Fundamentals of Modern Optics

Transcription

1 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx 1 Fundamentals of Modern Optics Winter Term 2014/2015 Prof. Thomas Pertsch Abbe School of Photonics Friedrich-Schiller-Universität Jena Table of content 0. Introduction Ray optics - geometrical optics (covered by lecture Introduction to Optical Modeling) Introduction Postulates Simple rules for propagation of light Simple optical components Ray tracing in inhomogeneous media (graded-index - GRIN optics) Ray equation The eikonal equation Matrix optics The ray-transfer-matrix Matrices of optical elements Cascaded elements Optical fields in dispersive and isotropic media Maxwell s equations Adaption to optics Temporal dependence of the fields Maxwell s equations in Fourier domain From Maxwell s equations to the wave equation Decoupling of the vectorial wave equation Optical properties of matter Basics Dielectric polarization and susceptibility Conductive current and conductivity The generalized complex dielectric function Material models in time domain The Poynting vector and energy balance Time averaged Poynting vector Time averaged energy balance Normal modes in homogeneous isotropic media Transversal waves Longitudinal waves Plane wave solutions in different frequency regimes Time averaged Poynting vector of plane waves Beams and pulses - analogy of diffraction and dispersion Diffraction of monochromatic beams in homogeneous isotropic media Propagation of Gaussian beams Gaussian optics Gaussian modes in a resonator Pulse propagation... 87

2 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx (The Kramers-Kronig relation, covered by lecture Structure of Matter) Diffraction theory Interaction with plane masks Propagation using different approximations The general case - small aperture Fresnel approximation (paraxial approximation) Paraxial Fraunhofer approximation (far field approximation) Non-paraxial Fraunhofer approximation Fraunhofer diffraction at plane masks (paraxial) Fraunhofer diffraction pattern Remarks on Fresnel diffraction Fourier optics - optical filtering Imaging of arbitrary optical field with thin lens Transfer function of a thin lens Optical imaging Optical filtering and image processing The 4f-setup Examples of aperture functions Optical resolution The polarization of electromagnetic waves Introduction Polarization of normal modes in isotropic media Polarization states Principles of optics in crystals Susceptibility and dielectric tensor The optical classification of crystals The index ellipsoid Normal modes in anisotropic media Normal modes propagating in principal directions Normal modes for arbitrary propagation direction Normal surfaces of normal modes Special case: uniaxial crystals Optical fields in isotropic, dispersive and piecewise homogeneous media Basics Definition of the problem Decoupling of the vectorial wave equation Interfaces and symmetries Transition conditions Fields in a layer system matrix method Fields in one homogeneous layer The fields in a system of layers Reflection transmission problem for layer systems General layer systems Single interface Periodic multi-layer systems - Bragg-mirrors - 1D photonic crystals Fabry-Perot-resonators Guided waves in layer systems Field structure of guided waves Dispersion relation for guided waves Guided waves at interface - surface polariton Guided waves in a layer film waveguide how to excite guided waves

3 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx 3 8. Statistical optics - coherence theory Basics Statistical properties of light Interference of partially coherent light

4 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx 4 0. Introduction 'optique' (Greek) lore of light 'what is light'? Is light a wave or a particle (photon)? D.J. Lovell, Optical Anecdotes Light is the origin and requirement for life photosynthesis 90% of information we get is visual A) Origin of light atomic system determines properties of light (e.g. statistics, frequency, line width) optical system other properties of light (e.g. intensity, duration, ) invention of laser in 1958 very important development Schawlow and Townes, Phys. Rev. (1958). laser artificial light source with new and unmatched properties (e.g. coherent, directed, focused, monochromatic) applications of laser: fiber-communication, DVD, surgery, microscopy, material processing,...

5 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx 5 Fiber laser: Limpert, Tünnermann, IAP Jena, ~10kW CW (world record) B) Propagation of light through matter light-matter interaction effect dispersion diffraction absorption scattering governed by frequency spatial center of wavelength spectrum frequency frequency spectrum matter is the medium of propagation the properties of the medium (natural or artificial) determine the propagation of light light is the means to study the matter (spectroscopy) measurement methods (interferometer) design media with desired properties: glasses, polymers, semiconductors, compounded media (effective media, photonic crystals, meta-materials) Two-dimensional photonic crystal membrane.

6 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx 6 C) Light can modify matter light induces physical, chemical and biological processes used for lithography, material processing, or modification of biological objects (bio-photonics) Hole drilled with a fs laser at Institute of Applied Physics, FSU Jena.

7 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx 7 D) Optics in our daily life A small story describing the importance of light for everyday life, where all things which rely on optics are marked in red.

8 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx 8 E) Optics in telecommunications transmitting data (Terabit/s in one fiber) over transatlantic distances 1000 m telecommunication fiber is installed every second.

