Phase, Phase radius, Gaussian beam, divergence angle, pilot beam, phase data, Rayleigh range……In this example, we will share how to interpret the optical physical optics parameters in a physical optics propagation

The lens example that we’ll evaluate is shown below; it’s a two-lens system. The first lens collimates the beam and the second lens focuses the beam. Both lenses are aspheric singlets that carry an r4 aspheric term to correct spherical aberration. There is a small central obscuration in the collimated portion of the beam. The wavelength for the system is set to 1 um.

The final irradiance the POP as below:

How to setup the POP analysis

We want to launch a Gaussian beam into the system, as if it’s fed by a fiber optic. In the System Explorer shown below, the Aperture Type is set to Object Space NA and the Aperture Value to 0.05. This corresponds to a beam divergence angle of about 2.9° and a Gaussian beam waist semi-diameter of about 6.4 µm, using the equations below.

Despite these settings, it is important to note that the layout plot shows geometric rays only! The layout plot is not a representation of a true Gaussian beam.

In the POP setting.

Beams are always centered on the chief ray for the selected field and wavelength. Therefore, the data in the beam file is positioned relative to the chief ray. The center point in the beam file is at the coordinate (nx/2+1, ny/2+1).

The analysis window may include some of the following data:

- Display X Width/ Y Height: The width and height of the beam array in lens units at the end surface.
- Peak Irradiance: The maximum irradiance in power per area.
- Total Power: The total power in the beam.
- Pilot Beam Data: The Pilot Beam data includes the radial beam size, beam waist, position, and Rayleigh range.
- Beam Width: The X, Y beam width calculated using second moment method.

It should be noted that because of the complexity of the POP calculations, we must always examine the beam information at every surface to ensure the calculation is behaving correctly.

In the text file, it is the same as graph report.

Let’s switch to the POP report, and use surface 2 data to interpret and verify a few key parameters：

Let’s first understand what is pilot beam:

This criterion is named Gaussian pilot beam and fixes the best propagation method (among angular spectrum, Fresnel and Fraunhofer diffraction) by looking at the behavior of a Gaussian beam piloted from the aperture position and the observation position. A pilot beam is used to assist the physical optics propagation algorithm in determining which propagation algorithm to select. The pilot beam is an ideal Gaussian beam, with a waist, beam size, phase radius, and relative z position. The initial parameters may be generated by fitting the Gaussian beam equations to the initial distribution. The pilot beam is then propagated from surface to surface. At each surface, new beam parameters, such as the new waist, phase radius, or position are computed.

- Consider a Gaussian beam with waist ω0 . The Rayleigh range is given by

So from the data, the Rayleigh range in this example is : 0.127mm, which is the same as the starting piolet beam Rayleign.

- The phase radius of curvature of the beam is a function of the distance from the beam waist, z:

Z is 24.261mm, in this case, the phase radius of curvature is : 24.261687mm

Note: because the sampling is low, the phase is not well represented.

Note that radius is infinite at z = 0, reaches a minimum of 2zr at z = zr , and asymptotically approaches infinity as z approaches infinity.

- The phase of the Gaussian beam along the axis is defined by the Gouy shift, given by

In this case, the phase at beam waist is 0 ( surface 1), and when the beam reach to surface 2, the phase is 0+1.5655 radians. it matches with the result.

- The beam size is also a function of the distance from the waist:

Let’s calculate the beam size at surface 2: it is 1.2142mm, which is quite close the report.

Note for large distances the beam size expands linearly.

- The divergence angle of the beam is given by

In this case, the divergence angle is 2.86519 degree, or 0.05 radians, which is the same as surface 1 data.

The properties of the pilot beam as above are then used to determine if the actual distribution is inside or outside the Rayleigh range, and what propagation algorithms are appropriate. After passing through an aperture that significantly truncates the beam, such as a pinhole aperture, it may be required to recompute the pilot beam parameters, we will share with you in more details in the future regarding to this topic.

Note: Although the pilot beam is an ideal Gaussian beam, it is different from the Paraxial Gaussian Beam. The pilot beam are propagated by real rays, which are called probing rays. Using probing rays accounts for all situations namely when propagating through non-ideal optics.

On the other hand, Paraxial Gaussian Beam uses only paraxial data when propagating beams. For this reason, the pilot beam will give slightly different result compared to the Paraxial Gaussian Beam. The difference can happen even in a refraction-free air propagation, where the slopes of these real rays differ very slightly from the paraxial rays that are linear.

Design file used in this example, Please download it here: How to interpret the physical optics data.

Reference Source:

- https://www.zemax.com/
- Zemax Optical Design Program User’s Guide, Zemax Development Corporation
- https://en.wikipedia.org/wiki/Main_Page