Introduction to Optics

Transcription

1 Introduction to Optics Dunlap Institute Instrumentation Summer School 2016 Paul Hickson vista.ac.uk

2 Outline What is optics? Branches of optics and a very-brief history Geometrical optics Fermat s principle, optical path length Reflection and refraction at surfaces Gaussian optics - imaging, magnification, optical invariants Images, pupils, thin lenses, thick lenses, mirrors Image quality - Seidel aberrations, aspheres Physical optics Huygens principle, Kirchoff s diffraction integral Fresnel and Fraunhoffer diffraction, Fourier optics Coherence - spatial and temporal Image quality - wavefront error, Strehl ratio An interesting application - liquid-mirror telescopes Dunlap Institute Summer Instrumantation School

3 What is optics? The study of the propagation and interaction of light (including other forms of electromagnetic radiation) Imaging and dispersion of light short wavelength limit is geometrical optics Wave nature of light physical optics Propagation of light through a turbulent medium atmospheric optics, adaptive optics Interaction of light and matter quantum optics, photonics Coherent states, nonlinear effects quantum optics, nonlinear optics Dunlap Institute Summer Instrumantation School

4 Some milestones in optics -700 First lenses appear in Assyria and Egypt -300 Euclid - law of reflection 984 Ibn Sahl - law of refraction (Snell s law) 1609 Galileo - first astronomical telescope 1662 Fermat - Fermat s principle 1666 Newton - dispersion of light, reflecting telescope 1678 Huygens - wave nature of light 1814 Fraunhofer - spectroscopy 1816 Young, Fresnel - interference and diffraction 1841 Gauss - imaging theory 1843 Petzval, Seidel, Abbe - aberration theory 1873 Maxwell, Heaviside - classical EM wave theory 1883 Kirchoff - diffraction theory, radiation laws Dunlap Institute Summer Instrumantation School

5 Contemporary developments 1900 Planck, Einstein, Dirac - quantum nature of light 1960 Sudarshan, Glauber, Mandel - quantum optics 19xx Michelson, Lyot, Fabry, Pérot - interferometers, coronographs 19xx Richey, Chrétien, Schmidt, Paul, Baker, Maksutov, Korsch - telescope design 1953 Babcock - concept of adaptive optics 1957 Leighton - first tip-tilt correction 1961 Rytov,Tatarskii - theory of wave propagation in turbulent media 197x Development of adaptive optics Dunlap Institute Summer Instrumantation School

6 Some references E. Hecht - Optics M. Born and E. Wolf - Principles of Optics L. Mandel and E. Wolf - Optical Coherence and Quantum Optics R. N. Wilson - Reflecting Telescope Optics I. Basic Design Theory and its Historical Development R. J. Sasiela - Electromagnetic wave propagation in turbulence: evaluation and application of Mellin transforms Dunlap Institute Summer Instrumantation School

7 Geometrical Optics K Thompson 2013, Adv Opt Techn, 2, 89 Dunlap Institute Summer Instrumantation School

8 Fermat s principle Light travels between two points along a path that requires the least time - Fermat Light travels more slowly in a medium that has a higher index of refraction n, v c{n 6 time 9 n. This is equivalent to minimizing the optical path length ż l n ds. Fermat s principle is related to the principle of least action in classical mechanics. At a deeper level, it follows from Feynman s path integral formulation of quantum mechanics, ż x f, t f x i, t i Dxptqe isrxptqs{. Max Planck Institute for Gravitational Physics Dunlap Institute Summer Instrumantation School

9 Huygens principle Wavefronts are surfaces of constant phase, perpendicular to rays. Every point on a wavefront becomes the source of spherical secondary waves. These secondary waves interfere constructively to form a new wavefront. Constructive interference requires that the change in optical path length between two wavefronts be constant, karagioza.com Dunlap Institute Summer Instrumantation School

10 Reflection and refraction at a surface By symmetry, the incident, reflected and refracted rays all lie in the same plane. The optical path difference between wavefronts must be the same in each medium, so n 1 λ 1 n 2 λ 2. From the figure, we see that the angles of incidence and refraction, measured from the normal to the interface, are related by λ 1 sin θ 1 λ 2 sin θ 2, 6 sin θ 2 sin θ 1 λ 2 λ 1 n 1 n 2, which gives us Snell s law, n 1 sin θ 1 n 2 sin θ 2. Dunlap Institute Summer Instrumantation School

