Haidinger's fringes

Haidinger's fringes are a system of concentric circular interference fringes produced by the interference of two beams whose path difference depends on their angle of inclination.

They are also known as fringes of equal inclination.

Formation in a Michelson Interferometer

In a Michelson interferometer, let \(d\) be the separation between the two reflecting surfaces and let a ray make an angle \(\theta\) with the normal.

The optical path difference between the two interfering beams is:

\[ \Delta = 2d\cos\theta \]

Condition for Bright Fringes

For constructive interference, the condition for bright fringes is:

\[ 2d\cos\theta = n\lambda \]

where:

• \(d\) = separation between the reflecting surfaces
• \(\theta\) = angle of inclination of the ray
• \(n\) = order of the fringe
• \(\lambda\) = wavelength of light

Condition for Dark Fringes

For destructive interference, the condition for dark fringes is:

\[ 2d\cos\theta = \left(m+\frac{1}{2}\right)\lambda \]

Why are they called fringes of equal inclination?

For a particular fringe order \(m\), the equation is:

\[ 2d\cos\theta = m\lambda \]

If \(d\), \(m\), and \(\lambda\) are constant, then \(\theta\) must also be constant. Therefore, all rays forming one fringe have the same angle of inclination.

Therefore: Haidinger's fringes are called fringes of equal inclination.

Important Characteristics

• They are generally concentric circular fringes.
• They are localized at infinity.
• They are observed when the telescope is focused at infinity.
• Their formation depends on the angle of inclination \(\theta\).
• The optical path difference is \(2d\cos\theta\).