Wavefront and Surface Measurement with Interferometers and Shack Hartmann Sensors


Fizeau Interferometer for Wavefront and Surface Measurement

Set up of a Fizeau interferometer
Set up of a Fizeau interferometer

Fizeau Setup
The Most Commonly Used Interferometer Setup

The last surface of the beam shaping optics is the so called Fizeau surface. It has to have the same shape as the sample to be tested (commonly spherical or flat) and is placed concentrically into the optical path, so the individual rays intersect perpendicular to the Fizeau surface. The majority of the light passes the Fizeau surface and is reflected at the test surface. The returning light interferes with the part of the light reflected at the Fizeau surface. So the Fizeau surface acts as beam splitter as well as reference surface. So the cavity is formed by the gap between the Fizeau and the test surface containing no additional optics. That is the reason why a Fizeau interferogram commonly directly shows the deviations of the test sample from the reference surface, i.e. Fizeau surface. The quality of the Fizeau surface determines the accuracy of the Fizeau interferometer. Fizeau surfaces are commonly available with a quality of λ/10 – λ/20 PV, better on request.



Twyman Green Interferometer for Wavefront and Surface Measurement

Set up of a Twyman-Green interferometer
Set up of a Twyman-Green interferometer

Twyman-Green Setup
The Most Flexible Interferometer Setup

A Twyman-Green interferometer is a modified Michelson interferometer. Here the beam splitter is separated from the reference surface. The advantage of this configuration is greater flexibility, because both interferometer arms can be modified independently of each other. So the intensity of the reference and test arm can be easily adapted to each other depending on different sample reflectivities in order to get maximum fringe contrast. This increases the range of applications enormously. Only a maximum fringe contrast enables a maximum resolution in depth. The reference surface can be a surface that is inexpensive and accurately producible regardless of the sample size. The adaption to the sample size is done by conventional beam shaping optics introduced to the test arm. Contrary to the beam shaping optics for Fizeau interferometers these optics do not require an expensive Fizeau surface as a final surface.

As a consequence of this flexibility, the interference patterns are not caused by the sample errors only but also by the aberrations of the additional optics in the individual interferometer arms. However, nowadays samples are no longer evaluated by visual inspection of the fringe pattern but by computer controlled analysis of the phase map causing the fringe pattern. During this analysis the aberrations of the additional optics can easily be taken into account. Finally, the software provides an objective digital measurement result.

TRIOPTICS Interferometer: µPhase®

Wavefront Measurement with Shack-Hartmann Sensors

Operating Principle of a Shack-Hartmann Sensor and Wavefront Analysis

Schematic setup of a Shack-Hartmann sensor with an a) incoming plane wavefront and (b) incoming diverging Wavefront
Schematic setup of a Shack-Hartmann sensor with an a) incoming plane wavefront and (b) incoming diverging Wavefront
Zernike analysis
Zernike analysis

The standard design of a Shack-Hartmann sensor consists of a CCD camera which is placed in the focal plane of a microlens array. An incoming wavefront is sampled by the lenses of the microlens array and the foci form a spot pattern on the camera which would be evenly spaced in case of a plane wavefront. Any aberration introduced by the sample lens leads to a curvature of the wavefront thus resulting in local wavefront tilts. These induce a measurable shift of each focus spot position.

A numerical integration of the obtained slope information allows for reconstruction of the wavefront profile with high accuracy. Using state of the art computers this wavefront reconstruction is done within the CCD camera frame rate i.e. within fractions of a second even if large, high resolution arrays are used.

High Dynamic Range Compared to Interferometers

The dynamic range of a Shack-Hartmann sensor heavily depends on the algorithm which assigns each measured spot to the corresponding microlens. A wavefront is reconstructed only when this correlation is kept. Especially in case of stronger curved wavefronts sophisticated algorithms are needed since the simple assignment of a predefined searching area in the CCD plane of the size of a microlens is not sufficient anymore. Modern techniques achieve wavefront dynamic ranges up to 1500 A.

Due to this high dynamic range Shack- Hartmann sensors are able to measure wavefronts with strong aberrations which are not accessible with interferometers anymore.

Real Time Wavefront Analysis
Zernike Polynomials

The measured wavefront is decomposed into a linear combination of Zernike polynomials which describe typical optical properties and errors of a lens or lens system as e.g. defocus, coma or astigmatism.

The polynomial decomposition gives a numerical representation of any kind of aberration of the sample. These have basically two sources: aberrations directly linked to the design of the lens, most likely spherical terms, and asymmetric contributions due to lens errors.

MTF, PSF and Strehl ratio

The effects of aberrations are also characterized by calculating the Point Spread Function (PSF), Modulation Transfer Function (MTF or Strehl ratio of the optical system which are obtained from the wavefront. The MTF is as well-known as the modulus of the Optical Transfer Function.

The wavefront measurement and its further analysis give a full spatially resolved description of the imaging characteristics of the lens under test.



Different Setups with Shack-Hartmann Sensors

Finite setup in transmission of a Shack Hartmann sensorn
Finite setup in transmission
Infinite setup for wavefront measurement in transmission
Infinite setup for wavefront measurement in transmission
Reverse infinite setup of a Shack Hartmann sensor in transmission
Reverse infinite setup in transmission
Setup in reflection with a Shack-Hartmann sensor
Setup in reflection with a Shack-Hartmann sensor

Measurement Setups in Transmission and Reflection

Different configurations of the setup can be chosen for measuring the wavefront. Most important for the choice of configuration is whether the optical properties - using the transmission mode - or the lens shape - using the reflection mode - shall be analyzed

Transmission Mode

Measurement in transmission provides information about the optical properties of the lenses or lens systems combining the influence of all surfaces as well as refractive index variations in the measured wavefront.

Basic Infinite Setup

In the basic transmission setup the sample lens is illuminated with collimated light. A lens in combination with a telescope is then used to collimate the beam again and image the wavefront onto the Shack-Hartmann sensor.

In this setup the sample lens can be easily adjusted in its lateral and height position to achieve the best focus position with respect to the sensor.

Reverse Infinite Setup

In this configuration  the sample lens is illuminated by a point light source in the focal plane of the lens. The exit pupil of the lens is imaged onto the wavefront sensor by a telescope.

The height position of the point light source, the lateral position of the sample lens and the image plane of the Shack- Hartmann sensor are chosen separately. The reverse infinite setup is used for the instruments WaveMaster® COMPACT and WaveMaster® PRO.

Finite Setup

In addition to the reverse setup, a collimating lens is added between sample lens and sensor. In this configuration the lens is illuminated and tested in a configuration which is equal or close to the conditions of its dedicated application.

This configuration is only available with the instrument recommended for research and development..

Reflection Mode

Measurements in reflection provide information about the topography of the sample surface. For this measurement the illumination unit with beam splitter is mounted in front of the  wavefront sensor. A combination of collimating lens and telescope is used to illuminate the sample and image the reflected wavefront onto the Shack-Hartmann sensor.

The reflection setup is available as an easy to attach module for all WaveSensor® products. For production testing WaveMaster® PRO Reflex is available while for laboratory applications WaveMaster® COMPACT Universal is available.

TROPTICS Shack-Hartmann Sensors: WaveSensor®&WaveMaster®