11 Reflection and refraction at a surface The reflected and transmitted amplitudes (for non-magnetic media) are given by the Fresnel equations. p-polarized (E vector parallel to plane of incidence): r p n 2 cos θ i n 1 cos θ t n 2 cos θ i ` n 1 cos θ t 2n 1 cos θ t t p n 2 cos θ i ` n 1 cos θ t s-polarized (E vector perpendicular): r s n 1 cos θ i n 2 cos θ t n 1 cos θ i ` n 2 cos θ t 2n 1 cos θ i t p n 1 cos θ i ` n 2 cos θ t r 2 ` n2 cos θ t n 1 cos θ i t 2 1. The fraction of power reflected is r 2» rpn 2 n 1 q{pn 2 ` n 1 qs 2 which is about 4% for an uncoated glass surface. Dunlap Institute Summer Instrumantation School

12 Images For simplicity, we shall consider only rotationally-symmetric optical systems, which have an optical axis. A real image is formed when light rays originating at points on the object pass through corresponding points in the image. A virtual image is an image from which rays appear to originate. An object and its images are said to be optically conjugate. Dunlap Institute Summer Instrumantation School

13 Pupils An aperture or stop, and all images of it, are called pupils. The first pupil encountered by rays entering the optical system is called the entrance pupil. The last pupil encountered by rays exiting the optical system is called the exit pupil. A ray passing through the centre of a pupil is called a chief ray. A ray passing through the edge of a pupil is called a marginal ray. Dunlap Institute Summer Instrumantation School

14 First-order theory Replace sin θ with θ in Snell s law. This is the paraxial approximation. Assume that the refraction or reflection takes place in a plane. This is the thin-lens approximation. From the figure we see that h i h o i o x f i f f from which Gauss s image formula immediately follows, 1 i ` 1 o 1 f Dunlap Institute Summer Instrumantation School

15 First-order theory Gauss s formula is easily rearranged to give Newton s formula, x i x o f 2 The transverse magnification is m h i {h o i{o and in the limit o Ñ 8, we have the image scale α h i{i h i h i 1 f radians/m. Dunlap Institute Summer Instrumantation School

16 Thick or composite lenses The principal planes (H 1, H 2 ) are planes at which parallel light, entering from either direction, appears to be bent. They intersect the optical axis at the nodal points (N 1, N 2 ). The effective focal lengths are measured from the each principal plane to the corresponding focal point. The thin-lens formulae can be used if one assumes that the principal planes are optically conjugate with a magnification of 1. Dunlap Institute Summer Instrumantation School

17 Paraxial ray tracing The principal planes of an optical system can be located in several ways. One method is that of gaussian reduction which is an algebraic techniques that iteratively reduces the system to the equivalent simple lens. An alternative is paraxial ray tracing, which follows a ray through the system, using the linear approximation for refraction and reflection. For an axisymmetric system, a ray passing through any plane perpendicular to the axis can be specified by two parameters, the distance y from the axis and the angle θ with respect to the axis. At the first surface of the system, let these parameters be y 1 and θ 1. Upon exiting the last surface, the ray has different parameters y 2 and θ 2. These parameters are related by a linear transformation. Dunlap Institute Summer Instrumantation School

18 Paraxial ray tracing We can write this relationship in matrix form by putting y and θ in a column vector and representing the optical system by a matrix, j j j y2 a b y1 c d θ 2 To find the focal length, send in a ray parallel to the axis (y 1 1, θ 1 0). Apply the system matrix and then propagate the output ray a distance x until it crosses the axis (y 2 0. This gives the condition j 0 θ 2 1 x 0 1 j a c which has the solution x a{c, θ 2 c. The focal length is f 2 y 1 {θ 2 1{c. θ 1 j j b 1 d 0 The principal plane H 2 is located at a distance f 2 x p1 aq{c in front of the last surface. (This distance may be negative, in which case the principle plane is outside the lens.) Dunlap Institute Summer Instrumantation School

