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**Course outline Maxwell Eqs., EM waves, wave-packets**

Gaussian beams Fourier optics, the lens, resolution Geometrical optics, Snell’s law Light-tissue interaction: scattering, absorption Fluorescence, photo dynamic therapy Fundamentals of lasers Lasers in medicine Basics of light detection, cameras Microscopy, contrast mechanism Confocal microscopy משואות מקסוול, גלים אלקטרומגנטים, חבילות גלים קרניים גאוסיניות אופטיקת פורייה, העדשה, הפרדה אופטיקה גיאומטרית, חוק סנל אינטראקציה אור-רקמה: פיזור, בליעה, פלואורסנציה, טיפול פוטו-דינמי עקרונות לייזרים לייזרים ברפואה עקרונות גילוי אור, מצלמות מיקרוסקופיה, ניגודיות מיקרוסקופיה קונפוקלית

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**Plane waves Reminder “wavelength”**

Solutions for the Helmholtz equation: (proof in next slide) The real electric field: is independent of r infinite field!

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**Gaussian 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 wave Slowly varying complex amplitude (in space)

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**The paraxial Helmholtz equation**

Substitute the paraxial wave into the Helmholtz equation: Paraxial wave

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Paraxial wave We now assume that the variation of A(r) with z is slow enough, so that: These assumptions are equivalent to assuming that and Transverse Laplacian: Paraxial Helmholtz equation:

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**Gaussian beams z0: “Rayleigh range” Paraxial Helmholtz equation**

One solution to the paraxial Helmholtz equation of the slowly varying complex amplitude A, has the form: Where z0: “Rayleigh range” and q(z) can be separated into its real and imaginary parts: Where W(z): beam width R(z): wavefront radius of curvature

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**Gaussian 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.

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**Gaussian beams - properties**

Intensity At any z, I is a Gaussian function of . On the beam axis: (Lorentzian) I 1/2 z0 z - Maximum at z=0 - Half peak value at z = ± z0 z=0 z=z0 z=2z0 1 1 1

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**Gaussian beams - properties**

Beam width

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**Gaussian beams - properties**

Beam divergence Thus the total angle is given by

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**Gaussian beams - properties**

Depth of focus z0: Rayleigh range The total depth of focus is often defined as twice the Rayleigh range.

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**Gaussian beams - properties**

Phase 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.

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**Gaussian beams - properties**

Wavefront ~ spherical wave ~ plane wave

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**Gaussian beams - properties**

Propagation W W0 R z 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 located to the right at a distance

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**Propagating through lens**

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. 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'.

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**Propagating through lens**

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.

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**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:

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**Gaussian beams - properties**

The ABCD law Reminder: where or: 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: 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.

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Chapter 2 Propagation of Laser Beams

Chapter 2 Propagation of Laser Beams

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