## T T4010 with T40 v 13[lp83v 1p

The infinite slope at the null thickness of the mirror edge allows use of a simply supported edge as the boundary condition, while the infinite thickness at the center is due to the imposed null paraxial curvature. The central thickness can be made finite (Fig. 5.12) in agreement with Rayleigh's quarterwave criterion (Lemaitre 14, 16 ). Tulip-form mirrors of the VTD class in Sect. 3.2 can exactly generate an Astm 5 mode. Setting q 0, n 4, and m 2 in (3.14), the roots obtained for generating the...

## Portrait Gallery

The following portraits of some pioneer elasticians, and some opticians and astronomers who used elasticity to improve the performance of astronomical telescopes and instrumentation, are a personal choice of the author. Fig. 1 Galileo Galilei (1564-1642) was a universal scientist who, from poor monocular opera glasses, invented the telescope in 1609 - just four centuries ago - and discovered with it Jupiter's four Galilean satellites. He introduced mathematics in the formulation of the basic...

## Off Axis Segments of a Conicoid Mirror

The shape of large primary mirrors of presently large segmented telescopes are either a sphere (Het, Salt), a hyperboloid slightly deviating from a paraboloid (Keck, Gtc) or an actively controlled pseudo-plane-aspheric surface (Lamost). For the next telescope generation - which will have aperture diameters in the class 25-50 m -, it is useful to determine the shape of a segment for a conicoid primary mirror which may appreciably deviate from a paraboloid. Still representing the optical shape of...

## End of European Renaissance and Birth of Telescopes

The emergence of small blown objects in glass originates in Phoenicia, Syria, and Egypt around 200BC and much earlier for non-blown objects. The magnification effects of transparent materials was known in Antiquity. The blown glass technique had passed through the ages via the Romans and settled in Venice before the first millennium. The first lenses - lense was the Latin name given to a biconvex disk - used as spectacles appear in Italy before 1300 for correcting presbyopia because hand-lenses...

## Diopter of Curvature c 1R

Let us consider a surface - a diopter - which separates two media of refractive indexes n and n' (Fig. 1.15), and set the origin of the z-axis at its vertex. The abscissa f OF of the object focus F in the object space is defined as the distance from the surface vertex O to the convergence point F of rays issued from a source point at infinity in the image space. Conversely, the abscissa f' OF' of the image focus F' in the image space, is defined from a source point at infinity in the object...

## A40 E

The uniform bending moment at the contour, Mr a , exactly cancels the curvature component of the flexure (see also Sect. 7.3). Examples of such configurations are displayed by Fig. 3.3. 3.3.2 Configurations in the VTD Class Interesting solutions in the VTD class can be derived by assuming a null rigidity at the contour, D a 0. This condition directly entails that Mr a 0 which is easy to satisfy in practice and corresponds to a simply supported edge or to a free edge. From (3.21), the condition...

## Glossary

Abbe's number of a glass see reciprocal dispersive power. Abbe sine condition or sine condition optical design condition which cancels spherical aberration and linear coma. In third-order theory of aberrations, a system which satisfies the sine condition is said to be aplanatic. Abbe sphere locus of the intersection of input and output conjugate rays in a centered system satisfying the sine condition. Acomatic grating diffraction grating which provides an aberration correction for coma. Active...

## Gaussian Optics and Conjugate Distances

Considering a curved surface (diopter) separating two optical media of refractive indexes n in the object space and n' in the image space, and an optical axis z taken perpendicular to this diopter with the origin O onto its surface, the first-order optics theory based on Snell's law allows determining the axial positions of the focii F and F' and of the conjugate distances of an object and its real or virtual image. The general form of a local surface is defined by its two principal curvatures....

## Descartes Ovoids And The Aplanatic Sphere

Fig. 9.4 A limit case of ovoids is a sphere (thick line) providing anastigmatic conjugates. This sphere, obtained for a 0 and which center C is located at PC P'C n2, satisfies the all-order correction of spherical aberration and coma, i.e. the Abbe sine condition Fig. 9.4 A limit case of ovoids is a sphere (thick line) providing anastigmatic conjugates. This sphere, obtained for a 0 and which center C is located at PC P'C n2, satisfies the all-order correction of spherical aberration and coma,...

## A2 P2 YXX2

One may notice that, in practice, the available cone deformation mode from elasticity remains necessarily small. Telescope grazing incidence mirrors are, at least, always optically surfaced from a passive cone geometry. 10.2.4 Radial Thickness Distributions and 4th-Degree Flexure We consider various cases of obtaining a fourth-degree flexure where a uniform load is applied all over the tubular surface. Uniform load in reaction with simply supported ends Similarly as in the latter Section, we...

