Near Field and Far Field

1. In diffraction optics,

Fresnel number:

F =

R2 λz

(1)

where R is aperture's radius.

Far field:
     • F << 1.
     • It is the Fraunhofer diffraction area.
     • The diffraction pattern remains its shape; only its size gets bigger at farther distance.  

Near field:
     • F ≥ 1.
     • It is the Fresnel diffraction area.
     • The diffraction pattern changes shape as a function of distance.

 There is also an "extreme near field" region where the field observed behind the aperture is simply a geometrical projection of the aperture. Geometrical optics zone is when F >> 1.

2. For laser beam,

Rayleigh range:

z0 =

πW02 λ

=

π w02 λ M2

(2)

where W₀ is 1/e² waist radius and w₀ = W₀/M is the 1/e² waist radius of the embedded fundamental Gaussian beam. (i.e., a Gaussian beam and its embedded Gaussian beam have the same Rayleigh range).

Near field is near the laser beam waist, within its Rayleigh range.
Far field is outside of the Rayleigh range.

References:
[1] http://en.wikipedia.org/wiki/Fresnel_number
[2] 黄婉云，《傅里叶光学教程》，1985年第1版. Page 95.

Zemax POP macro: normalized plot and calculation of beam size

Zemax Physical Optics Propagation (POP) generates cross-sectional beam profile.  Two functions are lacking that are useful to me:
1. An irradiance-normalized plot. (The absolute "watts-per-sq-millimeter" value does not matter to me almost all the time).
2. Zemax gives "second-moment" beam diameter but again, almost all the time I just need a simple 1/e^2 diameter regardless of perfect Gaussian beam or not. (The second-moment beam diameter definition heavily weighs the tails or outer wings of the intensity profile. Therefore beams with side-lobes will have second-moment width substantially larger than their central lobe widths. [1])

So I wrote a macro file to do these two tasks:

!-----------------------------------------------------------
!FIRST, SAVE SETTINGS IN POP WINDOW. 
!The "Show As" in "Display" tab must be set as "Cross X (or Y)".
!----------------------------------------------------------- 
!Input POP parameters saved in POP window:
!-----------------------------------------------------------
sampling = 512
zoom = 8

AA$ = $TEMPFILENAME()
GETTEXTFILE AA$, POP    # Import POP text data file
OPEN AA$

!-----------------------------------------------------------
! Create a loop to read POP data
!-----------------------------------------------------------
invalid_row = 13        # The first 13 rows are text, not data
For i,1,invalid_row,1
    READ a,b
Next i

Grid_size = sampling/zoom        # POP grid size (=Sampling/zoom)
DECLARE xx, DOUBLE, 1, Grid_size+1                   
DECLARE Irrad, DOUBLE, 1, Grid_size+1
For i,1,Grid_size+1,1
    READ a,b
    xx(i) = a
    Irrad(i) = b
    print xx(i), ", ", Irrad(i)
Next i
CLOSE

!-----------------------------------------------------------
! Find peak
!-----------------------------------------------------------
Max_i = 1
For j,2,Grid_size+1,1
!    PRINT "j=",j,"xx=",xx(j),"Irrad=",Irrad(j)
    IF ( Irrad(j)>Irrad(j-1) ) & ( Irrad(j)>Irrad(Max_i) ) THEN Max_i = j
Next j

