To attain the perfect collimated beam, a lateral shearing interferometer is necessary. By shifting the laser beam in the lateral direction compared to the original wavefront, the lateral shearing interferometry creates two mutually displaced equal-amplitude copies of the testing wavefront. Shear interferometers are convenient to use to observe interference and test the collimation of light, especially laser beams with coherence length significantly longer than the thickness of the shear plate, requiring two orthogonally sheared interferograms for the complete evaluation of wavefronts. Numerical analysis is done to interpret the intensity distribution of the fringe pattern and the aberration curve. The fringes directly map the lateral aberration of the lens under test.

Aim:

The aim is to test the laser beam collimation and the quality of the collimation lens using lateral shearing interferometer.

Apparatus:

The apparatus includes a He-Ne Laser (λ = 632.8 nm), beam expander, spatial filter, shear plate, and optical bench.

Theory:

This method involves duplicating a wavefront, laterally displacing it by a small amount, and then obtaining the interference pattern between the actual and displaced wavefronts. When the wavefront is almost planar, lateral shear is obtained by displacing the wavefronts in their own plane.

Spatial filtering is an optical method that is part of the Fourier optics domain and used to alter the structure of a beam. It is mainly used to clean up the output of lasers and remove the aberrations. In spatial filtering, a lens focuses the beam, and due to diffraction, the beam is not a proper plane wave and will produce a pattern of bright and dark regions in the focal plane.

A pinhole is put in front of the focal plane of the lens, and diffraction rings are observed. First, the diffraction rings will be observed, and as we move further, the diffraction rings disappear.

Shear plates are high-quality optical flats that combine two wavefronts. The wavefronts that need to be tested will incident on the shear plate at a 45-degree angle for maximum scattering. Constructive and destructive interference will lead to bright and dark fringes corresponding to the in-phase and out-of-phase parts of wavefronts. The uncoated front and rear surfaces of the shear plate will generate reflections of almost equal intensity, and these reflected wavefronts are called laterally sheared. The amount of shear can be controlled by the tilt and thickness of the plate.

Let W(x, y) be the wavefront error, which is the difference between the original wavefront and the desirable one. When a planar wavefront is sheared by a small amount "l," the error at every point on the shear wavefront is W(x-l, y).

The resulting wavefront will be the difference at P(x, y), given by:

  ΔW(x, y) = W(x, y) - W(x-l, y)

  ΔW(x, y) = nλ

Where n is the order of interference fringes.

λ is the wavelength.

$$\left( \frac{\partial W}{\partial x} \right)l = n\lambda$$

The above equation provides information that the lateral shear is a ray aberration, with $\left( \frac{\partial W}{\partial x} \right)$ in angular units. Accuracy of this increases as l → 0.

Lateral shear interferometry is widely used to obtain the lateral aberration of a lens. The zeroth-order fringe is an example of a wedge-plate lateral shear interferometer and directly shows the lateral aberration curve of a lens. The fringes in lateral shear interferogram show the differential of the wavefronts, and the aberrations are directly proportional to the gradient of the wavefront of the lens used as a collimator.

Interferograms for Spherical and Flat Wavefronts:

1. Defocusing: -

Wavefront error for defocusing

W(x, y) = D(x² + y²)

Where D is the magnitude of the aberration.

When the upper equation is differentiated with respect to x, it becomes an equation of straight line, represented by ΔW(x,y) = 2Dxl = nλ.

This equation shows a system of straight line fringes that are perpendicular to the x-direction and equispaced. These fringes appear at the overlapped area of wavefronts. If D = 0, then there is no defocusing and no fringes present.

For the tilt direction orthogonal to the lateral shear, the equation becomes ΔW(x,y) = Py = nλ where P is the angle between the original and sheared wavefronts. When defocusing and tilt are both present, the equation is ΔW(x,y) = 2Dxl + Py = nλ. This equation represents a system of straight line fringes when an optical system is collimated with respect to a point source and passes through the focus region.

In the case of spherical aberration, the wavefront error is W(x,y) = A(x² + y²)². The equation with defocusing present is ΔW(x,y) = 4A(x² + y² + 2D)xl= nλ.

Coma causes an error in wavefront, which is represented by W(x,y) = By(x² + y²) and ΔW(x,y) = 2Bxyl = nλ in the lateral interferogram. When defocusing is present with coma, the equation becomes ΔW(x,y) = 2Bxyl + 2Dxl = nλ.

Similarly, W(x,y) = C(x² + y²) represents astigmation. The equation with defocusing is ΔW(x,y) = 2Cxl + 2Dxl = nλ. If the lateral shift is in the y-direction, the equation is ΔW(x,y) = 2(D-C)yT = nλ, where l is the sagittal shear, and T is the tangential shear. If l and T are equal in magnitude, there are two values of D, D = ± C, and LSI fails to exhibit fringes.

The experimental setup of lateral shear interferometry using a He-Ne laser involves passing the output through a collimated lens and using a pinhole as a filter. The light is reflected back and forth in the plate because of Fresnel reflection, and the lateral displacement or shear is ‘l’. If the plate has a thickness of ‘t’ and refractive index ‘n,’ then the angle of incidence is given by l/t = sin(2i)(n^2−sin^2i)^1/2. The full width of the beam is ‘d,’ and the focal length of the lens is ‘f.’ The pinhole size can be calculated to be 1.22λf*d/l, where f* = f/d.

1. Our first task is to precisely align the laser.

2. Subsequently, the excess noise frequencies are filtered through a spatial filter to ensure clarity.

3. The beam is then directed through a collimated lens to achieve ultimate precision.

4. It is important to place the spatial filter's pinhole at the lens's front focal length during the collimation process. This allows us to accurately determine the lens' focal length and analyze its quality, identifying any aberrations.

5. Utilizing a shear plate, fringes appear after lateral shearing. Analysis of these fringes enables us to identify the corresponding aberrations.

6. In order to obtain straight line fringes when the beam is collimated, a wedge is built in the plate in an orthogonal direction to the shear.

7. Finally, the optical system is focused and defocused to accurately record the fringes.