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es gaussian flat top tem xy or any shape must follow the equation above provided that the beam radius uses the d4σ definition of the beam width using the 10 90 knife edge the d86 or the fwhm widths does not work complete e field beam profiling edit beam profilers measure the intensity e field 2 of the laser beam profile but do not yield any information about the phase of the e field to completely characterize the e field at a given plane both the phase and amplitude profiles must be known the real and imaginary parts of the electric field can be characterized using two ccd beam profilers that sample the beam at two separate propagation planes with the application of a phase recovery algorithm to the captured data the benefit of completely characterizing the e field in one plane is that the e field profile can be computed for any other plane with diffraction theory power in the bucket or strehl definition of beam quality edit the m 2 parameter is not the whole story in specifying beam quality a low m 2 only implies that the second moment of the beam profile expands slowly nevertheless two beams with the same m 2 may not have the same fraction of delivered power in a given area power in the bucket and strehl ratio are two attempts to define beam quality as a function of how much power is delivered to a given area unfortunately there is no standard bucket size d86 width gaussian beam width airy disk nulls etc or bucket shape circular rectangular etc and there is no standard beam to compare for the strehl ratio therefore these definitions must always be specified before a number is given and it presents much difficulty when trying to compare lasers there is also no simple conversion between m 2 power in the bucket and strehl ratio the strehl ratio for example has been defined as the ratio of the peak focal intensities in the aberrated and ideal point spread functions in other cases it has been defined as the ratio between the peak intensity of an image divided by the peak intensity of a diffraction limited image with the same total flux 8 9 since there are many ways power in the bucket and strehl ratio have been defined in the literature the recommendation is to stick with the iso standard m 2 definition for the beam quality parameter and be aware that a strehl ratio of 0 8 for example does not mean anything unless the strehl ratio is accompanied by a definition beam divergence edit main article beam divergence the beam divergence of a laser beam is a measure for how fast the beam expands far from the beam waist it is usually defined as the derivative of the beam radius with respect to the axial position in the far field i e in a distance from the beam waist which is much larger than the rayleigh length this definition yields a divergence half angle sometimes full angles are used in the literature these are twice as large for a diffraction limited gaussian beam the beam divergence is λ πw 0 where λ is the wavelength in the medium and w 0 the beam radius radius with 1 e 2 intensity at the beam waist a large beam divergence for a given beam radius corresponds to poor beam quality a low beam divergence can be important for applications such as pointing or free space optical communications beams with very small divergence i e with approximately constant beam radius over significant propagation distances are called collimated beams for the measurement of beam divergence one usually measures the beam radius at different positions using e g a beam profiler it is also possible to derive the beam divergence from the complex amplitude profile of the beam in a single plane spatial fourier transforms deliver the distribution of transverse spatial frequencies which are directly related to propagation angles see us laser corps application note 10 for a tutorial on how to measure the laser beam divergence with a lens and ccd camera beam astigmatism edit see also astigmatism astigmatism in a laser beam occurs when the horizontal and vertical cross sections of the beam focus at different locations along the beam path astigmatism can be corrected with a pair of cylindrical lenses the metric for astigmatism is the power of cylindrical lens needed to bring the focuses of the horizontal and vertical cross sections together astigmatism is caused by thermal lensing in nd yag slab amplifiers a slab that is sandwiched between two metal heat sinks will have a temperature gradient between the heat sinks the thermal gradient causes an index of refraction gradient that is very similar to a cylindrical lens the cylindrical lensing caused by the amplifier will make the beam astigmatic unmatched cylindrical lenses or error in placement of these optics propagation through a nonlinear uniaxial crystal common in nonlinear optic crystals the x and y polarized e fields experience different refractive indices not propagating through the center of a spherical lens or mirror astigmatism can easily be characterized by a ccd beam profiler by observing where the x and y beam waists occur as the profiler is translated along the beam path beam wander or jitter edit every laser beam wanders and jitters albeit a small amount the typical kinematic tip tilt mount drifts by around 100 μrad per day in a laboratory environment vibration isolation via optical table constant temperature and pressure and no sunlight that causes parts to heat a laser beam incident upon this mirror will be translated by 100 m at a range of 1000 km this could make the difference between hitting or not hitting a communications satellite from earth hence there is a lot of interest in characterizing the beam wander slow time scale or jitter fast time scale of a laser beam the beam wander and jitter can be measured by tracking the centroid or peak of the beam on a ccd beam profiler the ccd frame rate is typically 30 frames per second and therefore can capture beam jitter that is slower than 30 hz it cannot see fast vibrations due to one s voice 60 hz fan motor hum or other sources of fast vibrations fortunately this is usually not a great concern for most laboratory laser systems and the frame rates of ccds are fast enough to capture the beam wander over the bandwidth that contains the greatest noise power a typical beam wander measurement involves tracking the centroid of the beam over several minutes the rms deviation of the centroid data gives a clear picture of the laser beam pointing stability the integration time of the beam jitter measurement should always accompany the computed rms value even though the pixel resolution of a camera may be several micrometres sub pixel centroid resolution possibly tens of nanometer resolution is attained when the signal to noise ratio is good and the beam fills most of the ccd active area 11 beam wander is caused by slow thermalization of the laser laser manufacturers usually have a warm up specification to allow the laser to drift to an equilibrium after startup tip tilt and optical mount drift caused by thermal gradients pressure and loosening of springs non rigidly mounted optics vibration due to fans people walking sneezing breathing water pumps and movement of vehicles outside the laboratory misrepresentation of beam profiler measurements for laser systems edit it is to most laser manufacturers advantage to present specifications in a way that shows their product in the best light even if this involves misleading the customer laser performance specifications can be