4-Wave LateralShearing Interferometry Testing Technology: Glass Plate 4-Wave LateralShearing Interferometry Method

The plate shearing beamsplitter method is a 4-Wave LateralShearing Interferometry technique where a single plate serves as both the shearing and beamsplitting device. Its principle relies on the front and back surfaces of the plate performing amplitude division on the wavefront under test and introducing a lateral displacement (Figure 3-5(a)).

The system's shear ratio is influenced by the plate's thickness, d, and the angle of incidence, θ. For ease of optical alignment, θ is typically set at 45°. It's important to note that while the plate's thickness generates lateral shear, it also introduces a significant optical path difference between the two sheared wavefronts. Without white light compensation, a laser light source becomes necessary.

In 1964, Murty [8-9] from the University of Rochester in the United States first introduced a He-Ne laser to construct this simple single-plate device (with the overall optical path shown in Figure 3-6). He used it to test the exit wavefront of lenses (wavefronts with different beam aberrations, as shown in Figure 3-7, which aligns with the simulation results in Table 3-1). Consequently, this setup is sometimes referred to as the Murty Plate 4-Wave LateralShearing Interferometry Device.

Certainly, the Murty Plate 4-Wave LateralShearing Interferometry system, as a prototype structure, has some drawbacks and issues. A significant problem is its general inability to measure converging wavefronts, as it is typically limited to measuring approximate plane wavefronts corresponding to Figure 3-1(a). This limitation arises because the parallel plate is not a perfect imaging device; light reflected from the back surface of the plate undergoes deformation. The corresponding wavefront aberration is called shear distortion, which is mathematically nonlinear and difficult to calibrate.

The presence of shear distortion not only causes severe deformation of the wavefront reflected from the back surface of the plate but also means the shear ratio within the interference field is no longer constant. As shown in Figure 3-6, if the interference field aperture is D, the plate thickness is d, the refractive index is n, and the angle of incidence is θ, then the shear ratio β in a 4-Wave LateralShearing Interferometry system can be expressed as:

For a converging beam in a 4-Wave LateralShearing Interferometry system, the angle θ varies for rays at different spatial heights. If we set $d/D=1/5$, meaning the plate thickness is 1 mm, the interference field diameter is 5 mm, and the plate's refractive index $n=1.41$, the relationship between β (shear ratio) and θ (incidence angle) is shown in Figure 3-9. The shear ratio changes drastically with the angle of incidence, making wavefront reconstruction difficult even if shear distortion could be calibrated.For a converging beam in a 4-Wave LateralShearing Interferometry system, the angle θ varies for rays at different spatial heights. If we set $d/D=1/5$, meaning the plate thickness is 1 mm, the interference field diameter is 5 mm, and the plate's refractive index $n=1.41$, the relationship between β (shear ratio) and θ (incidence angle) is shown in Figure 3-9. The shear ratio changes drastically with the angle of incidence, making wavefront reconstruction difficult even if shear distortion could be calibrated.

This also indicates that 4-Wave LateralShearing Interferometry systems that use a plate for beamsplitting, including the Murty plate, aren't well-suited for testing converging wavefronts, and there's still no good solution for this. However, many issues present in the early Murty plate designs have been resolved as subsequent researchers continuously improved them, leading to the proposal of new systems. We'll summarize these below.

First, early 4-Wave LateralShearing Interferometry setups using parallel plates weren't easy to align. Take the optical path shown in Figure 3-6 as an example: before starting a measurement, you needed to adjust the microscope objective so the beam passing through the objective under test was approximately parallel, essentially focusing the microscope objective's output beam near the theoretical focal point of the test lens. However, during this adjustment, it was impossible to tell if you were pre-focus or post-focus, because the interference fringes looked identical (as shown in Figure 3-10).

