Introduction:
In interference, two coherent waves superpose and form an interference pattern. This pattern consists of alternating bright and dark fringes. The intensity of light at different points in the interference pattern can be calculated using the principle of superposition and the concept of interference.

Given:
- Two identical coherent waves are producing an interference pattern.
- We need to find the ratio of intensity at the center of a bright fringe to the intensity at a point one-quarter of the distance between two fringes from the center.

Explanation:
Interference Pattern:
- When two coherent waves superpose, they interfere constructively at certain points (bright fringes) and destructively at other points (dark fringes).
- The pattern formed due to this interference is characterized by alternating bright and dark fringes.
- The intensity of light at different points in the interference pattern depends on the superposition of the waves.

Intensity at the Center of a Bright Fringe:
- The center of a bright fringe is the point where the two waves superpose constructively, resulting in maximum intensity.
- At this point, the amplitude of each wave is the same, and they are in phase.
- When two waves of equal amplitude and in phase superpose, the resulting amplitude is twice the individual amplitude.
- Intensity is proportional to the square of the amplitude, so the intensity at the center of a bright fringe is four times the intensity of each individual wave.

Intensity at a Point One-Quarter of the Distance Between Two Fringes from the Center:
- When we move one-quarter of the distance between two fringes from the center, the waves are out of phase by half a cycle.
- At this point, the waves interfere destructively, resulting in minimum intensity.
- The amplitude of each wave is the same, but they are out of phase.
- When two waves of equal amplitude and out of phase superpose, the resulting amplitude is zero.
- Intensity is proportional to the square of the amplitude, so the intensity at this point is zero.

Ratio of Intensity:
- The ratio of intensity at the center of a bright fringe to the intensity at a point one-quarter of the distance between two fringes from the center is given by:
Ratio = (Intensity at the center of a bright fringe) / (Intensity at the point one-quarter of the distance between two fringes from the center)
= (4) / (0)
= undefined

Conclusion:
The ratio of intensity at the center of a bright fringe to the intensity at a point one-quarter of the distance between two fringes from the center is undefined because the intensity at the latter point is zero.