🔬Modern Optics Unit 7 – Gaussian Beams and Resonators

Fundamentals of Gaussian Beams

• Gaussian beams are solutions to the paraxial wave equation and represent the transverse electromagnetic (TEM) modes of a laser cavity
• Characterized by a Gaussian intensity profile, with the highest intensity at the center and decreasing radially outward
• Defined by their wavelength ($\lambda$), beam waist radius ($w_0$), and radius of curvature ($R(z)$)
• The electric field amplitude of a Gaussian beam is given by $E(r,z) = E_0 \frac{w_0}{w(z)} \exp(-\frac{r^2}{w^2(z)}) \exp(-i(kz - \psi(z)))$
  • $E_0$ represents the electric field amplitude at the beam waist
  • $w(z)$ is the beam radius at position $z$
  • $r$ is the radial distance from the beam axis
  • $k$ is the wave number ($2\pi/\lambda$)
  • $\psi(z)$ is the Gouy phase shift
• The intensity distribution of a Gaussian beam is given by $I(r,z) = I_0 (\frac{w_0}{w(z)})^2 \exp(-\frac{2r^2}{w^2(z)})$, where $I_0$ is the peak intensity at the beam waist
• Gaussian beams maintain their Gaussian profile during propagation, making them ideal for many applications (laser focusing, optical communication)

Propagation and Characteristics

• As a Gaussian beam propagates, its beam radius $w(z)$ changes according to $w(z) = w_0 \sqrt{1 + (\frac{z}{z_R})^2}$, where $z_R$ is the Rayleigh range
• The Rayleigh range $z_R$ is the distance from the beam waist at which the beam radius increases by a factor of $\sqrt{2}$ and is given by $z_R = \frac{\pi w_0^2}{\lambda}$
• The radius of curvature $R(z)$ of the wavefront changes during propagation according to $R(z) = z(1 + (\frac{z_R}{z})^2)$
  • At the beam waist ($z=0$), the wavefront is planar with $R(0) = \infty$
  • Far from the beam waist ($z \gg z_R$), the wavefront becomes approximately spherical with $R(z) \approx z$
• The Gouy phase shift $\psi(z)$ is an additional phase term that depends on the propagation distance and is given by $\psi(z) = \arctan(\frac{z}{z_R})$
• The divergence angle $\theta$ of a Gaussian beam is the half-angle of the cone formed by the beam as it propagates far from the waist and is given by $\theta = \frac{\lambda}{\pi w_0}$
• The depth of focus (or confocal parameter) is the distance over which the beam radius remains within a factor of $\sqrt{2}$ of its minimum value and is given by $2z_R$

Beam Waist and Divergence

• The beam waist $w_0$ is the smallest beam radius, located at $z=0$, where the wavefront is planar
• The beam waist radius $w_0$ and the wavelength $\lambda$ determine the Rayleigh range $z_R$ and the divergence angle $\theta$
• A smaller beam waist results in a shorter Rayleigh range and a larger divergence angle, while a larger beam waist leads to a longer Rayleigh range and a smaller divergence angle
• The minimum spot size achievable when focusing a Gaussian beam is limited by diffraction and is given by $w_0 = \frac{\lambda}{\pi\theta}$, where $\theta$ is the half-angle of the focusing cone
• The beam divergence can be reduced by expanding the beam before focusing, using a beam expander (telescope) with a magnification factor $M$
  • The expanded beam waist is given by $w_0' = Mw_0$
  • The reduced divergence angle is given by $\theta' = \frac{\theta}{M}$
• Gaussian beam divergence is an important consideration in applications such as laser cutting, drilling, and optical communication, where maintaining a small spot size over a long distance is crucial

ABCD Matrix Method

• The ABCD matrix method is a powerful tool for analyzing the propagation of Gaussian beams through optical systems
• An optical system can be represented by a 2x2 matrix $\begin{pmatrix} A & B \\ C & D \end{pmatrix}$, where the elements $A$, $B$, $C$, and $D$ depend on the properties of the optical components
• The ABCD matrix relates the input and output beam parameters $(r_1, \theta_1)$ and $(r_2, \theta_2)$ as follows: $\begin{pmatrix} r_2 \\ \theta_2 \end{pmatrix} = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} r_1 \\ \theta_1 \end{pmatrix}$
  • $r$ represents the radial distance from the beam axis
  • $\theta$ represents the angle between the ray and the optical axis
• The ABCD matrix for a sequence of optical elements is obtained by multiplying the matrices of the individual elements in the order they are encountered by the beam
• The complex beam parameter $q$ is defined as $\frac{1}{q(z)} = \frac{1}{R(z)} - i\frac{\lambda}{\pi w^2(z)}$ and can be used to simplify the analysis of Gaussian beam propagation
• The complex beam parameter at the output of an optical system is related to the input complex beam parameter by $q_2 = \frac{Aq_1 + B}{Cq_1 + D}$
• The ABCD matrix method enables the design and optimization of optical systems for specific Gaussian beam propagation requirements (laser beam shaping, mode matching)

