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**Chapter 2 Propagation of Laser Beams**

Lecture 1 Gaussian laser beams 2.1 Scalar wave equation Lasers are coherent electromagnetic waves. Control of laser beams is the main topic of this course. Laser beams usually have a concentrated profile in the transverse direction, which diverges slowly when propagating. The beam profiles in space can be obtained by solving the wave equation. In an isotropic, linear medium with no free charge: Suppose within one wavelength the fractional change of e and m are small (slow-varying refractive index),

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**Assume any Cartesian component of the electric field varies as then**

Helmholtz equation Lens-like (quadratic index) media: Consider a cylindrically symmetric lens-like medium n(r) Significances of lens-like media: k2 (or n2) describes the inhomogeneity of the medium. k2 =0 returns to homogeneous medium. Lens-like media simulate the behavior of many optics, including lenses and spherical mirrors. Lens-like media are an approximation of many optical fibers.

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**Solving the Helmholtz equation:**

Suppose the light mainly propagates in the z direction, let Suppose the solution of the field is cylindrically symmetric, then Let y be in the form , then Will pick up later.

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**2.2 Gaussian beams in a homogeneous medium**

Let q0 be purely imaginary (so that z =0 has the most intensity), and

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**Fundamental Gaussian laser beams:**

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**Lecture 2 Geometry of Gaussian laser beams**

Geometry of the fundamental Gaussian laser beams: Beam waist w0. The minimum spot size. The geometry of the beam is uniquely determined by the size and location of w0 (suppose l and n are given). Beam width , where the radial intensity drops to 1/e2= Also called spot size. Beam area = p w2(z). Hyperbolically depends on z. The hyperbola can be thought as the boundary of the beam. Rayleigh range , where the beam area is doubled compared to that at the beam waist, so the intensity is halved. Confocal parameter b=2zR can be thought as the focal range along the beam direction, where the intensity does not drop too much. The Rayleigh range is proportional to w02, indicating that smaller beam diffracts faster. Gouy phase Phase delay compared to a plane wave. The total phase delay from –to is p, among which p /2 occurs in the focal range.

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**Radius of curvature. The radius of curvature of the wave front**

Radius of curvature The radius of curvature of the wave front. Refer to a spherical wave at a far point: R(z) is infinity at z =0 and z =, where the wave fronts are flat. R(z) z when z is large, where the beam is close to a spherical wave starting from the focal point. Beam divergence Half apex angle of the hyperbolic beam boundary. Inversely proportional to w0, indicating that smaller beam diffracts faster.

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**2.4 High order Gaussian beam modes in homogeneous medium**

If we are not limited to , the Helmholtz equation in a homogeneous medium has the solution Hermite polynomials

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**Hermite-Gaussian laser modes**

Laguerre-Gaussian laser modes

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**Elliptical Gaussian laser beams:**

Elliptical Gaussian laser beams are often seen in the lab.

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**Lecture 3 Ray equation The electromagnetic wave in space is given by**

Suppose the fractional change of n (and thus E0) is small in one wavelength, then O P dr n(r) f (r)= const Ray dr is defined to be along f, and |dr|=ds. Here s is the distance traveled by the wave. Compare to Geometrical optics is a simplification to wave optics, essentially like classical mechanics to quantum mechanics. Ray equation:

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**Application of the ray equation**

Paraxial rays in a medium with n = n(x). Suppose the incident ray is in the x-z plan. x z Inferior mirage (road, desert) Example 1: Optical mirages. Superior mirage (sea, airplane) b, c are determined by initial conditions (e.g., the injection height and direction).

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**Example 2: Optical fiber with quadratic index media.**

Ray matrix of a quadratic index medium: Will pick up later.

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***(Reading) Lagrangian optics:**

Lagrangian optics is a branch of geometrical optics which deals with the propagation of rays in inhomogeneous media. Fermat’s principle (the principle of least time): Optical Lagrangian equations are equivalent to the ray equation.

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**Lecture 4 Ray matrices Ray matrices of optical elements:**

The propagation of a ray through many optical elements can be described by using ray matrices. A ray at position z is uniquely characterized by its height r(z) and its slope We therefore use a column matrix (a vector) to represent the ray: An optical element transfers an input ray vector into an output ray vector. r2 r2' r1' r1 In paraxial optics the optical element is represented by a 22 matrix: Note that there may be different definitions of ray vectors, e.g. (nr', r). The ray matrices will accordingly be different.

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**Examples of ray matrices. I). A thin lens with focal length f .**

II). Refraction r2 r2' r1' r1 n2 n1

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**Ray matrices of some common optical elements and media:**

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**Note that there may be different sign conventions for R and for the z axis after reflection.**

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**Ray matrices of combined optical elements:**

Example: Thin lens combination. Two lens separated by d. d Significance of the ray matrix of a lens-like media: Free propagation, lenses, and spherical mirrors (cases 1, 2, and 5) are equivalent to a specific lens-like medium. E.g, a lens corresponds to l0, and k2=k/fl. Single refraction (cases 3 and 4) may be equivalent to a combination of two successive lens-like media.

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**2.3 Fundamental Gaussian beams in a lens-like medium**

Lecture 5 The ABCD law 2.3 Fundamental Gaussian beams in a lens-like medium Fundamental Gaussian beams in a lens-like medium:

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**Physical significance of the complex beam parameter q(z):**

1) Define for a beam in a lens-like medium. Then The real part of 1/q(z) specifies the radius of curvature of the wave front R(z). The imaginary part of 1/q(z) specifies the spot size of the beam w(z). 2) In a homogeneous medium, The real part of q(z) specifies the location of the beam waist z=0. The imaginary part of q(z) specifies the beam waist w0.

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**Another derivation for the transformation of the complex beam parameter:**

Ray equation in a lens-like medium. Consider rays in an r-z plane. The ABCD law: The transformation of the complex beam parameter q(z) through an optical system can be described by: , where is the ray matrix of the optical system.

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**Applications of the ABCD law:**

Importance of the ABCD law: The law enables us to trace the Gaussian beam parameter q(z) through a sequence of optical elements. The beam spot size w(z), the beam radius R(z), the beam waist w0 and its location can be obtained at any point along the beam. I) Thin lens: Compare to the lens equation II) Focusing a Gaussian laser beam: A Gaussian laser beam with waist size w01 is incident at its waist on a thin lens with focal length f. Please find the location and size of the waist of the output beam. w03=? l =?

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**Planes 1, 2, 3 are defined in the figure. We want to find w03 and l.**

Strategy: We find q2 first. The real part of q2 then specifies the location l of the output beam waist. The imaginary part of q2 specifies the beam waist w03. Please note that to find w03 and l it is not necessary to calculate q3 (as our textbook does). They are already embedded in q2. This problem also applies to the focusing of a laser beam far from its beam waist, since q1 is again purely imaginary.

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**II) Eigen mode of Gaussian beams in a quadratic-index medium:**

Let us try to find a Gaussian laser beam whose beam parameter q(z) does not change in a quadratic-index medium specified by k2. In the eigen mode the spot size w is independent of z. This is caused by the focusing nature of the medium, which cancels the diffraction nature of the beam.

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