!-----------------------------------------------------------
! Normalize lineshape
!-----------------------------------------------------------
DECLARE Irrad_norm, DOUBLE, 1, Grid_size+1
For j,1,Grid_size+1,1
    Irrad_norm(j) = Irrad(j)/Irrad(Max_i)
    PRINT "j=",j," xx=",xx(j)," Irrad_norm=",Irrad_norm(j)
Next j
!-----------------------------------------------------------
! Find 1/e^2 radius at extreme left (linear interpolation)
!-----------------------------------------------------------
Radius_Li = 1
For j,1,Max_i,1
    IF ( Irrad_norm(j)<=0.135 ) & ( Irrad_norm(j+1)>=0.135 ) THEN GOTO 1    #once the condition is met, jump out loop.
Next j
LABEL 1
Radius_Li = j
R_L1 = xx(Radius_Li)
R_L2 = xx(Radius_Li+1)
Irrad_L1 = Irrad_norm(Radius_Li)
Irrad_L2 = Irrad_norm(Radius_Li+1)
R_L = (0.135-Irrad_L1)*(R_L1-R_L2)/(Irrad_L1-Irrad_L2) + R_L1
PRINT "At ", R_L, " Irradiance = 0.135"
!-----------------------------------------------------------
! Find 1/e^2 radius at extreme right (linear interpolation)
!-----------------------------------------------------------
Radius_Ri = 1
For j,Max_i,Grid_size,1
    IF ( Irrad_norm(j)>=0.135 ) & ( Irrad_norm(j+1)<=0.135 ) THEN Radius_Ri = j+1
Next j
R_R1 = xx(Radius_Ri)
R_R2 = xx(Radius_Ri-1)
Irrad_R1 = Irrad_norm(Radius_Ri)
Irrad_R2 = Irrad_norm(Radius_Ri-1)
R_R = (0.135-Irrad_R1)*(R_R1-R_R2)/(Irrad_R1-Irrad_R2) + R_R1
PRINT "At ", R_R, " Irradiance = 0.135"
!-----------------------------------------------------------
! Calculate 1/e^2 beam diameter
!-----------------------------------------------------------
dia = R_R - R_L
PRINT "Beam Diameter = ", dia
!-----------------------------------------------------------
! Plot normalized lineshape
!-----------------------------------------------------------
PLOT NEW
PLOT DATA, xx,Irrad_norm,Grid_size+1,0,0,0
PLOT BANNER, "Normalized POP beam profile"
PLOT RANGEX, xx(1), xx(Grid_size+1)
PLOT RANGEY, 0, 1
PLOT COMM1, $DATE()
PLOT COMM2, $FILEPATH()
PLOT COMM3, "1/e^2 Gaussian Diameter = ", $STR(dia)
PLOT GO

Save the above text as a "xxx.zpl" file in Zemax's ZPL folder (the folder path can be viewed/changed in File -> Preferences). To run this macro, press F9, choose this file and click "Execute".

Here is an example. The POP window generated by Zemax shows an aberrated beam profile:


My macro generates a plot shown below.  First, it is normalized.  Second, the 1/e^2 beam diameter is given.



Compare three beam sizes:
1. POP pilot beam radius (paraxial 1/e^2) = 0.0070
2. POP "second moment" beam radius = 0.0071
3. Simple 1/e^2 radius calculated by my POP macro = 0.0024
The three sizes are different. Most often I just need know the 3rd value.

Next: It would be nice to also generate a Gaussian-fit curve.  I will perhaps write a MATLab-linked code to do this task.

Reference
[1] A.E.Siegman, "How to measure laser beam quality". PDF available on web by googling it.

Zemax as Gaussian beam calculator

This is version 2 of the last note on this subject.

Example 1: Given an input Gaussian beam of waist radius W01, a lens of focal length f, what is the output beam's waist radius W02 and waist location z2?

To quickly calculate the paraxial result, a simple Zemax model can be created as follows. In this example, W01 = 0.5 mm, z1 = 10 mm, f = 50 mm and wavelength = 640 nm.

The aperture and field settings are irrelevant to Gaussian calculation; they are set for making the optical layout look good:

Pressing [Ctrl]+[b] opens the Paraxial Gaussian Beam Data window. Right-click mouse opens the setting window:

So the waist radius (1/e^2 value) is set to 0.5 mm at the surface 1. Press OK the paraxial Gaussian beam results are given:

Notice that the image plane, the geometrical focus, is not exactly at the Gaussian beam waist. So a simple Merit Function can be used to move the image plane to the waist:

Optimize it and the image plane is adjusted to be at the waist. As the result, the Gaussian beam focus is about 66 um in front of the geometrical focus:

Example 2: We still have the same input Gaussian beam W01 = 0.5 mm and the lens f = 50 mm, but we want the focused beam waist radius to be 5 um. To do this, another lens is needed to expand the input beam first; say we have a f = -10 mm lens in hand. Where does this lens need to be? (d = ?)

To solve this, simply insert the lens f1 = -10 mm into the Zemax model:

Add in a GBPW operand into the Merit Function to calculate Gaussian waist. The target value is 5 um:

After optimization, Zemax gives the correct thickness of lens 1 to be 80.74 mm and the Paraxial Gaussian result shows that the beam is focused to 5 um radius:

Note that these are the first-order, paraxial calculations. Real lenses have aberrations and usually cannot give exactly the same results and will usually yield strong side-lobes. The next step should be to replace with the real lens models and use POP for actual beam evaluation.