Basic Examples (4) 

Gaussian beam formula:

In[1]:=

Out[1]=

Sample beam intensity along x-axis:

In[2]:=

Out[2]=

In[3]:=

Out[3]=

Parameter table for given inputs:

In[4]:=

Out[4]=

Plot beam intersection along the optical axis:

In[5]:=

Out[5]=

Scope (15) 

Profiles (4) 

Available beam profiles:

In[6]:=

Out[6]=

---

Beam intensity as a function of coordinates:

In[7]:=

Out[7]=

Plot it along z-axis:

In[8]:=

Out[8]=

---

Two-dimensional cross-section of beam amplitude for given z:

In[9]:=

Out[11]=

---

When no parameters are supplied, QuantityVariables are used instead:

In[12]:=

Out[12]=

In[13]:=

Out[13]=

Coordinate systems (3) 

By default, Cartesian coordinate system is used:

In[14]:=

Out[15]=

Cylindrical coordinates are also supported:

In[16]:=

Out[16]=

The two systems are equivalent:

In[17]:=

Out[17]=

In[18]:=

Out[18]=

Computed parameters (4) 

Available computed parameters:

In[19]:=

Out[19]=

---

Summary of all properties:

In[20]:=

Out[20]=

---

Beam radius formula:

In[21]:=

Out[21]=

Beam radius as a function of longitudinal position:

In[22]:=

Out[22]=

Beam radius for a specific longitudinal position:

In[23]:=

Out[23]=

Numerical parameter values are supported:

In[24]:=

Out[24]=

---

Syntax for most parameters follows that for beam radius:

In[25]:=

Out[25]=

Input values (4) 

Input parameters can also have units:

In[26]:=

Out[26]=

---

Define dependent parameters by specifying beam waist w₀ and Rayleigh range z_R:

In[27]:=

Out[27]=

---

Input the two values as Association:

In[28]:=

Out[28]=

Short keys are supported:

In[29]:=

Out[29]=

---

When input as Association, parameters other than beam waist and Rayleigh range can be supplied:

In[30]:=

Out[30]=

Compare with:

In[31]:=

Out[31]=

Applications (2) 

Interactive calculator of Gaussian beam parameters:

In[32]:=

Out[32]=

Do not evaluate dependent parameters and use default values of independent parameters for nice rendering:

In[33]:=

Out[33]=

In[34]:=

Out[34]=

Compare with evaluated formula:

In[35]:=

Out[35]=

Properties and Relations (4) 

Formulas are normalized such that the total intensity for a given longitudinal position is unity:

In[36]:=

Out[36]=

In[37]:=

Out[37]=

Beam cross section is given by Gaussian (normal) distribution:

In[38]:=

Out[39]=

In[40]:=

Out[40]=

Intensity is the square of real amplitude:

In[41]:=

Out[41]=

Complex amplitude contains both the amplitude and phase:

In[42]:=

Out[42]=

Curvature is the reciprocal of the radius of curvature:

In[43]:=

Out[43]=

For numerically heavy calculations with Gaussian profile use GaussianMatrix:

In[44]:=

Out[46]=

Compare times:

In[47]:=

Out[47]=

Possible Issues (4) 

For convenience, radius of curvature at z=0 is set to infinity:

In[48]:=

Out[48]=

Note however that limits from above and below zero differ:

In[49]:=

Out[49]=

In[50]:=

Out[50]=

Not all sets of input parameters define the beam uniquely:

In[51]:=

Out[51]=

Supply all necessary parameters:

In[52]:=

Out[52]=

Some sets of input parameters are overdetermined:

In[53]:=

Out[53]=

Pick only the independent parameters:

In[54]:=

Out[54]=

Assignment of two different values to the same parameter is not allowed:

In[55]:=

Out[55]=

Pick only one of them:

In[56]:=

Out[56]=

Neat Examples (2) 

Plot high-intensity region around the waist of z-axis-aligned beam:

In[57]:=

Out[58]=

Plot intensity relative to the optical axis intensity:

In[59]:=

Out[60]=

Plot wavefronts (might take a couple of minutes to compute):

In[61]:=

Out[62]=

In[63]:=

Out[64]=