19 The system matrix The matrix for an optical system can be found by applying successive transformations as the ray passes through the system. It is easy to verify that the transformations are j 1 d propagation over axial distance d 0 1 refraction at a surface reflection at a surface a thin lens 1 0 j pn 1 n 2 q{rn 2 n 1 {n 2 1 j 0 2{R 1 1 j 0 1{f 1 Here R, the radius of curvature of the surface, is positive if convex (centre of curvature is after the surface) and negative if concave. Dunlap Institute Summer Instrumantation School

20 Paraxial ray tracing For example, the matrix for a thick lens of index of refraction n, radii R 1 and R 2 and thickness t is j j j t 1 0 pn 1q{R 2 n 0 1 pn 1q{nR 1 1{n j 1 tpn 1q{nR 1 t{n pn 1qp1{R 2 1{R 1 tpn 1q{nR 1 R 2 q 1 ` tpn 1q{nR 2 Applying the previous results, we find that the focal length is nr 1 R 2 f 2 pn 1qrtpn 1q ` npr 2 R 1 qs, and that the principal plane H 2 occurs at a distance f 2 tpn 1q{nR 1 before the last surface. Dunlap Institute Summer Instrumantation School

21 Focal ratio and numerical aperture The focal ratio (f-number) N of a telescope is the ratio of the effective focal length f to the diameter D of the entrance pupil, N f {D The numerical aperture is defined by NA n sin θ where n is the index of refraction of the medium containing the focal plane and θ is the angle made at the focal plane by a marginal ray. We see that if n 1 then NA» 1{2N. Dunlap Institute Summer Instrumantation School

22 Magnification formulae For a general axisymmetric optical system, the transverse magnification is given by m dy 2 dy 1 θ 1 θ 2. The longitudinal magnification is given by m L dz 2 dz 1 m 2. Dunlap Institute Summer Instrumantation School

23 The Lagrange invariant At any pupil, the angle subtended by rays passing through the pupil is inversely proportional to the pupil diameter, D 1 θ 1 D 2 θ 2 Another way to say this is that the product AΩ is constant, where A is the area of the pupil and Ω is the solid angle of the rays. The Lagrange invariant is often a limiting factor in the design of optical systems, as the following example shows. Dunlap Institute Summer Instrumantation School

24 The Lagrange invariant - an example Consider a 30-metre telescope with a field of view of 10 arcminutes. We want to equip it with a multi-object spectrograph that can cover the entire field. Suppose that for best efficiency, the rays must strike the spectrograph grating, which is located at a pupil, within 5 degrees of the optimal angle. How large must the grating be? The Lagrange invariant is L 30 ˆ 5{ m-degrees. The grating is located at a pupil, so L must be equal to the grating diameter d times 5 degrees. Hence 5d 2.5. so we need a grating that is 0.5 metres in diameter. Dunlap Institute Summer Instrumantation School

25 Third-order theory - optical aberrations A spherical wavefront, centred at a point P, will propagate to produce a perfect image at that point. Aberrations result if the actual wavefront deviates from this sphere. Let W pr, φ, σq, the wave aberration, represent the optical path length difference, measured in the pupil. Here σ represents the field angle of the source (the angle between the direction to the source and the optical axis). Dunlap Institute Summer Instrumantation School

26 Third-order theory - optical aberrations For an axially-symmetric optical system, W can be expanded in a series. To fourth order, the only terms allowed by symmetry are W pr, φ, σq a 0 ` a 1 r 2 ` a 2 r cos φ ` a 3 r 4 ` a 4 σr 3 cos φ ` a 5 r 2 σ 2 cos 2 φ ` a 6 σ 2 r 2 ` a 7 σ 3 r cos φ `. The term a 0 produces a constant phase shift which has no effect on the image. The terms a 1 and a 2 correspond to a focus error or tilt error (the point P is not at the focus of the wave). These terms can be removed by a correct choice of P. The remaining five terms correspond to the five Seidel aberrations. The coefficients can be calculated, for any optical system, by keeping terms to third order in Snell s law (sin θ» θ θ 3 {6). Most depend on the paraxial parameters of the system (positions and powers of surfaces) and also on any aspheric surface shapes. Dunlap Institute Summer Instrumantation School

