In some cases, our waist will be located in the focal plane of the lens. Only then can we say that the laser is truly collimated and only then can the aperture – divergence product be used to determine the beam quality. In many practical cases, this does not hold true.

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Laser divergencein vacuum

By looking in the vicinity of the focal plane for a minimum diameter spot, the waist as transformed by the lens, we can gain much more useful information. For real lasers, the waist is usually located near the focal plane and usually a little way after it. Note that the original beam divergence is eliminated in the expression that gives M2. It is expressed in terms of the beam diameter at the waist, the diameter of the beam at the lens and the distance of the waist from the lens. All these parameters are relatively easily measured. The focal length of the lens and the beam diameter at the lens need to be chosen such that the waist is much smaller than the beam diameter at the lens. Technically speaking, to achieve accurate measurements, the waist must be much greater than its Rayleigh range from the lens. This value is given by πd02/4λM2.

Just as different manufacturing processes produce parts at various tolerances, they are also capable of different roughnesses. Generally, these two characteristics are linked: manufacturing processes that are dimensionally precise create surfaces with low roughness. In other words, if a process can manufacture parts to a narrow dimensional tolerance, the parts will not be very rough.

Surface finish, also known as surface texture or surface topography, is the nature of a surface as defined by the three characteristics of lay, surface roughness, and waviness.[1] It comprises the small, local deviations of a surface from the perfectly flat ideal (a true plane).

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Laser divergencepdf

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Lowdivergence laser

Surface roughness, commonly shortened to roughness, is a measure of the total spaced surface irregularities.[1] In engineering, this is what is usually meant by "surface finish." A Lower number constitutes finer irregularities, i.e., a smoother surface.

In general, the cost of manufacturing a surface increases as the surface finish improves.[4] Any given manufacturing process is usually optimized enough to ensure that the resulting texture is usable for the part's intended application. If necessary, an additional process will be added to modify the initial texture. The expense of this additional process must be justified by adding value in some way—principally better function or longer lifespan. Parts that have sliding contact with others may work better or last longer if the roughness is lower. Aesthetic improvement may add value if it improves the saleability of the product.

Each manufacturing process (such as the many kinds of machining) produces a surface texture. The process is usually optimized to ensure that the resulting texture is usable. If necessary, an additional process will be added to modify the initial texture. The latter process may be grinding (abrasive cutting), polishing, lapping, abrasive blasting, honing, electrical discharge machining (EDM), milling, lithography, industrial etching/chemical milling, laser texturing, or other processes.

Laser divergencecalculator

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In the United States, surface finish is usually specified using the ASME Y14.36M standard. The other common standard is International Organization for Standardization (ISO) 1302:2002, although the same has been withdrawn in favour of ISO 21920-1:2021.[3]

Laserbeamdivergenceand spot size

Measuring the beam in the focal plane of the lens yields the beam’s ‘raw’ divergence, simply by dividing the beam diameter in the focal plane by the focal length. Since we only know for sure what the laser beam diameter is near its output aperture, this figure tells us little about the beam quality. We do not know the position of the waist of the original laser beam or its size. In many real lasers, it can be located behind the laser (or sometimes well in front of it) and is thus often smaller than the laser beam diameter measured at its output aperture. Only the true waist diameter can be combined with the divergence to give M2.

Surface texture is one of the important factors that control friction and transfer layer formation during sliding. Considerable efforts have been made to study the influence of surface texture on friction and wear during sliding conditions. Surface textures can be isotropic or anisotropic. Sometimes, stick-slip friction phenomena can be observed during sliding, depending on surface texture.

A true measure of the quality of a laser beam is the parameter known as M2, also referred to as a number of ‘times diffraction limited’. It describes how a beam will diverge compared to a theoretically ‘perfect’ laser beam of the same wavelength and initial size that has the lowest possible divergence given by diffraction theory. A theoretically perfect laser beam must have a Gaussian intensity distribution whereas most real laser beams do not. We therefore have to make some approximations when talking about M2. We measure the laser beam diameter as the diameter of a circle containing 1/e2 or about 87% of the power or energy and the divergence angle as the full angle of the cone containing the same power or energy. We use an ellipse to make the same figures where the beam is not nominally circular. Real lasers can have M2 values ranging from very little more than 1 (the theoretical minimum) to several tens or even hundreds. Many practical applications require the lowest possible value. The optical set up depicted here can be used to measure both the beam divergence and the M2 value of a real laser. A lens of focal length 1 or 2m is typically used, along with attenuation optics and the beam is imaged using a CCD and software.

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Angle ofdivergenceformula

Divergenceoflaserbeam formula

A practical example is as follows. An aircraft maker contracts with a vendor to make parts. A certain grade of steel is specified for the part because it is strong enough and hard enough for the part's function. The steel is machinable although not free-machining. The vendor decides to mill the parts. The milling can achieve the specified roughness (for example, ≤ 3.2 μm) as long as the machinist uses premium-quality inserts in the end mill and replaces the inserts after every 20 parts (as opposed to cutting hundreds before changing the inserts). There is no need to add a second operation (such as grinding or polishing) after the milling as long as the milling is done well enough (correct inserts, frequent-enough insert changes, and clean coolant). The inserts and coolant cost money, but the costs that grinding or polishing would incur (more time and additional materials) would cost even more than that. Obviating the second operation results in a lower unit cost and thus a lower price. The competition between vendors elevates such details from minor to crucial importance. It was certainly possible to make the parts in a slightly less efficient way (two operations) for a slightly higher price; but only one vendor can get the contract, so the slight difference in efficiency is magnified by competition into the great difference between the prospering and shuttering of firms.

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Beamdivergenceangle

Lay is the direction of the predominant surface pattern, ordinarily determined by the production method used. The term is also used to denote the winding direction of fibers and strands of a rope.[2]

Many factors contribute to the surface finish in manufacturing. In forming processes, such as molding or metal forming, surface finish of the die determines the surface finish of the workpiece. In machining, the interaction of the cutting edges and the microstructure of the material being cut both contribute to the final surface finish.[citation needed]

The irreducible divergence (full angle) is given by the expression 4M2λ/πd where d is the diameter of the laser beam measured at its waist. For reasons given above, this figure is often lower than the ‘raw’ divergence of the laser beam itself. To a first order approximation, the above expression holds true even if the laser beam is expanded or reduced by telescopes and the like.

Surface finish may be measured in two ways: contact and non-contact methods. Contact methods involve dragging a measurement stylus across the surface; these instruments are called profilometers. Non-contact methods include: interferometry, confocal microscopy, focus variation, structured light, electrical capacitance, electron microscopy, atomic force microscopy and photogrammetry.

Waviness is the measure of surface irregularities with a spacing greater than that of surface roughness. These irregularities usually occur due to warping, vibrations, or deflection during machining.[1]

Due to the abstractness of surface finish parameters, engineers usually use a tool that has a variety of surface roughnesses created using different manufacturing methods.[4]

In the discussion above, the beam divergence as measured in the focal plane of our lens frequently contains some reducible elements. These can include sphericity imposed by lensing in an amplifier after the rod (and not fully compensated by the beam expanding telescope) or sphericity resulting from improper collimation of an unstable resonator; with a fixed length and a finite number of resonator mirror curvatures, it is often impossible to achieve perfect collimation. With proper beam control after the laser, such as the transmitting optics in a LIDAR system, or the focusing lens in a machining head, this sphericity can be eliminated and the irreducible element, as predicted by M2, will determine respectively the angular resolution of the LIDAR system or the spot size at a work piece.

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