Computer-Generated Hologram (CGH) Optimization in QUADOA® Optical CAD
A computer-generated hologram (CGH) compensates the difference between the spherical test wavefront of an interferometer and the shape of an aspheric surface under test, so the interferogram shows no fringes (a null test). This tutorial designs such a CGH in QUADOA® Optical CAD for a Fizeau setup: a third sequence traces light backward from the surface under test, a radial phase function on the CGH substrate is optimized for a plane wavefront, and the residual wavefront error reaches 0.01 λ.
Key points
- The setup has a reference sequence (Fizeau surface) and a test sequence (through transmission sphere, CGH and surface under test).
- A third sequence starts at the surface under test with WF from Surface and runs in reverse direction.
- The CGH is a Radial phase function (12 polynomial terms, norm radius about 35 mm) on the CGH substrate.
- Optimize the absolute wavefront (OPD reference Abs over Img) with a wavefront error merit function.
- Remove defocus first by moving the test surface, then optimize the CGH coefficients.
- Result: 0.01 λ wavefront error and an interferogram without fringes.
Terms used in this article
- CGH
- Computer-generated hologram: a diffractive element whose phase function is calculated to shape a wavefront.
- Null test
- An interferometric test in which a perfect surface produces no fringes.
- Transmission sphere
- An interferometer lens that creates a spherical test wavefront; its last surface is the Fizeau reference surface.
- Radial phase
- A rotationally symmetric phase φ = Σ A2i·h2i, with h the radius divided by the norm radius; coefficients in waves.
The interferometer setup
The model contains a transmission sphere, a CGH substrate (still without a phase function) and the surface under test, here an aspheric surface; a freeform would work the same way. The interferometer on the left emits a plane wavefront. Two sequences are already defined:
- Reference sequence: from the interferometer through the optics, reflected at the Fizeau cavity surface and back.
- Test sequence: through the transmission sphere and the CGH, reflected at the surface under test and back.
Without the CGH, the two wavefronts differ strongly: the interferogram shows far too many fringes to evaluate.
Step 1: Create a design sequence from the test surface
- Click New Auto Sequence in the Sequence tab and hide the two existing sequences.
- Revert the sequence direction so the rays start on the right, at the surface under test.
- Set the Source Type to WF from Surface with an aperture radius of 25 mm, the radius of the surface under test. The rays now start perpendicular to the surface.
- Set the wavelength to HeNe (632.8 nm), the test wavelength.
The goal: optimize the CGH so that this light leaves the system on the left as a plane wavefront.
Step 2: Add the CGH phase function
- Right-click the second surface of the CGH substrate and choose Add Phase → Radial.
- Set the Norm Radius to the ray height, about 35 mm.
- Set the Polynomial Order to 12.
The radial phase describes a rotationally symmetric phase φ(h) = Σ A2i·h2i with h = r / rn; the coefficients are in waves. It is used for Fresnel zone plates and for rotationally symmetric CGHs for asphere null tests. For acylinders, the Axial phase is the counterpart; general freeform CGHs can use Zernike or XY polynomial phases.
Step 3: Set up the merit function
- Set the OPD Reference of the design sequence to Abs over Img, so the absolute wavefront is optimized without exit-pupil corrections.
- Create a merit function that minimizes the wavefront error (Wavefront RMS) for sequence 3.
- Increase the ray grid of the ray trace to 15, since a high polynomial order needs a fine sampling.
Step 4: Optimize in two steps
- Remove defocus: the initial wavefront error is dominated by defocus. Make the Z position of the surface under test variable and run the Local Optimizer. The surface moves to match the radius of the transmission sphere wavefront; only spherical aberration remains.
- Optimize the CGH: fix the Z position again, set all CGH coefficients as variables and run the Local Optimizer.
Result: a wavefront error of 0.01 λ. If that is not good enough, increase the polynomial order.
Step 5: Check the null test
Hide the design sequence and show the reference and test sequences again: their rays now match. The interferogram is completely dark, which means zero fringes. Tilting the surface under test slightly produces straight tilt fringes, confirming a matched CGH. For diffractive ghost orders of the CGH, use the Diffractive Ghost Wizard, which also handles double-pass CGHs.
Video tutorial: CGH design and optimization
This 7-minute video designs and optimizes the CGH for a Fizeau interferometer test described in this article.
Video length: 7:03 min · Also on YouTube: Designing a CGH for Interferometric Asphere Testing in QUADOA® Optical CAD
Frequently asked questions
A computer-generated hologram converts the spherical test wavefront of an interferometer into the shape of an aspheric or freeform surface, so a perfect surface produces no fringes (a null test).
The Radial phase, φ = Σ A2i·h^2i with h = r / norm radius and coefficients in waves. For acylinders, the Axial phase is used.
Starting with a wavefront from the test surface and requiring a plane wavefront at the interferometer turns the CGH design into a simple wavefront optimization.
Abs over Img, which evaluates the absolute wavefront at the last surface without exit-pupil corrections, as for interferometers and wavefront sensors.
Increase the polynomial order of the radial phase and the ray grid of the merit function, then optimize again.