To address this, some researchers proposed replacing the parallel plate with a wedge plate (where the wedge angle direction is orthogonal to the shear direction). This introduces a tilt E into the interference fringes, creating a background fringe pattern:

The advantage of this is that interference fringes are visible under any circumstances (even with almost no aberrations), and it also solves the adjustment problem. This is because the interference fringes before and after the focal point are no longer identical, allowing the direction of adjustment to be determined (as shown in Figure 3-11).

It's important to note that the 'tilted carrier' mentioned here is not the same concept as the X- or Y-axis tilt listed in Table 3-1. The former refers to a beam aberration carried by the wavefront under test, whose differential is a constant, appearing as background light in the interferogram. The latter involves changing the propagation direction of the sheared wavefronts during beamsplitting, meaning the two sheared wavefronts no longer propagate along the same direction, which then introduces a linear function into the differential wavefront in 4-Wave LateralShearing Interferometry.

Let's further discuss the introduction of a tilted carrier by a wedge plate. As shown in Figure 3-12, for a wedge plate with a wedge angle of α and a refractive index of n (the wedge angle is drawn very large here for illustrative purposes, but in practice, it's typically only 1' to 2'), the reflected light beams from the front and back surfaces have a tilt angle γ (an example ray is shown in the figure).

Upon derivation, when α is very small, γ can be considered linearly proportional to α:

You can select a wedge plate with a specific wedge angle based on the desired tilt amount γ, as determined by Equation (3-16). The choice of γ generally follows these principles: its lower limit must exceed the maximum tilt of the wavefront under test, and its upper limit is restricted by the sensor's Nyquist sampling theorem.

In 4-Wave LateralShearing Interferometry, intentionally introducing a tilt isn't just about generating fringes or simplifying alignment, as mentioned previously. From a Fourier optics perspective, it's equivalent to applying a spatial carrier frequency, which facilitates subsequent interferogram demodulation using spatial phase retrieval techniques.

When using a wedge plate to introduce tilt in 4-Wave LateralShearing Interferometry, there are two important points to consider

Varying Tilt for Converging Beams

First, Equation (3-16) indicates that γ (tilt amount) is related to the angle of incidence θ. This means that incident light rays at different angles will acquire different tilt amounts. Consequently, for converging wavefronts, the tilt introduced by switching to a wedge plate often varies across the entire aperture, complicating subsequent tilt removal. This further confirms that plate shearing is not suitable for testing converging wavefronts.

Shear Ratio Variation with Wedge Angle

Second, the introduction of a wedge angle can cause the shear ratio to vary within the interference field even when the wavefront under test is an approximate plane wave. However, this effect is negligible when the wedge angle α is small (generally less than 1 degree). More details on this can be found in the phase-shifting techniques section of Chapter 3.5.

Second, early Murty Plate 4-Wave LateralShearing Interferometry systems made it difficult to adjust the shear ratio. In practical measurements, you often need to choose an appropriate shear ratio based on the characteristics of the wavefront under test. While Equation (3-14) suggests that rotating the Murty plate can adjust the shear ratio to achieve different measurement resolutions and dynamic ranges, rotation offers lower adjustment precision compared to translation, and its maximum adjustment range is limited by the plate thickness. More importantly, the relationship shown in Equation (3-14) is nonlinear, making calibration quite challenging.

Because of this, a modified approach [10] was proposed: adding a second plate (or mirror) to adjust the shear amount by changing the distance between the two plates, as shown in Figure 3-13. In this setup, Plate 1 remains fixed, while Plate 2 is mounted on a sliding rail along the optical axis. Both plates are parallel to each other and positioned at 45° to the system's optical axis. When parallel light is incident along the system's optical axis, and assuming the axial distance from the back surface of Plate 1 to the front surface of Plate 2 is t, the shear ratio β can be expressed as:

In the equation, D and d are the spot diameter on the detector and the plate thickness, respectively. This allows for linear adjustment of the shear amount by moving Plate 2 along the guide rail to change t, and this method offers a relatively large adjustment range. Of course, just like the single-plate setup, the dual-plate device can also be modified to use wedge plates.

Insertable Plate Shearing Method