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Polarization Lateral Shearing Interferometry

Polarization-based lateral shearing interferometry utilizes the birefringence effect of crystal plates to generate two overlapping light beams. Currently, there are mainly two specific configurations: a single crystal plate and a Savart polarizing prism composed of two plates with orthogonal optical axes, as shown in Figures 3-20(a) and (b). These crystal plates generally use uniaxial crystals (such as calcite). Since the refraction angles of the ordinary and extraordinary rays (denoted as θo and θe, respectively) in such crystals usually do not differ significantly, they are suitable for cases requiring small shear amounts. The Savart polarizing prism, on the other hand, can offer shear in different directions depending on the orientation of the optical axes of the two crystals. A common characteristic of polarization-based quadriwave lateral shearing interferometry systems is that they are less affected by stray light. However, they cannot achieve large shear magnitudes.

Quadriwave Lateral Shearing Interferometry based on Birefringent Crystal Plate

The lateral shear generated by polarization devices is based on the birefringence phenomenon of crystals. Specifically, along a certain wave-normal direction within a crystal, two linearly polarized beams with orthogonal vibration directions can be formed. These two beams exhibit different refractive indices, light propagation velocities, and ray directions. From another perspective, when a linearly polarized beam enters a birefringent crystal, it splits at the interface into two refracted beams with distinct refractive indices, propagation directions, and polarization orientations. These beams experience a lateral displacement relative to each other, which depends on the crystal thickness, refractive index, optic axis orientation, and angle of incidence. This lateral displacement realizes lateral shearing, meaning that the birefringent crystal plate simultaneously performs both beam splitting and shearing.The following introduces the principle of polarization-based quadriwave lateral shearing interferometry using a birefringent crystal plate. As an example, a calcite (CaCO₃, no = 1.65578, ne = 1.48535) plate is discussed. Based on the orientation of the optical axis relative to the incident surface, the shearing–splitting configuration of the birefringent crystal plate generally operates in two modes: one where the crystal's optical axis lies within the incident plane, and the other where it is perpendicular to that plane.As shown in Figure 3-21, let the incident angle be θ, the incident light vector be K, the refraction angle of the ordinary ray (i.e., the angle between the o-ray wave normal Ko and the interface normal) be θo, and the refraction angle of the extraordinary ray be θe. The deviation angle α is defined as the angle between the extraordinary wave normal Ke and its corresponding ray direction Se (note: the deviation angle for the o-ray is 0°). The resulting lateral shear is denoted as S. (In this context, we describe the shear amount S rather than the shear ratio, since the pupil size of the test beam is not considered.)

As shown in Figure 3-21(a), when the optical axis lies within the incident plane, and specifically when the optical axis is perpendicular to the interface normal, the angle between the extraordinary ray direction $S_e$ and the interface normal is defined as $\theta' = \theta_e + \alpha$ . Under this condition, the lateral shear $S$ can be expressed as:

Obviously, $S$ is a variable dependent on the incident angle $\theta$ , so rotating the birefringent crystal plate allows tuning of the lateral shear. It should be noted that, for simplicity in the previous derivation, the orientation of the optical axis was fixed. However, during the rotation of the crystal plate, the direction of the optical axis also changes. Therefore, Equation (3-24) only provides an approximate expression.

Another special case is shown in Figure 3-21(b), where the optical axis of the crystal is perpendicular to the incident plane. In this configuration, the wave vectors of both the ordinary and extraordinary rays are orthogonal to the optical axis. Thus, for the extraordinary ray, its wave vector direction coincides with the ray direction (i.e., the deviation angle $\alpha$ is 0°). The analysis in this case is more straightforward, and the relationship between the lateral shear $S$ and the incident angle $\theta$ can be derived directly from Equations (3-20), (3-21), and (3-22), where the extraordinary refractive index $n_e(\theta) = n_e$ remains constant.

由式(3-24)和式(3-25)可得不同入射角下剪切量S的调节范围(相对于晶体平板厚度d)，如图3-22所示。

As shown in Figure 3-22, regardless of the configuration, the variation of the lateral shear $S$ $S$ exhibits two notable characteristics:
First, the change is nonlinear and non-monotonic, making calibration quite challenging.
Second, the maximum achievable shear $S$ $S$ is relatively limited—approximately 5% of the crystal plate's own thickness—which makes it unsuitable for testing wide beams.

