Instruments and techniques were developed to obtain the beam characteristics listed above. These include:
As of 2002, commercial knife-edge measurement systems cost $5,000–$12,000 USD and CCD beam profilers cost $4,000–9,000 USD. The cost of CCD beam profilers has come down in recent years, primarily driven by lower silicon CCD sensor costs, and as of 2008 they can be found for less than $1000 USD.
D4σ is shorthand for the diameter than is 4 times σ, where σ is the standard deviation of the horizontal or vertical marginal distribution. Mathematically, the D4σ beam width in the x-dimension for the beam profile is expressed as
is the centroid of the beam profile in the x-direction. The wings of the beam profile influence the D4σ value more than the center of the beam profile since the wings of the marginal distribution are weighted by the square of its distance, x2, from the center of the beam. If the beam does not fill more than a third of the beam profiler’s sensor area, then there will be a significant number of pixels at the edges of the sensor that register a small baseline value (the background value). If the baseline value is large or if it is not subtracted out of the image, then the computed D4σ value will be larger than the actual value because the baseline value near the edges of the sensor are weighted in the D4σ integral by x2. Therefore, baseline subtraction is necessary for accurate D4σ measurements. The baseline is easily measured by recording the average value for each pixel when the sensor is not illuminated. The D4σ width, unlike the FWHM and 1/e2 widths, is meaningful for multimodal marginal distributions — that is, beam profiles with multiple peaks — but requires careful subtraction of the baseline for accurate results. The D4σ is the ISO international standard definition for beam width.
Before the advent of the CCD beam profiler, the beam width was estimated using the knife-edge technique. The technique is as follows: slice a laser beam with a razor and measure the power of the clipped beam as a function of the razor position. The measured curve is the integral of the marginal distribution, and starts at the total beam power and decreases monotonically to zero power. The width of the beam is defined as either the distance between the points of the measured curve that are 10% and 90% (or 20% and 80%) of the maximum value. If the baseline value is small or subtracted out, the knife-edge beam width always corresponds to 60%, in the case of 20/80, or 80%, in the case of 10/90, of the total beam power no matter what the beam profile. On the other hand, the D4σ, 1/e2, and FWHM widths encompass fractions of power that are beam-shape dependent. Therefore, the 10/90 or 20/80 knife-edge width is a useful metric when the user wishes to be sure that the width encompasses a fixed fraction of total beam power. Most CCD beam-profiler software can compute the knife-edge width numerically.
The 1/e2 width is equal to the distance between the two points on the marginal distribution that are 1/e2 = 0.135 times the maximum value. If there are more than 2 points that are 1/e2 times the maximum value, then the two points closest to the maximum are chosen. The 1/e2 width only depends on 3 points on the marginal distribution, unlike D4σ and knife-edge widths that depend on the integral of the marginal distribution. 1/e2 width measurements are noisier than D4σ width measurements for each collected CCD frame. For multimodal marginal distributions (a beam profile with multiple peaks), the 1/e2 width usually does not yield a meaningful value and can grossly underestimate of the inherent width of the beam. For multimodal distributions, a D4σ width would be a better choice.
The American National Standard Z136.1-2007 for Safe Use of Lasers (p.6) defines the beam diameter as the distance between diametrically opposed points in that cross-section of a beam where the power per unit area is 1/e (0.368) times that of the peak power per unit area. This is the beam diameter definition that is used for computing the maximum permissible exposure to a laser beam. In addition, the Federal Aviation Administration also uses the 1/e definition for laser safety calculations in FAA Order 7400.2F, "Procedures for Handling Airspace Matters," February 16, 2006, p. 29-1-2.
The D86 width is defined as the diameter of the circle that is centered at the centroid of the beam profile and contains 86% of the beam power. The solution for D86 is found by computing the area of increasingly larger circles around the centroid until the area contains 0.86 of the total power. Unlike the previous beam width definitions, the D86 width is not derived from marginal distributions. The strange percentage of 86, rather than 50, 80, or 90, is chosen because a circular Gaussian beam profile integrated down to 1/e2 of its peak value contains 86% of its total power. The D86 width is often used in applications that are concerned with knowing exactly how much power is in a given area. For example, high-energy laser weapons and ladars are two applications that require precise knowledge of how much transmitted power actually illuminates the target.
Beam quality is important in many applications. In fiber-optic communications beams with an M2 close to 1 are required for coupling to single-mode optical fiber. Laser machine shops care a lot about the M2 parameter of their lasers because the beams will focus to an area that is M2 times larger than that of a Gaussian beam with the same wavelength and D4σ waist width; in other words, the fluence scales as 1/M2. The general rule of thumb is that M2 increases as the laser power increases. It is difficult to obtain excellent beam quality and high average power (100 W to kWs) due to thermal lensing in the laser gain medium.
The M2 parameter is determined experimentally as follows:
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.
The M2 parameter is not the whole story in specifying beam quality. A low M2 only implies that the second moment of the beam profile expands slowly. Nevertheless, two beams with the same M2 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 M2, 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. 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 M2 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.
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.
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 can’t 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.
Beam wander is caused by:
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:
The knife-edge slit technique is a method of quantifying the width of a laser beam. The technique involves chopping the beam with a knife and measuring 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 for several directions, the original beam profile can be reconstructed – but not in real time – using algorithms developed for x-ray tomography. Instead of a knife, sometimes a scanning slit is used. 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. The advantages of the knife-edge technique are that it can measure very small spot sizes down to 1 μm, and can be used for high power lasers. The disadvantages are that the knife-edge technique does not offer real-time readout, measures the 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), and does not work for pulsed laser sources. Knife-edge beam profiling instruments often only work for continuous wave lasers because of the extra complexity of synchronizing the rotating knife-edge and the laser pulses.
