3.2 Beam divergence and focusing

Beam divergence and focusing are crucial concepts in laser engineering. They determine how laser beams spread and concentrate, impacting applications from cutting to imaging. Understanding these principles helps engineers design systems that deliver precise, high-intensity beams for various uses.

This topic covers beam propagation, waist size, and focusing techniques. It explores factors affecting divergence, measurement methods, and strategies to minimize spread. Key applications like material processing and microscopy showcase the importance of controlled beam behavior in modern technology.

Beam divergence fundamentals

• Beam divergence is a critical parameter in laser engineering that describes how quickly a laser beam spreads out as it propagates through space
• Understanding beam divergence is essential for designing laser systems that can deliver focused, high-intensity beams for various applications
• Key concepts in this section include Gaussian beam propagation, beam waist, Rayleigh range, and far-field divergence angle

Gaussian beam propagation

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• Gaussian beams are the most common type of laser beam, characterized by a transverse intensity profile that follows a Gaussian distribution
• As a Gaussian beam propagates, its wavefront curvature and beam width change according to the beam parameter $q(z)$, which depends on the propagation distance $z$
• The complex beam parameter is given by $\frac{1}{q(z)} = \frac{1}{R(z)} - i\frac{\lambda}{\pi w^2(z)}$, where $R(z)$ is the wavefront radius of curvature, $w(z)$ is the beam width, and $\lambda$ is the wavelength

Beam waist and Rayleigh range

• The beam waist $w_0$ is the location where the Gaussian beam has its minimum width and planar wavefront
• The Rayleigh range $z_R$ is the distance from the beam waist where the beam width increases by a factor of $\sqrt{2}$ and is given by $z_R = \frac{\pi w_0^2}{\lambda}$
• The beam width at any position $z$ can be calculated using $w(z) = w_0\sqrt{1 + (\frac{z}{z_R})^2}$

Far-field divergence angle

• In the far-field region ($z \gg z_R$), the beam width increases linearly with distance, and the divergence can be characterized by the far-field divergence angle $\theta$
• The far-field divergence angle is given by $\theta = \frac{\lambda}{\pi w_0}$, which shows that a smaller beam waist results in a larger divergence angle
• The divergence angle is often expressed in terms of the full angle, $\theta_{full} = 2\theta$, which is the angle between the $1/e^2$ intensity points

Beam parameter product

• The beam parameter product (BPP) is a measure of the beam quality and is defined as the product of the beam waist radius and the far-field divergence angle: $BPP = w_0\theta$
• For an ideal Gaussian beam, the BPP is equal to $\frac{\lambda}{\pi}$, which is the minimum possible value
• The BPP is conserved during beam propagation through lossless optical systems, making it a useful parameter for characterizing beam quality

Factors affecting beam divergence

• Several factors influence the divergence of a laser beam, including wavelength, aperture size, and beam quality
• Understanding these factors is crucial for designing laser systems with desired divergence properties and optimizing beam delivery for specific applications

Wavelength dependence

• The far-field divergence angle is directly proportional to the wavelength, as given by $\theta = \frac{\lambda}{\pi w_0}$
• Shorter wavelengths result in smaller divergence angles for a given beam waist, making them advantageous for applications requiring low divergence (visible and UV lasers)
• Longer wavelengths, such as those in the infrared range, tend to have larger divergence angles, which can be a challenge for beam delivery and focusing

Aperture size effects

• The beam waist size is determined by the aperture size of the laser cavity or the focusing optics
• A larger aperture results in a larger beam waist and, consequently, a smaller divergence angle, as evident from the relation $\theta = \frac{\lambda}{\pi w_0}$
• Aperture size is often limited by practical constraints such as the size of the gain medium, cavity design, and available optics

Beam quality and M² factor

• Real laser beams often deviate from the ideal Gaussian profile due to factors such as higher-order modes, wavefront distortions, and aperture truncation
• The beam quality factor, M², quantifies the deviation of a real beam from an ideal Gaussian beam, with M² = 1 representing a perfect Gaussian beam
• The divergence angle of a real beam is M² times larger than that of an ideal Gaussian beam with the same waist size, given by $\theta_{real} = M^2\frac{\lambda}{\pi w_0}$

