Introduction: What Is Apodization in Optical Design?
Apodization describes how the beam amplitude (or intensity) varies across the entrance pupil of an optical system. In Zemax OpticStudio, the apodization factor defines the rate at which beam amplitude decreases with radial pupil coordinate, normalized to unity at the pupil center.
Apodization plays a critical role in:
- Diffraction behavior
- Point Spread Function (PSF) shape
- Energy distribution in the focal plane
- Aberration weighting across the pupil
This article demonstrates how changing the apodization type affects pupil illumination and diffraction performance, using a simple optical system example.
Uniform Illumination and the Airy Diffraction Pattern
Consider an optical system represented by a paraxial lens surface, modeling a highly corrected imaging system that is uniformly illuminated across its full aperture.
With uniform apodization:
- The entrance pupil is illuminated by a top-hat (constant amplitude) function
- The resulting FFT Point Spread Function (PSF) is the classic Airy pattern
- This follows directly from Fourier optics: The Fourier transform of a circular top-hat function is a Bessel function
To visualize low-intensity diffraction features, the PSF is typically displayed on a logarithmic intensity scale, which highlights the outer diffraction rings (“feet”).
Gaussian Apodization and Beam Truncation
Next, we change the pupil illumination to a Gaussian apodization, where the beam intensity decreases smoothly from the center toward the edge of the pupil.
Meaning of the Apodization Factor
For Gaussian apodization:
- The apodization factor controls how rapidly amplitude decays radially
- An apodization factor of 3 means the entrance pupil diameter corresponds to √3 × (1/e²) beam radius
- This results in minimal truncation at the aperture stop
Because the Fourier transform of a Gaussian is also a Gaussian, the FFT PSF becomes a single-lobed Gaussian profile with strongly suppressed diffraction rings.
Key Observation
The lens itself has not changed—only the illumination function across the pupil. Since little optical energy reaches the pupil edge, far less energy is diffracted into sidelobes, effectively “cutting the feet” of the diffraction pattern.
Apodization vs Spatial Filtering
The term apodization is sometimes loosely applied to spatial filtering, such as:
- Using a pinhole to transmit only the central Airy lobe
- Blocking higher-order diffraction rings
However, in optical design terminology, apodization more broadly refers to any pupil illumination function, regardless of whether the resulting PSF is single-lobed or multi-lobed.
This distinction is important because:
- Optical aberrations vary over the pupil
- Apodization determines which pupil zones contribute most strongly to image formation
- It effectively weights aberrations spatially
Supported Apodization Functions in OpticStudio
OpticStudio supports several built-in apodization models, allowing realistic simulation of illumination conditions.
1. Uniform Apodization
- Constant amplitude across the pupil
- Simulates uniform illumination
- Typical for distant objects and infinity-conjugate imaging systems
Apodization factor = 0 corresponds to uniform illumination.
2. Gaussian Apodization
- Amplitude decreases radially according to a Gaussian function
- Used to study truncated Gaussian beams
Key interpretations:
-
Apodization factor = 1.0
→ Amplitude falls to 1/e at pupil edge
→ Intensity falls to 1/e² ≈ 13% -
Valid range: ≥ 0
-
Values above ~4.0 are not recommended
(ray sampling becomes insufficient near the pupil edge)
This model is especially relevant for:
- Laser optics
- Fiber-coupled systems
- Beam-shaping analysis
3. Cosine-Cubed Apodization
- Models illumination from a point source incident on a flat pupil plane
- Intensity varies as cos³(θ), where θ is the ray angle relative to the optical axis
- Only valid for on-axis or near-axis field points and systems where the pupil diameter is small compared to source distance
4. Surface Transmittance (Custom Apodization)
Arbitrary apodization defined by:
- Ray coordinates
- Direction cosines
- Surface parameters
- Look-up tables or DLL-based functions
This method enables:
- Realistic modeling of measured pupil transmission
- Simulation of complex apodizers, masks, or diffractive elements
Why Apodization Matters in Optical Design
Apodization directly influences:
- Diffraction-limited resolution
- PSF sidelobe suppression
- Contrast and stray-light performance
- Aberration sensitivity across the pupil
It is particularly critical in:
- High-performance imaging systems
- Laser beam delivery
- Astronomical optics
- Lithography and metrology systems