## Atmospheric Dispersion Compensation

Fig. 1.37 Solid-folded (Left) and semi-solid (Right) Schmidt spectrograph cameras If a concave back-surface mirror has a shorter radius on the front than on the back, it is called a Mangin mirror. Given a wavelength, the spherical aberration caused when light passes through the front is balanced by that caused by the reflection at the spherical back. Mangin mirrors are used in some spectrographs as the camera mirror. If the glass is made thick, a very fast f-ratio can be designed such as solid...

## Refractive Telescopes

Galileo Galilei (1564-1642) heard from France, in 1609, that Lippershey in Holland had constructed a sort of enlarging monocular. This device, made of a single lens of positive power at the first end of a tube and of a negative lens on a sliding tube, was in fact a chance arrangement of eyeglass lenses available on the market which thus may only magnify distant objects by two or three times. It must be considered as a poor half part of our ancient opera glasses, and was totally useless for...

## Astronomy And Astrophysics Library

Burton, Charlottesville, VA, USA and Leiden, The Netherlands M. A. Dopita, Canberra, Australia T. Encrenaz, Meudon, France E. K. Grebel, Heidelberg, Germany B. Leibundgut, Garching, Germany A. Maeder, Sauverny, Switzerland V. Trimble, College Park, MD, and Irvine, CA, USA Artist view of the Thirty Meter Telescope project (Tmt) based on a three-mirror design (Courtesy of the Tmt Observatory Corporation) Artist view of the Thirty Meter Telescope project (Tmt) based on a three-mirror design...

## Schmidt Correctors and Diffraction Gratings Aspherized by Active Optics

5.1 Various Types of Aspherical Schmidt Correctors Active optics methods were originally applied to the figuring of refractive Schmidt correctors. Further developments followed with the development of reflective correctors and reflective-diffractive correctors. The various ways to obtain each of the three corrector types are different. 1. The methods developed for refractive plate correctors have been called elastic relaxation figuring or more commonly stress figuring. 2. Two different methods...

## Active Optics with Multimode Deformable Mirrors MDM Vase and Meniscus Forms

7.1 Introduction - Clebsch-Seidel Deformation Modes This chapter treats the general case of obtaining aspherical mirrors such as off-axis mirrors locally defined from an axisymmetric shape, or on-axis mirrors for aberration correction of a non-centered system. This requires the simultaneous correction of several wavefront modes such as 1st-order curvature, 3rd-order coma, astigmatism and spherical aberration, 5th-order triangle, etc. Although a mirror belonging to the variable thickness...

## X

Fig. 8.20 Mirror thickness geometries T(p) or T(p, 6) for atmospheric field stabilization holed mirrors. (A) Moderately lightweight axisymmetric mirror with conical edges optimized for a minimal axisymmetric flexure - as part of a torus surface - with a ring support. (B) Increased lightweight three-fold symmetry mirror with linear prismatic edges From this statement, one can derive thickness distributions for minimal flexures in kg gravity cases such as for holed field stabilization mirrors....

## Conicoids

The conicoids are rotational symmetry aspherics whose meridian sections belong to the conic family. These shapes are represented by (1 + k) z2 - 2Rz + r2 0, (1.37) where R 1 c is the radius of curvature and k the conic constant, also called the Schwarzschild constant, which characterizes the asphericity since k 0 generates a sphere. This equation may be written whose first terms of the expansion are 1 2 1 + k 4 (1 + k)2 6 5(1 + k)3 8 z r2 + 77- r4 + v ' r6 + v ' r8 + (1.38b) 2 R 8 R3 16 R5 128...

## Q

Which links the mirror thickness t (r) to the central force F40. For instance, the all-reflective Schmidt telescope Faust at f 1.5 has been built for an ultraviolet space survey of extended objects. A stainless steel Fe87Cr13 primary of 18 cm clear aperture has been built and polished flat at rest. The thickness distribution was dimensioned by to 5.6 mm. Its edge was simply supported by a radially thin cylindrical collar linked to a rigid ring of thicknesses tz 25 mm and tr 14 mm....