9 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx 9 F) Optics in medicine, life sciences

10 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx 10 G) Optical sensors and light sources new light sources to reduce energy consumption new projection techniques Deutscher Zukunftspreis IOF Jena + OSRAM

11 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx 11 H) Micro- and nano-optics ultra small camera Insect inspired camera system develop at Fraunhofer Institute IOF Jena

12 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx 12 I) Relativistic optics

13 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx 13 K) What is light? electromagnetic wave propagating with speed of c= 3*10 8 m/s leading to evolution of: amplitude and phase complex description polarization vectorial field description coherence statistical description Spectrum of Electromagnetic Radiation Region Wavelength (nanometers) Wavelength (centimeters) Frequency (Hz) Energy (ev) Radio > 10 8 > 10 < 3 x 10 9 < 10-5 Microwave x x Infrared x x x Visible x x x x Ultraviolet x x x X-Rays x x Gamma Rays < 0.01 < 10-9 > 3 x > 10 5

14 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx 14

15 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx 15 L) Schematic of optics quantum optics electromagnetic optics wave optics geometrical optics geometrical optics λ << size of objects daily experiences optical instruments, optical imaging intensity, direction, coherence, phase, polarization, photons wave optics λ size of objects interference, diffraction, dispersion, coherence laser, holography, resolution, pulse propagation intensity, direction, coherence, phase, polarization, photons electromagnetic optics reflection, transmission, guided waves, resonators laser, integrated optics, photonic crystals, Bragg mirrors... intensity, direction, coherence, phase, polarization, photons quantum optics small number of photons, fluctuations, light-matter interaction intensity, direction, coherence, phase, polarization, photons in this lecture electromagnetic optics and wave optics no quantum optics subject of advanced lecture

16 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx 16 M) Literature Fundamental 1. Saleh, Teich, 'Fundamenals of Photonics', Wiley, Mansuripur, 'Classical Optics and its Applications', Cambridge, Hecht, 'Optik', Oldenbourg, Menzel, 'Photonics', Springer, Lipson, Lipson, Tannhäuser, 'Optik'; Springer, Born, Wolf, 'Principles of Optics', Pergamon 7. Sommerfeld, 'Optik' Advanced 1. W. Silvast, 'Laser Fundamentals', 2. Agrawal, 'Fiber-Optic Communication Systems', Wiley 3. Band, 'Light and Matter', Wiley, Karthe, Müller, 'Integrierte Optik', Teubner 5. Diels, Rudolph, 'Ultrashort Laser Pulse Phenomena', Academic 6. Yariv, 'Optical Electronics in modern Communications', Oxford 7. Snyder, Love, 'Optical Waveguide Theory', Chapman&Hall 8. Römer, 'Theoretical Optics', Wiley,2005.

17 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx Ray optics - geometrical optics (covered by lecture Introduction to Optical Modeling) The topic of Ray optics geometrical optics is not covered in the course Fundamentals of modern optics. This topic will be covered rather by the course Introduction to optical modeling. The following part of the script which is devoted to this topic is just included in the script for consistency. 1.1 Introduction Ray optics or geometrical optics is the simplest theory for doing optics. In this theory, propagation of light in various optical media can be described by simple geometrical rules. Ray optics is based on a very rough approximation (λ 0, no wave phenomena), but we can explain almost all daily life experiences involving light (shadows, mirrors, etc.). In particular, we can describe optical imaging with ray optics approach. In isotropic media, the direction of rays corresponds to the direction of energy flow. What is covered in this chapter? It gives fundamental postulates of the theory. It derives simple rules for propagation of light (rays). It introduces simple optical components. It introduces light propagation in inhomogeneous media (graded-index (GRIN) optics). It introduces paraxial matrix optics. 1.2 Postulates A) Light propagates as rays. Those rays are emitted by light-sources and are observable by optical detectors. B) The optical medium is characterized by a function n(r), the so-called refractive index (n(r) 1 - meta-materials n(r) <0) c n = c C) optical path length delay i) homogeneous media nl ii) inhomogeneous media B n n() r ds A c n speed of light in the medium

18 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx 18 D) Fermat s principle B d n() r ds = 0 A Rays of light choose the optical path with the shortest delay. 1.3 Simple rules for propagation of light A) Homogeneous media n = const. minimum delay = minimum distance Rays of light propagate on straight lines. B) Reflection by a mirror (metal, dielectric coating) The reflected ray lies in the plane of incidence. The angle of reflection equals the angle of incidence. C) Reflection and refraction by an interface Incident ray reflected ray plus refracted ray The reflected ray obeys b). The refracted ray lies in the plane of incidence. The angle of refraction θ 2 depends on the angle of incidence θ 1 and is given by Snell s law: n sin θ = n sin θ no information about amplitude ratio. 1.4 Simple optical components A) Mirror i) Planar mirror Rays originating from P 1 are reflected and seem to originate from P 2.