27 Spherical aberration a 3 r 4 independent of field angle σ proportional to NA 4 can be removed by an aspheric surface (eg. a parabolic mirror) N. Mansurov telescope-optics.net Dunlap Institute Summer Instrumantation School

28 Coma a 4 σr 3 cos φ increases linearly with field angle proportional to NA 3 zero if the system satisfies the Abbe sine condition a coma-free system is aplanatic telescope-optics.net Dunlap Institute Summer Instrumantation School

29 Astigmatism a 5 σ 2 r 2 cos 2 φ Proportional to square of field angle Proportional to NA 2 Focal distance depends on position angle in the pupil (φ) An astigmatism-free system is astigmatic telescope-optics.net Dunlap Institute Summer Instrumantation School

30 Field curvature a 6 σ 2 r 2 Focal surface is curved. Depends only on powers of optical surfaces via the Petzval formula wikipedia Dunlap Institute Summer Instrumantation School

31 Distortion a 7 σ 3 r cos φ Images are sharp, but displaced from their correct positions One has pincushion or barrel distortion depending on the sign of this term barrel distortion pincushion distortion Dunlap Institute Summer Instrumantation School

32 Chromatic aberration The index of refraction in glasses depends on wavelength. This is dispersion. Dispersion can be compensated at particular wavelengths by combining different types of glass in the optical design. An example is a achromatic doublet often used as the objective lens in small telescopes. Dunlap Institute Summer Instrumantation School

33 Abbe sine condition An optical system will be free from coma if y R sin θ Where R is a constant, the radius of the Abbe sphere. Dunlap Institute Summer Instrumantation School

34 Petzval formula The radius of curvature R f of the focal surface is given by 1 ÿ p1{n i q n f R f R i i Where R i is the radius of the i-th optical surface and p1{n i q 1{n i`1 1{n i is the change in the reciprocal of the refractive index, across the surface. For reflecting surfaces, 1 R f 2 ÿ i 1 R i. Here, R i is positive for a convex surface and negative for a concave surface, as seen by an observer traveling with the light. Dunlap Institute Summer Instrumantation School

35 Correcting aberrations It is generally possible to correct one aberration for each optical surface. A parabolic mirror corrects spherical aberration only. A Cassegrain (two-mirror) telescope corrects only spherical aberration, but, A Richey-Chrétien (two-mirror) telescope corrects both spherical aberration and coma. A Baker-Paul (three-mirror) telescope corrects spherical aberration, coma and astigmatism, and in some cases field curvature too. A Schmidt telescope (spherical mirror + thin lens) corrects all Seidel aberrations except field curvature. Dunlap Institute Summer Instrumantation School

36 Diffraction Dunlap Institute Summer Instrumantation School

37 Kirchoff s integral theorem Start with the wave equation for a complex scalar amplitude Ψpx, tq in vacuum, ˆ 2 1 B 2 c 2 Bt 2 Ψpx, tq 0. The substitution Ψpx, tq Upxqe iωt gives the Helmholtz equation ` 2 ` k 2 Upxq 0. where k ω{c 2π{λ is the wave number. For a unit source located at a point P, which we can take as the origin of the coordinate system, we need the inhomogeneous equation ` 2 ` k 2 U 0 pxq δpxq. Its solution is the Greens function U 0 prq eikr 4πr Dunlap Institute Summer Instrumantation School

38 Kirchoff s integral theorem Combining the two Helmholtz equations we see that U 0 2 U U 2 U 0 Uδpxq. The left hand side can be written as a pure divergence, pu 0 U U U 0 q Uδpxq. Integrating this over an arbitrary volume containing P and applying Gauss s theorem gives Kirchoff s integral theorem UpPq 1 ż U B ˆeikr j eikr BU ds 4π S Bn r r Bn where the integration is over an arbitrary surface enclosing P and n is a unit vector normal to the surface. Dunlap Institute Summer Instrumantation School

39 Kirchoff s diffraction formula Apply this to an aperture illuminated by a unit point source located at a distance s from the aperture. Assuming that r and s are much larger than λ, and that U U 0 everywhere on S except in the aperture, one obtains Kirchoff s diffraction formula UpPq 1 ż e ikpr`sq rcospn, sq cospn, rqs ds. 2iλ S rs Dunlap Institute Summer Instrumantation School

