Gaussian Beam Size

A.
For circular fundamental mode Gaussian beam, irradiance (intensity) profile is
     I(r) = exp(-2r²)                                              (1)
Here r is scaled by w (1/e² radius), i.e., when r=1, the irradiance drops to 1/e²=13.5% of the peak value.

1. D86 diameter
Assuming we measure power with a variable aperture whose diameter is ρ, the contained power is an integration of the irradiance:

P(ρ) = ∫

ρ 0

exp(-2r2)rdr

∫

2π 0

dθ =

π 2

[1-exp(-2ρ2)]

(2)

The total power is when ρ=∞: P_tot = P(ρ=∞) = π/2. Then
    P(ρ)/P_tot = 1-exp(-2ρ²)                                                                              (3)
The D86 diameter is when 86.5% of the total power is contained. From the above equation, D86 diameter is equal to the 1/e² diameter because P(ρ=1)/P_tot = 1-1/e² = 0.865.

2. D63 diameter
I see nowhere uses D63 diameter except one: the IEC Laser Safety Standard. The D63 diameter is when 63.2% of the total power is contained in a variable aperture. From Eq.(3), P(ρ=1/$\sqrt{2}$)/P_tot = 0.632. So D63 diameter is 1/$\sqrt{2}$=0.707 times the D86 diameter. From Eq.(1), I(1/$\sqrt{2}$)=1/e. So D63 diameter is the width at 1/e intensity points.

3. Knife edge width
Now we measure power with a traveling knife edge along the x-axis, the transmitted power is an integration of the irradiance:

P(a) = ∫

a -∞

exp(-2x2)dx

∫

+∞ -∞

exp(-2y2)dy =

π 4

[1+erf($\sqrt{2}$ a)]

(4)

where erf is error function. The total power P_tot=P(∞)=π/2. So
    P(a)/P_tot = 1/2 [1+erf($\sqrt{2}$ a)]                                                         (5)
From Eq.(5), P(-0.5)/P_tot = 0.159 and P(0.5)/P_tot = 0.841. This means that the knife edge traveling distance between the 15.9% and 84.1% power levels is equal to the 1/e² radius.
Alternatively, if one measures 10% - 90% power levels, the distance is equal to 1.281*1/e² radius.

4. Full width or half width at half maximum
Let I(r) = exp(-2r²) = 0.5, we get ln(0.5) = -2r², or r = $\sqrt{\mathrm{(ln2)/2}}$ = 0.5887. So when 1/e² radius is 1, the half width at half maximum is 0.5887.

5. D4 sigma (2nd moment) diameter
This is the ISO standard (ISO 11146-1). For the fundamental Gaussian, this definition and the traditional 1/e² definition are identical. This definition heavily weights the tails or outer wings of the intensity profile; so for non-ideal beams having side-lobes, the 2nd moment diameter can be substantially larger than their central lobe diameter. I will not go through the math detail here.

B.
For asymmetric 1-D Gaussian beam (Gaussian form at x, arbitrary form at y), irradiance (intensity) profile is
     I(x,y) = exp(-2x²) I(y)                                              (6)
Here x is scaled by w_x (1/e² half-width). I use "width" to replace "diameter" (and "half-width" to replace "radius") for non-circular beam.

1. D86 width
Instead of a circular aperture, a variable 1-D rectangular aperture is used here. Assuming a is the half-width of the aperture, the transmitted power is

P(a) = ∫

a -a

exp(-2x2)dx

∫

+∞ -∞

I(y)dy =

$\sqrt{\mathrm{\pi /2}}$

erf($\sqrt{2}$ a)

∫

+∞ -∞

I(y)dy

(7)

Divided by total power P_tot=P(∞),
   P(a)/P_tot = erf($\sqrt{2}$ a)                                                        (8)
Solve for erf($\sqrt{2}$ a)=0.865, we get a=0.747. So for 1-D Gaussian beam, D86 width is 0.747 times the 1/e² width.

2. D63 width
Using Eq.(8), solve for erf($\sqrt{2}$ a)=0.632, we get a=0.450. So for 1-D Gaussian beam, D63 width is 0.45 times the 1/e² width. Also for 1-D Gaussian beam, D63 width is 0.45/0.707=0.636 times smaller than that of the 2-D circular Gaussian beam.

3. Knife edge width
From Eq.(4), since the knife edge scan is a 1-D scan, the knife edge width is the same for 1-D and 2-D Gaussian.

4. FWHM
FWHM is an "intensity point relative to peak" type of width, it has nothing to do with contained power, so it is the same for 1-D and 2-D beams, i.e., it is still 0.5887 times the 1/e² width.

5. Second-moment width
2nd-moment width is inherently defined for separate x and y, so it is the same for 1-D and 2-D beams.

Summary:
Use this table to convert between different Gaussian beam sizes:

	Circular 2D Gaussian	1D Gaussian
1/e2 half-width	1	1
D86 half-width	1	0.747
1/e half-width	0.707	0.707
D63 half-width	0.707	0.45
Knife edge width 15.9%-84.1% clip	1	1
Knife edge width 10%-90% clip	1.281	1.281
HWHM	0.5887	0.5887

Note: for ideal fundamental Gaussian form only.

References:
[1] User Manual, ModeMaster PC - M² beam propagation analyzer. Coherent inc.
[2] A.E.Siegman, "How to Measure Laser Beam Quality". PDF copy available online.