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:
The disadvantages of the CCD camera technique are that attenuation is required for high power lasers, and CCD sensor size limited to about 1 inch.
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 , where is the number of pixels in the wings. Without baseline subtraction, the D4σ width is overestimated.
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.
For laser beam profiling with CCD sensors, typically two types of attenuators are used: neutral density filters, and wedges or thick optical flats.
Optical wedges and reflections from uncoated optical glass surfaces are used to attenuate high power laser beams. About 4% is reflected from the air/glass interface and several wedges can be used to greatly attenuate the beam to levels that can be attenuated with ND filters. The angle of the wedge is typically selected so that the second reflection from the surface does not hit the active area of the CCD. The farther the CCD is from the wedge, the smaller the angle required. Wedges have the disadvantage of both translating and bending the beam direction — paths will no longer lie on convenient rectangular coordinates. Rather than using a wedge, an optical-quality thick glass plate tilted to the beam can also work — actually, this is the same as a wedge with a 0° angle. The thick glass will translate the beam but it will not change the angle of the output beam. The glass must be thick enough so that the secondary reflection does not illuminate the active area of the CCD. The Fresnel reflection of a beam from a glass plate is different for the s- and p-polarizations (s is parallel to the surface of the glass, and p is perpendicular to s) and changes as a function of angle of incidence – keep this in mind if you expect that the two polarizations have different beam profiles. To prevent distortion of the beam profile, the glass should be of optical quality — surface flatness of λ/10 (λ=633 nm) and scratch-dig of 40-20 or better. A half-wave plate followed by a polarizing beam splitter form a variable attenuator and this combination is often used in optical systems. The variable attenuator made in this fashion is not recommended for attenuation for beam profiling applications because: (1) the beam profile in the two orthogonal polarizations may be different, and (2) the polarization beam cube may have a low optical damage threshold value. Inexpensive cube polarizers are formed by cementing two right angle prisms together. The glue does not stand up well to high powers — the intensity should be kept under 500 mW/mm2. Single-element polarizers are recommended for high powers.
There are two competing requirements that determine the optimal beam size on the CCD detector. One requirement is that the entire energy — or as much of it as possible — of the laser beam is incident on the CCD sensor. This would imply that we should focus all the energy in the center of the active region in as small a spot as possible using only a few of the central pixels to ensure that the tails of the beam are captured by the outer pixels. This is one extreme. The second requirement is that we need to adequately sample the beam profile shape. As a general rule of thumb, we want at least 10 pixels across the area that encompasses most, say 80%, of the energy in the beam. Therefore, there is no hard and fast rule to select the optimal beam size. As long as the CCD sensor captures over 90% of the beam energy and has at least 10 pixels across it, the beam width measurements will have some accuracy.
The larger the CCD sensor, the larger the size of beam that can be profiled. Sometimes this comes at the cost of larger pixel sizes. Small pixels sizes are desired for observing focused beams. A CCD with many megapixels is not always better than a smaller array since readout times on the computer can be uncomfortably long. Reading out the array in real-time is essential for any tweaking or optimization of the laser profile.
A far-field beam profiler is nothing more than profiling the beam at the focus of a lens. This plane is sometimes called the Fourier plane and is the profile that one would see if the beam propagated very far away. The beam at the Fourier plane is the Fourier transform of the input field. Care must be taken in setting up a far-field measurement. The focused spot size must be large enough to span across several pixels. The spot size is approximately fλ/D, where f is the focal length of the lens, λ is the wavelength of the light, and D is the diameter of the collimated beam incident upon the lens. For example, a helium-neon laser (633 nm) with 1 mm beam diameter would focus to a 317 μm spot with a 500 mm lens. A laser beam profiler with a 5.6 μm pixel size would adequately sample the spot at 56 locations.
The prohibitive costs of CCD laser beam profilers in the past have given way to low-cost beam profilers. Low-cost beam profilers have opened up a number of new applications: replacing irises for super-accurate alignment and simultaneous multiple port monitoring of laser systems.
In the past, alignment of laser beams was done with irises. Two irises uniquely defined a beam path; the farther apart the irises and the smaller the iris holes, the better the path was defined — that is, only a few light rays could be drawn through both irises. The smallest aperture that an iris can define is about 0.8 mm. In comparison, the centroid of a laser beam can be determined to sub-micrometre accuracy with a laser beam profiler. The laser beam profiler's effective aperture size is three orders of magnitude smaller than that of an iris. Consequently, the ability to define an optical path is 1000 times better when using beam profilers over irises. Applications that need microradian alignment accuracies include earth-to-space communications, earth-to-space ladar, master oscillator to power oscillator alignment, and multi-pass amplifiers.
Experimental laser systems benefit from the use of multiple laser beam profilers to characterize the pump beam, the output beam, and the beam shape at intermediate locations in the laser system, for example, after a Kerr-lens modelocker. Changes in the pump laser beam profile indicate the health of the pump laser, which laser modes are excited in the gain crystal, and also determine whether the laser is warmed up by locating the centroid of the beam relative to the breadboard. The output beam profile is often a strong function of pump power due to thermo-optical effects in the gain medium.