40 Approximations to Kirchoff s diffraction formula Fresnel diffraction - z " w Fraunhoffer diffraction - z " πw 2 {λ where w is the aperture width. Dunlap Institute Summer Instrumantation School

41 Fraunhoffer diffraction Let Upxq be the amplitude of an incident wave at a point x in an aperture A. From Kirchoff s integral theorem, the amplitude at a point P on a sphere of radius R centred on the aperture is given by UpPq» 1 ż iλ A Upxq eikr d 2 x. r The distance r is given by the cosine rule, and can be expanded in a series r a R 2 ` x 2 2R x R R x R ` x 2 pr xq2 ` 2R 2R 3 `. The first term is just a constant phase. The second term can be written as α x, where α R{R is the angular position of P. Dunlap Institute Summer Instrumantation School

42 Fraunhoffer diffraction In the limit R Ñ 8 the quadratic and higher terms vanish and we are left with Fraunhoffer s diffraction formula, Upαq λ ir ż Upuqe 2πiα u d 2 u, A where u x{λ is the position in the aperture in units of wavelength, and α is the angular position in the far field. The integral can be recognized as a Fourier transform, so we see that: The angular distribution of the amplitude in the far-field diffraction pattern is proportional to the Fourier transform of the amplitude distribution in the aperture. The intensity is equal to the square of the amplitude, I U 2. Dunlap Institute Summer Instrumantation School

43 Fourier imaging In a telescope, the image is optically conjugate to the object, which is at infinity (practically). Therefore, the amplitude in the focal plane is related to the amplitude in the pupil by Fraunhoffer diffraction. Specifically, the amplitude in the focal plane is proportional to the Fourier transform of the amplitude in any pupil of the system. For example, if the pupil is a uniformly-illuminated disk of diameter D, Upαq 9 ż D{2λ 0 e 2πiα u d 2 u 2J 1pπDα{λq πdα{λ The square of this is the Airy profile (Airy disk) j 2 2J1 pπdα{λq I pαq 9 πdα{λ The angular diameter of this image, at half maximum intensity (FWHM), is ε 0.976λ{D. Dunlap Institute Summer Instrumantation School

44 Fresnel diffraction In the near field, the quadratic terms must be included. This gives the Fresnel approximation. In this approximation, the amplitude in a plane x parallel to the aperture and located at a distance z becomes ż Upxq» eikz Upx 1 qe ik x 1 x 2 {2z d 2 x 1. iλz A which is the formula for Fresnel diffraction. This is equivalent to convolving Upxq by the Fresnel kernel Kpxq 1 2 iλz eikx {2z, spie.org and multiplying by an overall phase factor e ikz to account for propagation in the z direction. Dunlap Institute Summer Instrumantation School

45 Fresnel diffraction Ñ Ñ timaras.com Dunlap Institute Summer Instrumantation School

46 Babinet s principle Ñ It follows from the linearity of Maxwell s equations, and Fourier transforms, that the diffracted far-field intensity pattern of an obstruction is the same as that of the complimentary aperture. For example, the diffraction produced by a spider secondary support structure is the same as that produced by a pair of crossed slits of the same length and width. This shows that the light loss due to a thin obstruction is twice that due to the geometrical area alone. Dunlap Institute Summer Instrumantation School

47 Diffraction gratings Diffraction gratings can be understood using Huygens principle. For a beam entering at angle α and observed from angle β, the condition for constructive interference is sin β sin α nλ d where d is the groove separation or pitch and n is an integer called the order. This is the grating equation. wikipedia Dunlap Institute Summer Instrumantation School

48 Coherence We are all familiar with lasers, but there are other forms of coherent light. Light will not interfere unless it is coherent. Essentially this means that there must be a well-defined phase relationship over the region of interest. Temporal coherence requires that the optical path difference between two interfering beams be less than the coherence length λ2 { λ, where λ is the range of wavelengths present. Spatial coherence requires that the transverse separation between two interfering beams be less than λ{α, where α is the angular size of the source. First light of ESO s 4LGS laser guide star system Dunlap Institute Summer Instrumantation School