19 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx 19 ii) Parabolic mirror Parallel rays converge in the focal point (focal length f). Applications: Telescope, collimator iii) Elliptic mirror Rays originating from focal point P 1 converge in the second focal point P 2 iv) Spherical mirror Neither imaging like elliptical mirror nor focusing like parabolic mirror parallel rays cross the optical axis at different points connecting line of intersections of rays caustic

20 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx 20 parallel, paraxial rays converge to the focal point f = (-R)/2 convention: R < 0 - concave mirror; R > 0 - convex mirror. for paraxial rays the spherical mirror acts as a focusing as well as an imaging optical element. paraxial rays emitted in point P1 are reflected and converge in point P (imaging formula) z z ( R) 1 2 paraxial imaging: imaging formula and magnification m = -z2 /z1 (proof given in exercises) B) Planar interface Snell s law: n 1 sin θ 1 = n 2 sin θ 2 for paraxial rays: n1θ 1= n2θ2 external reflection ( n 1 < n 2 ): ray refracted away from the interface internal reflection ( n 1 > n 2 ): ray refracted towards the interface total internal reflection (TIR) for: π n2 θ 2 = sin θ 1 = sin θ TIR = 2 n 1

21 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx 21 C) Spherical interface (paraxial) paraxial imaging n n n y θ θ (*) n n R n1 n2 n2 n1 + (imaging formula) z z R m 1 2 n z n z 1 2 = (magnification) 2 1 (Proof: exercise) if paraxiality is violated aberration rays coming from one point of the object do not intersect in one point of the image (caustic) D) Spherical thin lense (paraxial)

22 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx 22 two spherical interfaces (R 1, R 2, ) apply (*) two times and assume y=const ( small) 2 1 y f = f R1 R2 θ θ with focal length: ( n 1) z2 + (imaging formula) m = (magnification) z z f z 1 2 (compare to spherical mirror) 1.5 Ray tracing in inhomogeneous media (graded-index - GRIN optics) n() r - continuous function, fabricated by, e.g., doping curved trajectories graded-index layer can act as, e.g., a lens Ray equation Starting point: we minimize the optical path or the delay (Fermat) 1 computation: B d n() r ds = 0 A B A ( ) L = n r s ds

23 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx 23 variation of the path: r() s +δr () s B d L = d nds + ndds A B A d n= grad n dr ( r r) ( r) 2 2 d ds = d + dd d ( dr) 2dr d r ( d r) ( dr) = + d + d dr ddr ds 1+ 2 ds ds ds dr ddr ds 1+ ds ds ds dr ddr = ds ds ds B dr ddr d L = grad n d r + n ds ds ds A integration by parts and A,B fix B d dr = grad n n d ds ds ds r δ L = 0 for arbitrary variation Possible solutions: A) trajectory A d dr grad n= n ray equation ds ds 2 2 x(z), y(z) and ds = dz 1+ ( dx dz) + ( dy dz) solve for x(z), y(z) paraxial rays (ds dz )

24 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx 24 d dx dn nxyz (,, ) dz dz dx d dy dn nxyz (,, ) dz dz dy B) homogeneous media straight lines C) graded-index layer n(y) - paraxial, SELFOC for n(y)-n 0 <<1: dy paraxial 1 and dz dz ds n( y) = n0 ( 1 α y) ny ( ) n 0 1 y 2 α 2 for a d dy d dy d y d y dn y n( y) n( y) n( y) 2 2 ds ds dz dz = dz dz n y dy 2 d y dz 2 2 = α y θ0 yz ( ) = y0 cosα z+ sin αz α 1 ( ) ( ) dy θ ( z) = = y0αsin α z+θ0cosαz dz The eikonal equation bridge between geometrical optics and wave eikonal S(r) = constant planes perpendicular to rays from S(r) we can determine direction of rays grad S(r) (like potential) ( ) = n( ) 2 2 grads r r Remark: it is possible to derive Fermat s principle from eikonal equation geometrical optics: Fermat s or eikonal equation ( r B B ) ( r ) = grad ( r ) = ( r ) S S S ds n ds B A eikonal optical path length phase of the wave A A

25 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx Matrix optics technique for paraxial ray tracing through optical systems propagation in a single plane only rays are characterized by the distance to the optical axis (y) and their inclination (θ) two algebraic equation 2 x 2 matrix Advantage: we can trace a ray through an optical system of many elements by multiplication of matrices The ray-transfer-matrix in paraxial approximation: y = Ay + Bθ θ = Cy + Dθ A=0: same θ 1 same y 2 focusing D=0: same y1 same θ 2 collimation y A B y A B = C D M = C D 2 1 θ2 θ Matrices of optical elements A) free space 1 d M = 0 1 B) refraction on planar interface

26 Script Fundamentals of Modern Optics, FSU Jena, Prof. T. Pertsch, FoMO_Script_ s.docx M = 0 n1 n 2 C) refraction on spherical interface D) thin lens 1 0 M = ( n2 n1) nr 2 n1 n M = 1 f 1 E) reflection on planar mirror 1 0 M = 0 1 F) reflection on spherical mirror (compare to lens) 1 0 M = 2 R Cascaded elements yn + 1 A B y1 A B = = N + 1 C D M θ θ 1 C D M=M N.M 2 M