Optical Resonators and Cavity Modes

• Optical resonators, also known as optical cavities, are essential components of laser systems and consist of two or more mirrors that confine and amplify light
• The most common types of optical resonators are the Fabry-Perot cavity (two parallel mirrors) and the ring cavity (three or more mirrors in a closed loop)
• Resonators support standing waves, or modes, that satisfy the condition for constructive interference after a round trip in the cavity
• The resonance condition for a Fabry-Perot cavity is given by $2L = m\lambda$, where $L$ is the cavity length, $m$ is an integer (mode number), and $\lambda$ is the wavelength
• The frequency spacing between adjacent longitudinal modes (free spectral range, FSR) in a Fabry-Perot cavity is given by $\Delta\nu_{FSR} = \frac{c}{2L}$, where $c$ is the speed of light
• Transverse electromagnetic (TEM) modes describe the transverse intensity distribution of the beam in the cavity and are labeled as TEM$_{mn}$, where $m$ and $n$ are integers representing the number of nodes in the horizontal and vertical directions, respectively
  • The fundamental mode, TEM$_{00}$, has a Gaussian intensity profile
  • Higher-order modes, such as TEM$_{01}$, TEM$_{10}$, and TEM$_{11}$, have more complex intensity distributions and are generally less desirable for most applications
• The stability and mode structure of an optical resonator depend on the curvature and spacing of the mirrors, as described by the stability criteria

Stability Criteria for Resonators

• The stability of an optical resonator determines whether the beam remains confined within the cavity after multiple round trips
• A resonator is considered stable if the beam parameters (radius and radius of curvature) reproduce themselves after each round trip
• The stability criterion for a two-mirror resonator is given by $0 \leq (1 - \frac{L}{R_1})(1 - \frac{L}{R_2}) \leq 1$, where $L$ is the cavity length, and $R_1$ and $R_2$ are the radii of curvature of the mirrors
• Resonators can be classified into three categories based on their stability:
  • Stable resonators: Satisfy the stability criterion and confine the beam within the cavity (e.g., confocal, concentric, and plane-parallel resonators)
  • Unstable resonators: Do not satisfy the stability criterion and allow the beam to expand with each round trip (e.g., convex-plane and concave-convex resonators)
  • Marginally stable resonators: Lie on the boundary between stable and unstable regions (e.g., hemispherical and concentric resonators)
• The stability diagram is a graphical representation of the stability regions in the $(g_1, g_2)$ plane, where $g_1 = 1 - \frac{L}{R_1}$ and $g_2 = 1 - \frac{L}{R_2}$
• The beam waist size and location within a stable resonator can be calculated using the cavity parameters ($L$, $R_1$, $R_2$) and the wavelength $\lambda$
• Stable resonators are preferred for most laser applications due to their ability to maintain a well-defined beam profile and efficient energy extraction

Applications in Laser Systems

• Gaussian beams and optical resonators are fundamental to the operation and design of various laser systems
• In gas lasers (e.g., CO2 and HeNe lasers), the cavity length and mirror curvatures are chosen to achieve a stable resonator configuration that supports the desired transverse mode (usually TEM$_{00}$)
• Solid-state lasers (e.g., Nd:YAG and Ti:Sapphire lasers) employ stable resonators with appropriate mode-matching techniques to efficiently pump the gain medium and extract the laser beam
• Semiconductor lasers (e.g., laser diodes) have short cavity lengths and high gain, requiring careful design of the resonator to achieve stable operation and good beam quality
• Fiber lasers utilize the waveguiding properties of optical fibers to confine the beam and support stable resonator modes
• Mode-locking techniques, such as active and passive mode-locking, are used to generate ultrashort pulses by establishing a fixed phase relationship between the longitudinal modes in the resonator
• Q-switching is another technique used to generate high-energy pulses by modulating the cavity losses and allowing the population inversion to build up before releasing the stored energy in a short pulse
• Gaussian beam optics is crucial for designing efficient beam delivery systems, such as those used in laser material processing, laser surgery, and optical communication

Practical Considerations and Limitations

• Real laser beams often deviate from the ideal Gaussian profile due to various factors, such as gain saturation, thermal lensing, and imperfect optical components
• The $M^2$ factor, or beam quality factor, is a measure of how much a real beam deviates from an ideal Gaussian beam, with $M^2 = 1$ representing a perfect Gaussian beam
• Higher $M^2$ values indicate a larger beam divergence and a larger focused spot size compared to an ideal Gaussian beam with the same waist size
• Thermal lensing occurs in solid-state lasers when the gain medium experiences a temperature gradient due to non-uniform pumping or cooling, leading to a refractive index gradient that acts as a lens and affects the beam profile and resonator stability
• Optical aberrations, such as spherical aberration, coma, and astigmatism, can distort the beam profile and degrade the focusing performance of the laser system
• Misalignment and vibrations of the optical components can cause beam pointing instability and reduce the overall efficiency of the laser system
• Nonlinear optical effects, such as self-focusing and self-phase modulation, can occur at high beam intensities and affect the beam profile and pulse characteristics
• Proper design, alignment, and maintenance of the laser system are essential to minimize these practical limitations and ensure optimal performance
• Advanced techniques, such as adaptive optics and real-time beam monitoring, can be employed to correct for beam distortions and maintain a high-quality Gaussian beam profile