## Q 2 oi

M -- x2J + Fw + M0, x2 < 1 4. (1.145) After substitution, the differential equation of the bending is d2w F q (i2 , M0 J 2--w - x2 + . (1.146) Loaded plate without edge force in x-direction (F 0) Let us consider a uninform load q applied all over the plate and bending moments M0 applied to the long edges. If no force acts in the x-direction at the long edges, then the first corresponding boundary at those edges is a freedom to move in the x-direction. Various boundary cases are shown in Fig....

## Dcr

Where F na q and the products qA40 and FA40 are negative. VTD 3 - Uniform load and free edge The input of A1, A2, and A3 in the load (3.19c) provides A3 q 64A40 and after substituting the Ai in (3.24b), the force Vr is If the second term on the right is considered as qa2 2r or F 2nr with F na2q, the expression of Vr represents the net shearing forces generated by a uniform load q reacting with a central force F which corresponds to a free edge condition Vr a 0 so, retaining this case, we set A2...

## Descartes Ovoids

Fig. 9.3 Descartes ovoids for various values of k b a. These curves provide stigmatic lenses at all orders. The thick curve is the Pascal limacon kn 1 associated with the refractive index n Fig. 9.3 Descartes ovoids for various values of k b a. These curves provide stigmatic lenses at all orders. The thick curve is the Pascal limacon kn 1 associated with the refractive index n 9.1.3 Aplanatic and Anastigmatic Singlet Lenses A special case of a singlet lens with spherical surfaces is when one of...

## Non Axisymmetric Surfaces and Zernike Polynomials

Non-axisymmetric optical surfaces are used for non-centered systems. Such surfaces may also represent a wavefront shape including some of the optical aberrations. Taking the origin of the azimuth angle 9 in the z, x plane by setting x r cos 9, y r sin 9, the general shape may be represented by Z XZn,m X (An,mrn cosmQ + Bn,mrn sinmQ), (1.40a) with n, m positive integers, n + m even, m < n and where An,m, Bn,m are coefficients. A particular class of non-centered systems presents a symmetry...

## Own Weight Flexure and Figure Control of Telescope Mirrors

8.1 Primary Mirror Support Systems Against Gravity 8.1.1 Introduction Foucault solved the problem of maintaining a mirror with a high reflective coating by introducing glass mirrors, which then can be easily re-silvered after chemical removal of the tarnished coating. This ended the era of speculum metal mirrors which required, when tarnished, a repolishing within a seeing limited - or diffraction limited - criterion. The chemical process was later replaced by the vacuum deposition process by...

## Theory of Shells and Aspherization of Axisymmetric Mirrors Meniscus Vase and Closed Forms

6.1 Active Optics Aspherization of Fast f-Ratio Mirrors We consider in this Chapter axisymmetric mirror substrates characterized by highly curved surfaces i.e. surfaces with a significant cambrure. The elasticity theory of thin plates basically assumes a plane middle surface. Thus, its validity field remains limited to mirrors of moderate f-ratio. For mirrors faster than f 4 or f 3, the theory of shallow spherical shells, in French coques surbaissees, allows taking into account the significant...

## Z Z f f

1 Introduction to Optics and Elasticity x Medium ( ') 1 Introduction to Optics and Elasticity x Medium ( ') Fig. 1.16 Conjugates and transverse magnification Fig. 1.16 Conjugates and transverse magnification The conjugate distances from the vertex diopter and those from the location of the focii are related by

## Aspherization of Concave Spheroid Mirrors

The active optics aspherization of a concave spheroid mirror requires some conditions relative to the sign and amplitude of the consecutive polynomial terms which represent the mirror figure. In certain cases, it is clear that those conditions entail that the active optics co-addition law (6.59) cannot provide a spherical surface for which an elastic flexure exists. Table 6.5 Normal thickness distribution f,, for in situ parabolization of an f 1.75 holed vase shell. Mirror clear aperture 186...

## Gka

Because of the smoothly decreasing profile for low values of r and vertical tangents at the substrate edge r a, we call this thickness distribution a cycloid-like form (Fig. 2.2). Conclusion for VTD solution Type 1 Variable curvature mirrors are obtained by uniform loading q and reaction at the edge provided a cycloid-like thickness t T20 to is such as

## The Persian Mathematicians and Mirrors

After the end of the Alexandrine superiority in 641, a scientific renaissance took place in Persia - mainly in Baghdad - where most of the main Greek works were translated into Arabic during the following centuries (Rouse Ball 9 ). Some advances in algebra had already been assimilated by the Persians along with mathematical developments previously achieved in India, such as decimal numbering and the important invention by Bramaguptas in 629 of the 0 symbol for zero. In 820 Al-Khwarizmi...