49 Michelson stellar interferometer These coherence conditions were used by Michelson in 1919 to measure the angular sizes of nearby stars. caltech.edu J. Monnier dept.astro.lsa.umich.edu Dunlap Institute Summer Instrumantation School

50 Image quality A ideal telescope, observing a point source with no atmosphere, would produce an image (the point spread function or PSF) that is the Fraunhoffer diffraction pattern of the telescope aperture. For a circular aperture, this is the Airy disk. Wavefront distortion blurs the image. As a result, the central intensity I p0q of the PSF is lower than that of the ideal telescope, I 0 p0q. This is quantified by the Strehl ratio S I p0q I 0 p0q, which has a maximum possible value of 1. gatinel.com Dunlap Institute Summer Instrumantation School

51 Image quality The Strehl ratio is related to the RMS value of W px, yq, the optical path difference (also called the wavefront error) σ. If the wavefront error is not too large, (Á 0.01) the Maréchal approximation can be used to predict the Strehl ratio, S» e k2 σ 2. Galactic centre seen by old (left) and new laser AO system at Keck Observatory. The Strehl ratio is a factor of 2 higher in the right panel (simulation by TMT project) Dunlap Institute Summer Instrumantation School

52 An interesting application: liquid mirror telescopes R. Wood, 1909 E. Borra 1989 Emilio Segre Visual Archives, Am. Instiute Physics E. Borra, U. Laval Dunlap Institute Summer Instrumantation School

53 Liquid mirrors Principle: the surface of a liquid, rotating uniformly about a vertical axis in a constant gravitational field, is a paraboloid. Why? In equilibrium, the surface must be an equipotential, otherwise there would be a transverse force that would redistribute the liquid. Thus gz 1 2 ω2 r 2 const gz 0. The first term is the gravitational potential, the second is the centrifugal potential (whose gradient is the centrifugal force). Therefore, z z 0 ` ω2 2g r 2 If the liquid is reflective, it can be used to focus light. Dunlap Institute Summer Instrumantation School

54 The Large Zenith Telescope P. Hickson, UBC 6-metre Large Zenith Telescope (P. K. Chen) P. Hickson, UBC Dunlap Institute Summer Instrumantation School

55 Science with zenith-pointing telescopes As the Earth rotates, such telescopes observe a long strip of sky each night, using time-delay integration (TDI) readout. This is very efficient, as no time is wasted moving the telescope and acquiring fields. Coadding data gives deep images with a very uniform background. By subtracting data, one can detect and study variable objects: quasars and gravitational lenses supernovae microlensing high-proper motion objects low-surface brightness galaxies AGN variability variable stars serendipitous phenomena Dunlap Institute Summer Instrumantation School

56 The International Liquid Mirror Telescope Project A collaboration between Belgium, India and Canada, to build and operate a 4-meter astronomical LMT at Devethal Peak in the Indian Himalayas. ILMT project Dunlap Institute Summer Instrumantation School

57 The ILMT prime focus corrector The ILMT corrector not only removes telescope coma, astigmatism, distortion and field curvature, but by tilting and decentering the optical elements, it introduces asymmetric distortion that corrects star-trail curvature. ILMT project Dunlap Institute Summer Instrumantation School

58 Project status Work is progressing on the observatory. First light is expected in Dunlap Institute Summer Instrumantation School

59 Test your understanding 1. Derive the laws of reflection and refraction directly from Fermat s principle. 2. Show that a flat window, of thickness t, placed before the focus moves the focal point away from the telescope a distance tpn 1q{n. 3. Using first-order theory, show that the Lagrange invariant is constant for an arbitrary axisymmetric optical system. 4. Show that e ikr {4πr is a solution of the inhomogeneous Helmholtz equation (slide 37). 5. In the Fresnel approximation, show that propagating from plane A to plane B and then again from plane B to plane C is the same as propagating from plane A directly to plane C (slides 44 and 45). 6. Derive the grating equation. 7. The Earth s gravitational field is not constant, but has approximate radial symmetry, and decreases with distance from the centre of the Earth. What effect does this have on the surface of a rotating liquid mirror? What aberration(s) are introduced by this? Dunlap Institute Summer Instrumantation School