Presentation on theme: "Course outline Maxwell Eqs., EM waves, wave-packets"— Presentation transcript:
1Course outline Maxwell Eqs., EM waves, wave-packets Gaussian beamsFourier optics, the lens, resolutionGeometrical optics, Snell’s lawLight-tissue interaction: scattering, absorption Fluorescence, photo dynamic therapyFundamentals of lasersLasers in medicineBasics of light detection, camerasMicroscopy, contrast mechanismConfocal microscopyמשואות מקסוול, גלים אלקטרומגנטים, חבילות גליםקרניים גאוסיניותאופטיקת פורייה, העדשה, הפרדהאופטיקה גיאומטרית, חוק סנלאינטראקציה אור-רקמה: פיזור, בליעה, פלואורסנציה, טיפול פוטו-דינמיעקרונות לייזריםלייזרים ברפואהעקרונות גילוי אור, מצלמותמיקרוסקופיה, ניגודיותמיקרוסקופיה קונפוקלית
2Plane waves Reminder “wavelength” Solutions for the Helmholtz equation:(proof in next slide)The real electric field:is independent of rinfinite field!
3Gaussian beams – the paraxial wave Spherical wave:Plane wave:A paraxial wave is a plane wave traveling mainly along the z direction (e-ikz, with k=2π/λ), modulated by a complex envelope that is a slowly varying function of position, so that its complex amplitude is given by:‘Carrier’ plane waveSlowly varying complex amplitude (in space)
4The paraxial Helmholtz equation Substitute the paraxial wave into the Helmholtz equation:Paraxial wave
5Paraxial waveWe now assume that the variation of A(r) with z is slow enough, so that:These assumptions are equivalent to assuming thatandTransverse Laplacian:Paraxial Helmholtz equation:
6Gaussian beams z0: “Rayleigh range” Paraxial Helmholtz equation One solution to the paraxial Helmholtz equation of the slowly varying complex amplitude A, has the form:Wherez0: “Rayleigh range”andq(z) can be separated into its real and imaginary parts:Where W(z): beam widthR(z): wavefront radius of curvature
7Gaussian beams The full Gaussian beam: With beam parameters: A0 and z0 are two independent parameters which are determined from the boundary conditions. All other parameters are related to z0 and by these equations.
8Gaussian beams - properties IntensityAt any z, I is a Gaussian function of . On the beam axis:(Lorentzian)I1/2z0z- Maximum at z=0- Half peak value at z = ± z0z=0z=z0z=2z0111
15Gaussian beams - properties PropagationWW0RzConsider a Gaussian beam whose width W and radius of curvature R are known at a particular point on the beam axis.The beam waist radius is given bylocated to the right at a distance
16Propagating through lens Gaussian beams - propertiesPropagating through lensThe complex amplitude induced by a thin lens of focal length f is proportional to exp(-ik2/2f). When a Gaussian beam passes through such a component, its complex amplitude is multiplied by this phase factor. As a result, the beam width does not change (W'=W), but the wavefront does.Consider a Gaussian beam centered at z=0, with waist radius W0, transmitted through a thin lens located at position z. The phase of the emerging wave therefore becomes (ignore sign):Where The transmitted wave is a Gaussian beam with width W'=W and radius of curvature R'. The sign of R is positive since the wavefront of the incident beam is diverging whereas the opposite is true of R'.
17Propagating through lens Gaussian beams - propertiesPropagating through lensThe magnification factor M plays an important role: The waist radius is magnified by M, the depth of focus is magnified by M2, and the divergence angle is minified by M.
18Gaussian beams - properties Beam focusingFor a lens placed at the waist of a Gaussian beam (z=0), the transmitted beam is then focused to a waist radius W0’ at a distance z' given by:
19Gaussian beams - properties The ABCD lawReminder: where or:The ABCD LawThe q-parameters, q1 and q2, of the incident and transmitted Gaussian beams at the input and output planes of a paraxial optical system described by the (A,B,C,D) matrix are related by:Example: transmission Through Free SpaceWhen the optical system is a distance d of free space (or of any homogeneous medium), the ray-transfer matrix components are (A,B,C,D)=(1,d,0,1) so q2 = q1 + d.*Generality of the ABCD lawThe ABCD law applies to thin optical components as well as to propagation in a homogeneous medium. Since an inhomogeneous continuously varying medium may be regarded as a cascade of incremental thin elements, the ABCD law applies to these systems as well, provided that all rays (wavefront normals) remain paraxial.