## Optical Design with the Schmidt Concept Telescopes and Spectrographs

4.1.1 The Class of Two-Mirror Anastigmatic Telescopes The basic principle of the wide-field telescope invented by the Estonian optician and astronomer Bernhard Schmidt in 1928 ( 67-70 , E. Schmidt 72 ), is that a single concave and spherical mirror used with a pupil stop at its center of curvature has no unique axis and therefore yields equal size images at all points of its field of view. In the third-order theory, the mounting is free from Coma 3 and astigmatism Astm 3 all images have the...

## Fermats Principle

In 1657, in a letter to Cureau 73 , Pierre de Fermat stated that the law of refraction might be deduced from a minimum principle, and this similarly as the remark by Heron of Alexandria (100 AD) for the reflection on a plane. In 1662, considering optical paths between two points in different media, he proved that a ray follows the optical path which takes the least time since this also entails the direct recovering of Snell's law. In all generality, Fermat's principle of least time may be...

## Kelvin Functions

Denoting x r i, the four yn(x) functions in (6.15) are Kelvin functions. These are real and imaginary parts of zero-order Bessel functions I0( ix) and K0( fix), defined from Dwight 5 and Abramovitz & Stegun 1 by I0(Vix) ber x + i bei x, Ko(Vix) ker x + i kei x. The Kelvin functions are represented by the following series b 1 _ (x 2)4 + (x 2)8 ber x 1 (2 )2 + (4 )2 kerx - ln ( ) + 7 berx + beix V1 + 2) (2 )2 + + 2 + 3 + 4) (4 )2 keix - ln ( ) + 7 beix - f berx where y is the Euler constant y...

## Slitless Spectrograph

If a dispersive element is located before the focal plane of a telescope, the processor is called a slitless spectrograph. Objective grating If a concave reflective grating is substituted to the primary mirror of a small telescope, one obtains the basic form of slitless spectroscopy with an objective grating. For an object at infinity and given a wavelength, the design with a paraboloid reflective grating working at a normal diffraction angle is rigourously stigmatic on the axis similarly a...

## V2 1 V2

Various t2 ti, v, b a solution sets for a vase-form geometry satisfying 5.22c are given in Table 5.3. The uniform load q is applied to the inner part of the vase form and its reaction arises at the contour r rm a. For a best validity of this CTD, the junction between the two thicknesses t1 and t2 is realized with a radius of curvature equal to thickness t1 Fig. 5.11 see also Fig. 7.9-Up . Spherical aberration corrections including Sphe 5, Sphe 7 and higher-order terms can be achieved with a...

## Displacement Vector and Strain Tensor

Let us consider any point of a solid whose coordinate is x, y, z when in state without stress, and denote r its position vector from the origin. When the solid is deformed by external forces, or internal forces such as those exerted by a gravity field, the position vector becomes r'. The displacement vector u representing the flexure is where its cartesian components ux x,y,z , uy x,y,z and uz x,y,z may be denoted ui, If dl is the length separating two adjacent points, dl2 dx2 dy2 dz2 dx2, this...

## Pupils and Principal Rays

It is useful to define an aperture area where all the field beams enter into an optical system this is called the entrance pupil, or input pupil. In the next medium we obtain an image pupil. At the system output medium we obtain the exit pupil, or output pupil, which is the system conjugate of the entrance pupil. An aperture stop physically defines the beam area in which all beams are allowed to pass, then defining all the pupils of the system. A field stop physically defines the linear size of...

## Human

If the processor is a human eye, objects laying over a large field can be correctly imaged for distances varying from infinity topunctumproximum 25 cm . The optics standard parameters of the human eye generally assume that the low gradient index of the lens may be approximated by a constant index. Le Grand 70 proposed nd 1.44 or 1.45 for this index we have found that raytraces are more realistic with nd 1.45 for the lens Table 1.7 . Table 1.7 Optical parameters of the standard human eye working...

## D

Where Q are unknown constants that are determined from the boundaries. The terms r2 ln r and ln r allow treating cases with a ring force or a central force, and holed plates. Starting from the stress-relations 1.124b , and similarly as for rectangular plates, we derive the three stress components that differ from zero, namely orr, att and azr. This allows determining the bending and twisting moments, per unit length, as f d2w v dw d2w 1 dw Mr D I r 2 --T gt Mt D v-y - , Mrt 0. 1.181 dr2 r dr...

