Laser Beam Collimation

This post discusses one question: how to achieve the best collimation of a laser beam?

To collimate a laser diode or fiber's output using a lens, a common belief is to make the output beam's waist location as distant as possible form the lens.  For example, see Saleh and Teich [1].  Well, this is only approximately correct.

The best collimation is achieved when the output beam has either:
(1) the smallest divergence angle; or
(2) the largest waist size; or
(3) the longest Rayleigh range.
The above three conditions are equivalent, i.e., achieving any single one means the other two are achieved simultaneously.

To reach the above conditions, the input waist needs be at exactly the object focal plane of the collimating lens. Then the output beam waist will be at exactly the conjugate (image) focal plane [2]:

Figure 1: Geometry of the perfect collimation.

I plot the output waist location (relative to the image focal plane) and the output waist diameter as a function of the input waist location (relative to the object focal plane). This plot shows that (a) output waist size is at maximum and (b) waist location is at the image focal plane when input beam waist is at the object focal plane. (Assuming input beam waist diameter = 5 um, lens focal length = 10 mm, wavelength = 1 um.)

Figure 2: Output beam properties vs. input beam waist location. Notice my custom sign rule: if "relative to" is at left of the reference point, the sign is plus. For example, positive waist location means it is at the left side of the focal plane.

This figure also indicates that the output beam's waist location cannot be placed at infinity.  The farthest distance one can make is when the input beam's waist is located at Z_R + f from the lens (Z_R is input beam's Rayleigh range and f is focal length). This is the common belief that the best collimation should be reached; however, the output waist size is not the largest.



References:
[1] Saleh and Teich, Fundamentals of Photonics, 2nd Ed. Page 90.
[2] Sidney A. Self, "Focusing of spherical Gaussian beams," Applied Optics 22, 658 (1983).

Diffraction theory summary

This is a summary of the scalar diffraction theory after reading Ref. 1 and 2.

0. Scalar diffraction theory neglects the vectorial nature of the e&m wave. Two conditions need be met for scalar theory to produce accurate result: (1) the diffracting aperture larger than wavelength; and (2) the observing field not too close to the aperture. (Ref.1, p.35)

1. Scalar diffraction formula with least approximation:

In figure above, Σ is the diffration aperture, P₁ is a point in it and U₀(P₁) is the complex amplitude waiting to be diffracted, P is the point we want to solve. The complex amplitude at P is:

U(P) =

1 jλ

∫∫ U0(P1)

ejkr r

cos(n, r) ds

(1)

The approximation needed here is r >> λ . This is the Rayleigh-Summerfeld solution.

In frequency domain,  let G_z(f_x, f_y) and G_z=0(f_x, f_y) be the Fourier transform of U(P) and U₀(P₁) respectively; they are the amplitudes of each fundamental plane-waves (decomposed by Fourier transform).  By solving Helmholtz wave equation, Eq.(1) in frequency domain becomes (Ref.2, p.83):
     G_z(f_x, f_y) = G₀(f_x, f_y) exp[jkz(1-α²+β²)^1⁄₂]                     (2)
where
    f_x = α/λ,  f_y = β/λ                                                                  (2a)
are the spatial frequencies on 2-D planes normal to the z-axis; they depend on these plane-wave's propagation direction and α, β are the directional cosines relative to x and y axes respectively.  Eq.(1) analyzed in frequency domain is called "Angular Spectrum" approach (Ref.1, p.55).

2. Fresnel diffraction:
If further approximation is made:
    cos(n,r) ≈ 1                                                                                (3)
then

U(x, y) =

ejkz jλz

∫∫ U0(x1, y1) exp[jk

(x-x1)2+(y-y1)2 2z

] dx1dy1

(4)

2.1
Eq.(4) can be written as:

U(x, y) =

ejkz jλz

∫∫ U0(x1, y1) h(x-x1, y-y1) dx1dy1

(5)

=

ejkz jλz

U0(x, y) ∗ h(x, y)

(5a)

where

h(x, y) = exp(jk

x2+y2 2z

)

(5b)

Eq.(5)-(5b) have the following physical meaning:
• Eq.(5b): the impulse response h is the complex amplitude of the spherical wave from a unit point source at (x, y) = (0, 0).
• Eq.(5a): the diffracted field is simply a convolution between the initial field U₀ and the impulse response h.
• Fresnel diffraction is a space-invariant linear system.

2.2
Eq.(4) can also be written as:

U(x, y) =

ejkz jλz

exp(jk

x2+y2 2z

) F[U0(x1, y1) exp(jk

x12+y12 2z

)]

(6)

Eq.(6) means:
• The diffracted field is a Fourier transform of U₀ multiplied by a diverging spherical wave.

3. Fraunhofer diffraction:
If an even more stringent approximation is made:

z >>

(x12+y12)max 2λ

(7)

then

U(x, y) =

ejkz jλz

exp(jk

x2+y2 2z

) F[U0(x1, y1)]

(8)

So Fraunhofer diffraction is simple: the diffracted field is simply a Fourier transform of the initial field.

Eq.(7) is more often expressed by "Fresnel number":

F =

R2 λz

(7a)

where R is aperture's radius. Fraunhofer diffraction is when F << 1. 

In summary,
     • When diffraction distance >> λ , scalar theory can be used. In frequency domain, the angular spectrum approach is very physically intuitive.
     • When diffraction angle is small (Eq.(3)), Fresnel diffraction is used.
     • When Eq. (3) and (7) are both satisfied, Fraunhofer diffraction can be used.

References:
[1] Goodman, "Introduction to Fourier Optics", 2nd Ed.
[2] 黄婉云，《傅里叶光学教程》，1985年第1版.

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.