Beam focusing principles

• Focusing a laser beam is essential for achieving high intensities and small spot sizes required in many applications
• This section covers the fundamental principles of beam focusing, including focal length, spot size, and Gaussian beam focusing

Focal length and spot size

• The focal length $f$ of a lens determines the distance from the lens where the beam is focused to its smallest size, known as the spot size
• The spot size $w_f$ at the focus is related to the input beam size $w_0$ and the focal length by $w_f = \frac{\lambda f}{\pi w_0}$
• A shorter focal length or a larger input beam size results in a smaller focused spot size

Thin lens approximation

• The thin lens approximation is a simplified model for describing the focusing properties of lenses when the lens thickness is much smaller than the focal length
• In this approximation, the focusing power of a lens is given by $\frac{1}{f} = (n - 1)(\frac{1}{R_1} - \frac{1}{R_2})$, where $n$ is the refractive index of the lens material, and $R_1$ and $R_2$ are the radii of curvature of the lens surfaces
• The thin lens approximation is useful for quick calculations and understanding basic focusing principles but may not be accurate for thick lenses or high-precision applications

Gaussian beam focusing

• When a Gaussian beam is focused by a lens, the focused beam also maintains a Gaussian profile, with its waist located at the focal plane
• The focused beam waist size $w_f$ is given by $w_f = \frac{\lambda f}{\pi w_0}$, where $w_0$ is the input beam waist size and $f$ is the focal length of the lens
• The Rayleigh range of the focused beam $z_{Rf}$ is given by $z_{Rf} = \frac{\pi w_f^2}{\lambda}$, which determines the depth of focus

Spherical aberrations

• Spherical aberrations occur when a lens fails to focus all rays to a single point due to the non-ideal shape of the lens surfaces
• Spherical aberrations cause the focal spot to be larger and distorted, reducing the peak intensity and beam quality
• Aspherical lenses and multi-element lens systems can be used to minimize spherical aberrations and improve focusing performance

Focusing elements and systems

• Various optical elements and systems are used to focus laser beams, each with its own advantages and limitations
• This section covers the most common focusing elements, including lenses, achromatic doublets, and beam expanders

Positive and negative lenses

• Positive lenses, also known as converging lenses, have a positive focal length and focus collimated beams to a spot (biconvex, plano-convex)
• Negative lenses, or diverging lenses, have a negative focal length and diverge collimated beams (biconcave, plano-concave)
• The focal length of a lens depends on its shape, thickness, and refractive index

Spherical vs aspherical lenses

• Spherical lenses have surfaces with a constant radius of curvature, which is easier to manufacture but can introduce spherical aberrations
• Aspherical lenses have surfaces with a variable radius of curvature, designed to minimize spherical aberrations and improve focusing performance
• Aspherical lenses are more complex to manufacture and are typically more expensive than spherical lenses

Achromatic doublets

• Achromatic doublets are two-element lenses designed to minimize chromatic aberrations, which occur when different wavelengths are focused at different positions
• An achromatic doublet consists of a positive and a negative lens made from materials with different dispersion properties (crown and flint glass)
• Achromatic doublets are useful for focusing multi-wavelength or broadband laser sources

Beam expanders and collimators

• Beam expanders are optical systems that increase the size of a laser beam while maintaining its collimation
• Galilean beam expanders use a negative lens followed by a positive lens, while Keplerian beam expanders use two positive lenses with a focus between them
• Beam collimators are similar to beam expanders but are designed to produce a collimated output beam from a diverging input beam (laser diode collimation)

Applications of focused beams

• Focused laser beams find numerous applications in various fields, exploiting the high intensity, small spot size, and precise control offered by laser focusing
• This section highlights some key applications of focused laser beams in material processing, microscopy, data storage, and printing

Laser material processing

• Focused laser beams are widely used in material processing applications, such as cutting, drilling, welding, and surface modification
• The high intensity of the focused beam enables localized heating, melting, or vaporization of the material, allowing for precise and efficient processing
• Examples include laser cutting of metals and polymers, laser welding of dissimilar materials, and laser surface hardening

Confocal microscopy

• Confocal microscopy is a powerful imaging technique that uses focused laser beams to achieve high-resolution, three-dimensional imaging of biological samples
• The focused beam is scanned across the sample, and the fluorescence signal is collected through a pinhole, which rejects out-of-focus light
• Confocal microscopy enables imaging of thick samples, live cells, and dynamic processes with sub-micrometer resolution

