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Description
Laser beams are often described using a beam quality metric called the "M2" value, which attempts to capture the characteristic beam propagation properties of a real source in comparison to an ideal TEM00 Gaussian mode. It is important to note that the M2 description of a laser beam is incomplete since it does not specify the beam profile. However, the M2 metric can serve as a useful approximation when detailed information is not readily available. For an overview of beam quality metrics, and the M2 parameter in particular, please refer to Siegman's paper, "How to (maybe) Measure Laser Beam Quality".
The M2 method relies on the fact that the second moment of the beam's irradiance distribution, measured transverse to the direction of propagation, rigorously propagates in free space regardless of the specific characteristics of the beam's profile or modal composition. Furthermore, since the second moment can be taken as a metric for the width of a beam, the M2 method then allows for rigorous propagation and evaluation of the beam's width.
Since the M2 beam parameter fundamentally describes the deviation of an arbitrary beam from an ideal TEM00 mode Gaussian, it is useful to have the concept of an "embedded Gaussian" inside of the aberrated beam being defined. Consider an aberrated beam at wavelength l whose divergence angle is defined by its standard deviation, s, in radians. The ideal, embedded TEM00 mode Gaussian will have the following properties:
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Qe
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Divergence angle in radians
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2*s
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W0e
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Beam waist radius at z=0
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l / (p * tan(Qe))
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We(z)
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Beam radius at z
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l * z / (p * W0e)
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Relationships between the embedded Gaussian and the standard deviation of the divergence angle
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The aberrated beam will have the following relationships with the embedded Gaussian:
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W0a
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Beam waist radius at z=0
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Sqrt(M2) * W0e
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s20a
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Second moment at z=0
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(W0a)2 / 4
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Wa(z)
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Beam radius at z
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Sqrt(M2) * We(z)
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s2a(z)
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Second moment at z
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(Wa(z))2 / 4
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Relationships of the aberrated beam widths and second moments to the M2 and embedded Gaussian
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The M2 laser beam source is created in FRED as a superposition of TEMmn Hermite-Gaus modes whose composite beam has the requested M2 value to better than 1% for modest divergence angles and M2 <= 3. Because the source is intended to be coherent and FRED treats coherent rays of the same wavelength fully coherent with respect to each other, the coherence between the individual TEMmn modes is broken by making slight changes to the wavelength of each individual mode. The user can retrieve the mode composition by right mouse clicking on the source node in the object tree and selecting "Detailed Report" from the context menu. As part of the output, the individual TEMmn modes, amplitudes and unique wavelengths will be listed.
Navigation
This feature can be accessed in the following ways:
•Menu > Create > Source Primitive > M-Squared Laser Beam (coherent)
•Right mouse click on the Optical Sources folder, select Create New Source Primitive > M-Squared Laser Beam (coherent)
•Toolbar button: .png)
Controls
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Control
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Inputs / Description
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Defaults
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Logical Parent
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Read-only. Specifies the source's parent node on the tree.
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Optical Sources
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Name
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Name of the source as it will appear on the tree view.
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M-Squared Laser Beam N
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Description
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Description string that will be visible on the tree view.
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Parameter Attributes
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0
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Total power of the source specified in Watts.
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1.0
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1
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Number of sample points across the full width of the beam in the X direction. The total number of rays generated by the source is related to both the number of sample points and the divergence angle, with the number of sampling points being used as a guide for FRED's internal field synthesis routine. It is recommended to change the number of sample points in steps of 2n.
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32
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2
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Number of sample points across the full width of the beam in the Y direction. The total number of rays generated by the source is related to both the number of sample points and the divergence angle, with the number of sampling points being used as a guide for FRED's internal field synthesis routine. It is recommended to change the number of sample points in steps of 2n.
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32
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3
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Divergence angle of the aberrated beam in the X direction whose interpretation is given by parameter 5. Must be a value greater than 0 and less than 90.
When using the Variance Power Half Width (second moment) as the angle interpretation (parameter 5), the beam divergence angle should be calculated in the following manner. A beam with semi-width W0_x at Z=0 is desired. The divergence angle will be:
angle_x = 0.5 * Sqrt(M2_x) * arctan( lambda / PI / W0_x )
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10
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4
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Divergence angle of the aberrated beam in the Y direction whose interpretation is given by parameter 5. Must be a value greater than 0 and less than 90.
