Gaussian_Beams Flashcards
What is the Helmholtz equation?
The Helmholtz equation is ∇²U + k²U = 0, where U is the wave function and k is the wave number.
What assumption is made in the paraxial approximation?
The paraxial approximation assumes that the amplitude A does not change significantly as z varies over a scale of one wavelength, implying small angles between rays and the propagation axis.
Explain the paraxial Helmholtz equation.
The paraxial Helmholtz equation is ∇²_T A - 2ik ∂A/∂z = 0, where ∇²_T is the transverse Laplacian, and it describes the slowly varying envelope approximation (SVEA) of the wave function.
What is the complex envelope of a Gaussian beam?
The complex envelope of a Gaussian beam is A_G(x, y, z) = A1/q(z) * e^(-ik(x² + y²)/2q(z)), with q(z) = z + iz₀.
How is the beam width W(z) of a Gaussian beam defined?
The beam width W(z) is given by W(z) = W₀ * sqrt(1 + (z/z₀)²), where W₀ is the beam waist and z₀ is the Rayleigh range.
Describe the radius of wavefront curvature R(z) for a Gaussian beam.
The radius of wavefront curvature R(z) is defined as R(z) = z * [1 + (z₀/z)²], where z₀ is the Rayleigh range.
What is the Hermite-Gaussian function Gl(u)?
The Hermite-Gaussian function Gl(u) is defined as Gl(u) = Hl(u)e^(-u²/2), where Hl(u) are Hermite polynomials.
What are Laguerre-Gaussian beams and when are they used?
Laguerre-Gaussian beams are solutions to the paraxial Helmholtz equation in cylindrical coordinates, often used in situations with cylindrical symmetry, such as in optical resonators.
What is a Bessel beam and how does it differ from Gaussian beams?
A Bessel beam is a solution to the Helmholtz equation with a non-diffracting intensity profile, characterized by Bessel functions. Unlike Gaussian beams, Bessel beams do not diffract and carry infinite power.
Diffraction: Bending around obstacle or spreading beam width
How are Bessel beams used in optical tweezers?
Bessel beams are used in optical tweezers due to their constant gradient as a function of z, allowing them to guide particles over long distances with precision.