## Equivalence of the Etendue and Lagrange Invariants

Assuming an optical system with both ends in the same medium n' n such as in air, an equivalence between the Lagrange invariant and the Etendue can be derived from the above relations. If the entrance pupil is a square aperture, then A 4x2, and if the field is a square solid angle, O 4 ax, then we obtain from 1.36a and 1.34 the equivalence If the entrance pupil and field of view are circular, then A nx2 and O 2n 1 -cos pmax . If the field angle is small then O K p nax and we obtain similarly...

## Lens in Air or in Vacuum

Let ci 1 Ri, c2 1 R2 and d be the curvatures and axial thickness of a lens with refractive index n. From 1.20a to 1.21 we obtain K1 n 1 R1 and K2 n 1 R2, hence by substitution in the above equation, the power of a lens in air is K 77 7 n-1 hr - 1.28a However, the refractive index of a vacuum is approximatively na 1.0003 under normal conditions of pressure and temperature. But it is a universal practice in optical design to specify the refractive indices of glasses with respect to air, therefore...

## Etendue Invariant

Let us consider a source of light - or focal plane - whose area A uniformly emits radiations of intensity Iv within a frequency band 5v. Assuming that the emitting surface is centered and perpendicular to the axis of an optical system, the total amount of energy per second W reaching the entrance pupil of area A, located at distance d from the source, is represented by the inverse square law This relationship is well known in spectroscopy and photometry Pecker and Schatzman 120 , Sterken and...

## Etendue Invariant and Lagrange Invariant 161 Lagrange Invariant

In general, an optical surface introduces aberrations at the focused image which deteriorate its sharpness. At first, this alteration of image quality is due to primary aberrations cf. Sect. 1.8 and their analytic calculation require using the Lagrange invariant. Summarizing the results in the previous section, the two forms of the Lagrange invariant H are In ray tracing analysis some other authors, such as for instance Welford 167 and Wilson 170 , define the Lagrange invariant as the opposite...

## Preface

Astronomical Optics and Elasticity Theory is intended to serve both as a text and as a basic reference on active optics methods. Mainly elaborated for astronomy, and following a conceptual idea originated by Bernhard Schmidt, the first developments of active optics began in the 1960s. These methods allow one to transform by a highly continuous process a spherical surface into the desired aspherical surface, as well as to correct tilt and decentering errors between telescope mirrors, to control...

## Contents

1 Introduction to Optics and 1.1 Optics and Telescopes - Historical 1.1.1 The Greek Mathematicians and 1.1.2 The Persian Mathematicians and 1.1.3 End of European Renaissance and Birth of Telescopes 5 1.1.4 Refractive Telescopes 1.1.5 Reflective Telescopes 1.2 Snell's Law and Glass 1.3 Fermat's 1.4 Gaussian Optics and Conjugate 1.4.1 Diopter of Curvature c 1.4.2 Mirror in Medium 1.4.3 Power of Combined 1.4.4 Lens in Air or in 1.4.5 Afocal Systems 1.4.6 Pupils and Principal 1.4.7 Aperture Ratio...

## Lagrange Invariant

A fundamental law of Gaussian optics can be derived from the properties of A, B points and their conjugates A',B' Fig. 1.16 . This law which links the transverse magnification M n' n to the aperture angle ratio u u', is the Lagrange invariant also known as Smith, Lagrange, or Helmholtz relationship. Let us consider the case of a single diopter, Eq. 1.25a may be written Now from the object point A, the ray of aperture angle u meets the x,y plane at the height x uz and goes through the image...

## In

C, 1.636 c3 -2.200 c. -0.741 c, 2.175 c3 3.663 c. -0.205 Fig. 1.5 Doublet-lens objectives achromatized for an object at infinity in the spectral range Xc 486 Xf 656 nm , the blue and red hydrogen lines, and Xd 587nm, the yellow helium line. Optimizations with Kerber's condition of focus defined from ray height at y 3 2 on the first surface, the entrance pupil. Effective focal length f 1. Focal-ratio f 16. The curvatures are exaggerated on the drawings. Left Clairaut's algebraic conditions, like...

## Reflective Telescopes

Nicolas Zucchi 132, 170 made the first attempt to build a reflecting telescope in 1616, i.e. soon after Galileo developed the refractor. He states that he procured a bronze concave mirror executed by an experienced and careful artist in the trade and used it directly with a Galilean eyepiece. In order to avoid obstruction by the observer's head, his design introduced a significant beam deviation at the mirror, similarly to the front-view type later introduced by W. Herschel. Depending on the...