Course outline 1. Maxwell Eqs., EM waves, wave-packets 2. Gaussian beams 3. Fourier optics, the lens, resolution 4. Geometrical optics, Snell’s law 5. Light-tissue interaction: scattering, absorption Fluorescence, photo dynamic therapy חבילות גלים, גלים אלקטרומגנטים, משואות מקסוול.1 קרניים גאוסיניות.2 הפרדה, העדשה, אופטיקת פורייה.3 חוק סנל, אופטיקה גיאומטרית.4 , פלואורסנציה, בליעה, פיזור:רקמה- אינטראקציה אור.5 דינמי-טיפול פוטו 6. Fundamentals of lasers עקרונות לייזרים.6 7. Lasers in medicine לייזרים ברפואה.7 8. Basics of light detection, cameras 9. Microscopy, contrast mechanism 10.Confocal microscopy מצלמות, עקרונות גילוי אור.8 ניגודיות, מיקרוסקופיה.9 מיקרוסקופיה קונפוקלית.10 Reminder Plane waves Solutions for the Helmholtz equation: (proof in next slide) The real electric field: E r E0e ik r H r H 0eik r E r , t Re E r eit Re E0eik r eit E0 is independent of r infinite field! r k: r k: “wavelength” E r , t E0 cos k r t E r , t0 E0 cos k r t0 E r , t0 constant k 2 k Gaussian beams – the paraxial wave A0 ikr e Spherical wave: U r r Plane wave: E r E0 eik r 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: ikz U r A r e ‘Carrier’ plane wave Slowly varying complex amplitude (in space) The paraxial Helmholtz equation Substitute the paraxial wave into the Helmholtz equation: 2U k 2U 0 2U 2U 2U 2 k U r 0 2 2 2 x y z 2 ik z 2 A r ik z 2 A r ik z A r e 2 ik z e e k A r e 0 2 2 2 x y z Paraxial wave U r A r eikz 2 A r ik z 2 A r ik z 2 A r ik z A r ik z e e e 2ik e k 2 A r eik z k 2 A r eik z 0 2 2 2 x y z z 2 A 2 A 2 A A 2 ik 0 2 2 2 x y z z A A A A 2 ik 0 2 2 2 x y z z 2 2 2 Paraxial wave U r A r eikz We now assume that the variation of A(r) with z is slow enough, so that: 2 A 2 k A z 2 2 A k A z 2 z These assumptions are equivalent to assuming that sin and tan 2 A 2 A A 2 2ik 0 2 x y z Paraxial Helmholtz equation: Transverse Laplacian: A 2 T A 2ik 0 z 2 2 A A T2 2 2 x y Gaussian beams A A 2ik 0 z Paraxial Helmholtz equation 2 T One solution to the paraxial Helmholtz equation of the slowly varying complex 2 amplitude A, has the form: ik A1 Ar e q z Where q z z iz0 and 2 q z z0: “Rayleigh range” 2 x2 y2 q(z) can be separated into its real and imaginary parts: 1 1 i q z R z W 2 z Where W(z): beam width R(z): wavefront radius of curvature Gaussian beams The full Gaussian beam: 2 2 W0 W 2 z ikz ik 2 R z i z U r A0 e e W z With beam parameters: z W z W0 1 z0 2 A0 A1 iz0 z0 2 R z z 1 z z z tan 1 z0 W0 z0 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. Gaussian beams - properties Intensity I r U r I 0 A0 2 2 W0 U r A0 e W z 2 W0 I , z I0 e W z 22 W 2 2 W 2 z e z W z W0 1 z0 z ikz ik 2 z0 2 R z z 1 z At any z, I is a Gaussian function of . On the beam axis: z tan 1 2 W0 I0 I 0, z I 0 2 W z 1 z z0 I (Lorentzian) W0 z z0 z0 1/2 0 z0 z - Maximum at z=0 - Half peak value at z = ± z0 z=0 z=z0 1 1 z=2z0 1 2 2 R z i z Gaussian beams - properties Beam width W0 U r A0 e W z 2 W 2 z e z W z W0 1 z0 2 z0 2 R z z 1 z z W z W0 1 z0 1 e line 1 e2 in intensity 2W0 2 z tan 1 W0 z0 z z0 ikz ik 2 2 R z i z Gaussian beams - properties Beam divergence z z0 z W z W0 1 z0 W W z 0 z z0 2 2 W 2 z e z W z W0 1 z0 ikz ik 2 2 R z 2 z0 2 R z z 1 z W02 z0 W0 W0 0 2 z0 W0 W0 Thus the total angle is given by W0 U r A0 e W z 4 20 2W0 z tan 1 W0 z z0 z0 2W0 2 0 4 i z Gaussian beams - properties Depth of focus z0 W0 W02 2 z0 2 W0 U r A0 e W z 2 W 2 z e z W z W0 1 z0 2 z0 2 R z z 1 z z tan 1 W0 z z0 z0 z0: Rayleigh range The total depth of focus is often defined as twice the Rayleigh range. ikz ik 2 2 R z i z Gaussian beams - properties Phase W0 U r A0 e W z W0 U r A0 e W z 2 W 2 z e ikz ik 2 2 R z i z z 0, z kz tan z0 1 2 W 2 z e z W z W0 1 z0 ikz ik 2 2 R z 2 z0 2 R z z 1 z z tan 1 W0 z z0 z0 A. Ruffin et al., PRL (1999) The total accumulated excess retardation as the wave travels from - to is . This phenomenon is known as the Gouy effect. i z Gaussian beams - properties Wavefront 2 z k , z kz tan 1 z0 2R z W0 U r A0 e W z 2 W 2 z e z W z W0 1 z0 ikz ik 2 2 R z i z 2 z0 2 R z z 1 z z tan 1 W0 z z0 z0 ~ spherical wave ~ plane wave z0 2 R z z 1 z z R z z Gaussian beams - properties Propagation W0 U r A0 e W z 2 W 2 z e z W z W0 1 z0 W W0 R ikz ik 2 2 R z i z 2 z0 2 R z z 1 z z tan 1 z W0 z z0 z0 Consider 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 by W0 W 1 W R 2 located to the right at a distance z R 1 R W 2 2 2 Gaussian beams - properties Propagating through lens The 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. W0 U r A0 e W z 2 W 2 z e ikz ik 2 2 R z i z 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): kz k 2 2R k 2 2f kz k 2 2R ' Where 1 1 1 R' R f 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'. Gaussian beams - properties Propagating through lens The 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. Gaussian beams - properties Beam focusing For 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: W0 ' z' W0 z 1 z0 f 2 f z 1 f z0 M z 0 f 0 f f f W0 W z0 W02 0 W0 f 0 f 2 1 1 z0 f z 2 W0 f 0 f z0 f W0 Gaussian beams - properties The ABCD law Reminder: A Ar 1 e q z ik 2 2 q z , where q z z iz0 1 1 i or: q z R z W 2 z The ABCD Law The 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: Aq1 B q2 Cq1 D Example: transmission Through Free Space When 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 law The 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.