clarified by asking questions such as is the specification typical or worst case performance what beam width definition was used is the m 2 parameter for both vertical and horizontal cross sections or just for the better cross section was m 2 measured using the iso standard technique or some other way e g power in the bucket over how long was the data taken to come up with the specified rms beam jitter rms beam jitter gets worse as the measurement interval increases what was the laser environment optical table etc what is the warm up time needed to achieve the specified m 2 beam width divergence astigmatism and jitter techniques edit beam profilers generally fall into two classes the first uses a simple photodetector behind an aperture which is scanned over the beam the second class uses a camera to image the beam 12 scanning aperture techniques edit the most common scanning aperture techniques are the knife edge technique and the scanning slit profiler the former chops the beam with a knife and measures the transmitted power as the blade cuts through the beam the measured intensity versus knife position yields a curve that is the integrated beam intensity in one direction by measuring the intensity curve in several directions the original beam profile can be reconstructed using algorithms developed for x ray tomography the measuring instrument is based on high precision multiple knife edges each deployed on a rotating drum and having a different angle with respect to beam orientation scanned beam is then reconstructed using tomographic algorithms and provides 2d or 3d high resolution energy distribution plots because of the special scanning technique the system automatically zooms in onto the current beam size enabling high resolution measurements of sub micron beams as well as relative large beams of 10 or more millimeters to obtain measurement of various wavelength different detectors are used to allow laser beam measurements from deep uv to far ir unlike other camera based systems this technology also provides accurate power measurement in real time scanning slit profilers use a narrow slit instead of a single knife edge in this case the intensity is integrated over the slit width the resulting measurement is equivalent to the original cross section convolved with the profile of the slit this fusion between knife edge technology and tomographic algorithms creates a new field of beam profiling cket computerized knife edge tomography this creates capability of accurate measurement from a micron to over 10 millimeters with adaptable resolution over a wide spectrum range practically if a single surface detector exists for a certain wavelength region then using this technology an image like profile could be derived 13 these techniques can measure very small spot sizes down to 1 μm and can be used to directly measure high power beams they do not offer continuous readout although repetition rates as high as twenty hertz can be achieved also the profiles give integrated intensities in the x and y directions and not the actual 2d spatial profile integrating intensities can be hard to interpret for complicated beam profiles they do not generally work for pulsed laser sources because of the extra complexity of synchronizing the motion of the aperture and the laser pulses 14 ccd camera technique edit the ccd camera technique is simple attenuate and shine a laser onto a ccd and measure the beam profile directly it is for this reason that the camera technique is the most popular method for laser beam profiling the most popular cameras used are silicon ccds that have sensor diameters that range up to 25 mm 1 inch and pixel sizes down to a few micrometres these cameras are also sensitive to a broad range of wavelengths from deep uv 200 nm to near infrared 1100 nm this range of wavelengths encompass a broad range of laser gain media the advantages of the ccd camera technique are it captures the 2d beam profile in real time high dynamic range even a webcam s ccd chip has a dynamic range of around 2 8 15 software typically displays critical beam metrics such as d4σ width in real time sensitive ccd detectors can capture the beam profiles of weak lasers resolution down to about 4 μm depending on the pixel size in a special case a resolution of 1 1 μm was demonstrated 15 ccd cameras with trigger inputs can be used to capture beam profiles of low duty cycle pulsed lasers ccd s have broad wavelength sensitivities from 200 to 1100 nm the disadvantages of the ccd camera technique are attenuation is required for high power lasers ccd sensor size is limited to about 1 inch ccds are prone to blooming when used near the edge of their sensitivity e g close to 1100 nm 16 17 baseline subtraction for d4σ width measurements edit the d4σ width is sensitive to the beam energy or noise in the tail of the pulse because the pixels that are far from the beam centroid contribute to the d4σ width as the distance squared to reduce the error in the d4σ width estimate the baseline pixel values are subtracted from the measured signal the baseline values for the pixels are measured by recording the values of the ccd pixels with no incident light the finite value is due to dark current readout noise and other noise sources for shot noise limited noise sources baseline subtraction improves the d4σ width estimate as n displaystyle sqrt n where n displaystyle n is the number of pixels in the wings without baseline subtraction the d4σ width is overestimated averaging to get better measurements edit averaging consecutive ccd images yields a cleaner profile and removes both ccd imager noise and laser beam intensity fluctuations the signal to noise ratio snr of a pixel for a beam profile is defined as the mean value of the pixel divided by its root mean square rms value the snr improves as square root of the number of captured frames for shot noise processes dark current noise readout noise and poissonian detection noise so for example increasing the number of averages by a factor of 100 smooths out the beam profile by a factor of 10 attenuation techniques edit since ccd sensors are highly sensitive attenuation is almost always needed for proper beam profiling for example 40 db nd 4 or 10 4 of attenuation is typical for a milliwatt hene laser proper attenuation has the following properties it does not result in multiple reflections leaving a ghost image on the ccd sensor it does not result in interference fringes due to reflections between parallel surfaces or diffraction by defects it does not distort the wavefront and will be an optical element with sufficient optical flatness less than one tenth of a wavelength and homogeneity it can handle the required optical power for laser beam profiling with ccd sensors typically two types of attenuators are used neutral density filters and wedges or thick optical flats neutral density filters edit main article neutral density filter neutral density nd filters come in two types absorptive and reflective absorptive filters are usually made of tinted glass they are useful for lower power applications that involve up to about 100 mw average power above those power levels thermal lensing may occur causing beam size change or deformation because of the low thermal conductivity of the substrate usually a glass higher power may result in melting or cracking absorptive filter attenuation values are usually valid for the visible spectrum 500 800 nm and are not valid outside of that spectral region some fi...