Optical data storage

• Focused laser beams are used in optical data storage systems, such as CDs, DVDs, and Blu-ray discs, to read and write data
• The focused beam is used to selectively modify the optical properties of the recording medium (reflectivity, absorption, or phase) to represent binary data
• The small spot size of the focused beam allows for high data densities and fast read/write speeds

Laser scanning and printing

• Laser scanning and printing systems rely on focused laser beams to create high-resolution images and patterns
• In laser scanning, a focused beam is scanned across a photosensitive material (photoresist, photopolymer) to create a latent image, which is then developed
• Laser printing uses a focused beam to selectively discharge a photoconductor drum, which then attracts toner particles to form the printed image

Measuring beam divergence

• Accurate measurement of beam divergence is essential for characterizing laser beams and ensuring they meet the requirements of specific applications
• This section presents several common methods for measuring beam divergence, including knife-edge scanning, camera-based profiling, and wavefront sensing

Knife-edge scanning method

• The knife-edge scanning method is a simple and widely used technique for measuring beam width and divergence
• A knife-edge (razor blade) is scanned across the beam at different positions along the propagation direction, and the transmitted power is recorded
• The beam width at each position is determined from the power transmission curve, and the divergence is calculated from the change in beam width with distance

Camera-based beam profiling

• Camera-based beam profiling systems use a CCD or CMOS camera to capture the transverse intensity profile of the laser beam
• The camera is placed at different positions along the beam path, and the beam width is determined from the recorded intensity profiles
• Camera-based profiling provides a direct measurement of the beam intensity distribution and can reveal beam quality issues (asymmetry, higher-order modes)

Shack-Hartmann wavefront sensing

• Shack-Hartmann wavefront sensing is a technique for measuring the wavefront of a laser beam, which can be used to determine the beam divergence
• A Shack-Hartmann sensor consists of a lenslet array and a camera, which measure the local wavefront slopes across the beam
• The wavefront data can be used to calculate the beam parameters, including the waist size and location, and the far-field divergence angle

ISO 11146 standard

• ISO 11146 is an international standard that provides guidelines for measuring and specifying the beam width, divergence, and beam propagation factor (M²) of laser beams
• The standard defines the measurement procedures, calculation methods, and reporting requirements for beam characterization
• Adhering to the ISO 11146 standard ensures consistent and comparable beam measurements across different laboratories and applications

Minimizing beam divergence

• Minimizing beam divergence is crucial for applications that require long-range beam propagation, tight focusing, or efficient coupling into optical systems
• This section explores various strategies for reducing beam divergence, including cavity design, mode-matching, adaptive optics, and beam shaping

Optimizing cavity design

• The design of the laser cavity plays a significant role in determining the beam divergence and quality
• Stable cavity designs, such as the plano-concave or hemispherical configurations, promote the formation of low-order transverse modes with reduced divergence
• Unstable cavity designs, such as the confocal or concentric configurations, can provide high output power but may result in increased divergence and lower beam quality

Mode-matching techniques

• Mode-matching refers to the process of matching the size and divergence of the laser beam to the fundamental mode of the cavity or the optical system
• Proper mode-matching ensures efficient energy extraction, reduced losses, and improved beam quality
• Mode-matching can be achieved through the use of lenses, curved mirrors, or a combination of both, placed at specific distances to match the beam parameters

Adaptive optics for wavefront correction

• Adaptive optics (AO) is a technique for actively correcting wavefront distortions in real-time, which can arise from atmospheric turbulence, thermal lensing, or imperfect optics
• An AO system consists of a wavefront sensor (Shack-Hartmann), a deformable mirror, and a control system that adjusts the mirror shape to compensate for the measured distortions
• By correcting wavefront distortions, AO can significantly reduce beam divergence and improve beam quality, especially for high-power lasers or long-range propagation

Fiber coupling and beam shaping

• Coupling laser beams into optical fibers is an effective way to reduce divergence and maintain beam quality over long distances
• Single-mode fibers act as spatial filters, allowing only the fundamental mode to propagate, resulting in a near-diffraction-limited output beam
• Beam shaping techniques, such as using aspheric lenses, diffractive optical elements, or spatial light modulators, can be employed to transform the beam profile and reduce divergence before fiber coupling or focusing