When using the Variance Power Half Width (second moment) as the angle interpretation (parameter 5), the beam divergence angle should be calculated in the following manner. A beam with semi-width W0_y at Z=0 is desired. The divergence angle will be:
angle_y = 0.5 * Sqrt(M2_y) * arctan( lambda / PI / W0_y )
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10
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5
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Internally, FRED uses the 1/e semi-angle of the field amplitude when constructing the Gaussian beam. This parameter informs FRED as to the physical meaning of the divergence angle values specified in parameters 3 and 4 of the primitive and allows FRED to convert the supplied angle values to its internal 1/e amplitude semi-angle specification. Width designations using "power" correspond to the power profile of the Gaussian function and those using "amplitude" correspond to the amplitude profile of the Gaussian function.
For example, if this parameter specifies that the user-supplied divergence angles correspond to "Full width at 1/e amplitude", FRED will internally scale the supplied angles by 0.5 in order to arrive at the 1/e semi-angle in field amplitude used internally in the construction of the Gaussian beam.
Variance power specifications (second moment) are typically used for beams with M2 values greater than 1.
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Half width at 1/e amplitude point
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6
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M2 value of the beam in the X direction. The algorithm used works best for M2 < 3 and will deteriorate as the divergence angle and M2 values increase.
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1.0
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7
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M2 value of the beam in the Y direction. The algorithm used works best for M2 < 3 and will deteriorate as the divergence angle and M2 values increase.
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1.0
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Wavelength Attributes
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Single
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The individual modes comprising the M2 source will be generated at wavelength values offset by small deltas from the designated Single wavelength value. The wavelength units are microns.
Right mouse click on the source node on the tree and select "Detailed Report" to get a listing of the individual mode wavelengths and modal composition.
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Default wavelength Preference
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Source Draw Color
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The ray positions and ray trajectories will be rendered with the selected color.
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Polarization
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Polarization
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If checked, polarization data for the rays is maintained and stored.
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Unchecked
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Handedness
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Sets the handedness of the polarization state (relevant for non-linear polarization states). If the ray is propagating towards you, the electric field vector rotates in a clockwise direction for Right handedness and counter-clockwise for Left handedness.
Note that for linear polarization, the application of Left or Right handedness is arbitrary. The user may find that the UI display switches handedness depending on the angle of the linear polarization state entered, but this will have no impact on the resulting representation of the linear state.
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Right
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Ellipticity
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Sets the ellipticity of the polarization state, 0 represents linear polarization and 1 represents circular.
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0
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Angle
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Sets the angle of the polarization relative to the X axis.
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90
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OK
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Accept settings and close dialog box.
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Cancel
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Discard settings and close dialog box.
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Apply
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Accept settings and keep dialog box open.
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Help
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Access this Help page.
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Application Notes
Example: Symmetric Beam with M2 = 3.2
A multi-mode source with a center wavelength of 0.6328 microns has a beam semi-width of 2 mm at Z=0 and is characterized by an M2 value of 3.2. The following parameters are used to construct the source primitive having these attributes.
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Parameter
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Description
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Number of samples in X = 129
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Must be a power of 2^n + 1. A larger number indicates better sampling of the field at the expense of more rays.
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Number of samples in Y = 129
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Must be a power of 2^n + 1. A larger number indicates better sampling of the field at the expense of more rays.
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X divergence angle (deg) = 0.00516
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Calculated as 0.5*Sqrt(3.2)*arctan(0.6328 / PI / 2.0).
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Y divergence angle (deg) = 0.00516
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Calculated as 0.5*Sqrt(3.2)*arctan(0.6328 / PI / 2.0).
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Meaning of divergence angle = "Variance power half width"
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Sets the interpretation of the X and Y divergence angle values, which have been calculated for a variance power half-width specification.
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M2 value in X = 3.2
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Beam M2 value in the X profile.
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M2 value in Y = 3.2
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Beam M2 value in the Y profile.
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With sigma being the second moment of the irradiance profile, the beam at the source plane has a 2*sigma semi-width of 1.98 mm. At Z=200 meters, the beam has a 2*sigma semi-width of 65.3 mm. The M2 value is then calculated from the "near-field" and "far-field" beam semi-widths as (1.98 mm) * (65.3 mm) * PI / (0.0006328 mm) / (200,000 mm) = 3.2.
Related Topics
Source Primitives
Plane Wave (coherent)
Point Source (coherent)
Laser Beam (00 mode)
Astigmatic Gaussian Beam
Laser Diode Beam (coherent)
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Copyright © Photon Engineering, LLC
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