Savart Polariscope Double Shearing Method

Issues with Single Crystal Plate Lateral Shearing Interferometry

Insufficient Shear Rate: This is particularly problematic for measuring large-aperture wavefronts (Figure 3-22).

Non-Equal Optical Path Interference: An optical path difference exists between the ordinary (o-ray) and extraordinary (e-ray) light.

Savart Polariscope as a Solution

These issues can be addressed by employing a Savart polariscope composed of two crystal plates [25].

Research into the Savart polariscope began in the 1950s. Its structure consists of two identical uniaxial crystals. By rotating (or flipping) one of the crystals, the relative orientation of the optical axes of the two crystals can be changed. Based on the relative position of their optical axes, Savart polariscopes primarily exist in two forms, as illustrated in Figure 3-23.

In Figure 3-23(a), the optical axes of the two crystals are not coplanar. Consequently, when the e-ray (extraordinary ray) from the first crystal enters the second crystal, it transforms into an o-ray (ordinary ray), denoted as 'eo' light in the figure. The same applies to the o-ray from the first crystal, which is denoted as 'oe' light. Due to birefringence, these two beams undergo a shift in both the x and y directions upon exiting (i.e., they experience two shearing operations). When the two plates are identical, the magnitudes of these shifts are the same, resulting in a shearing direction at 45° to the positive x and y axes in the xOy plane. At this point, the path lengths traveled by the two beams within the Savart polariscope are also identical, thus satisfying the equal optical path condition. This structure of the Savart polariscope is more commonly encountered.

In Figure 3-23(b), the optical axes of the two crystals are coplanar. If light is directly incident, the ordinary (o) or extraordinary (e) light from the first crystal remains as o or e light, respectively, in the second crystal. While these two beams undergo lateral displacement upon exiting, they also acquire a significant tilt, causing the fringe density to exceed the detector's sampling limit. Therefore, a half-wave plate must be inserted between the two crystals. The fast axis of the half-wave plate should be oriented at 45° to both the o-light and e-light polarization directions. This rotates the polarization direction of the o (e) light by 90° after passing through the half-wave plate, transforming it into e (o) light. This setup achieves an effect similar to the structure in Figure 3-23(a), with the only difference being that the final eo and oe light beams have opposite polarization directions, and the shearing direction is along the y-axis in the figure.Compared to the previous structure (Figure 3-23(a)), this configuration can achieve a larger shear amount $S$ . However, the added half-wave plate introduces several problems. For example, it compromises the equal optical path property because an optical path difference is generated between the two beams inside the half-wave plate. More importantly, in multispectral systems, a half-wave plate cannot function properly across multiple spectral bands simultaneously. Consequently, this structure is generally not used in systems like interferometric imaging spectrometers.

There are also many other polarization devices that can be used for lateral shearing interferometry, such as the Wollaston prism. However, the structure and principle of generating lateral shear with a Wollaston prism are similar to the first type of Savart polariscope, and it is less commonly applied, so we won't elaborate further here.

It's important to note that in Figures 3-21 and 3-23, we observe that the polarization directions of the exiting beams are mutually orthogonal, which does not inherently satisfy the interference condition. Therefore, a polarizer is required, with its transmission axis typically oriented along the angular bisector of the two beams' polarization directions. Simultaneously, to ensure the visibility of interference fringes, the intensities of the two beams should be as consistent as possible. This necessitates that the polarization direction of the incident light also satisfies certain conditions, thus requiring a polarizer to be placed before the polarization shearing device [26]. Its polarization direction generally aligns with that of the analyzer. The polarizer and analyzer are typically inexpensive thin-film polarizers. In cases requiring higher performance, polarizing prisms like the Glan-Taylor prism can be used, offering an extinction ratio on the order of $1 0^{- 5}$ . Figure 3-24 illustrates a schematic of polarization lateral shearing interferometry with the addition of a polarizer and analyzer.

References

[24]Liu L, Zeng A, Zhu L, et al. Lateral shearing interferometer with variable shearing for measurement of a small beam[J]. Opt. Lett., 2014, 39(7): 1992-1995.

[25]Lin S T, Shih S H, Feng H N, et al. Phase-shifting Savart shearing interferometer[J]. Optical Engineering, 2006, 45(12):125602.

Lateral Shearing Interferometry - Polarization Lateral Shearing Interferometry

Savart Polariscope as a Solution

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Lateral Shearing Interferometry - Polarization Lateral Shearing Interferometry

Savart Polariscope as a Solution

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