Coherence condition in wave optics: for two electromagnetic waves to be considered coherent, their phase difference must be what?

Introduction / Context:
Coherence underpins interference and diffraction phenomena used in optical remote sensing, holography, and laser applications. Exam questions often probe whether you can distinguish temporal coherence (phase stability over time) from spatial coherence (phase correlation across space).

Given Data / Assumptions:

• Two waves are compared at one or more spatial points.
• Interference visibility requires a well-defined phase relationship during observation.
• Idealized assumptions ignore noise and turbulence for the conceptual definition.

Concept / Approach:
Temporal coherence requires the phase difference between two waves at a point to remain constant (or predictable) over the measurement time, enabling stable interference fringes. Spatial coherence concerns correlation of phase at different points in a wavefront. Many textbook definitions test temporal coherence as the minimal condition for two beams to produce steady fringes.

Step-by-Step Solution:
Identify essential requirement: a constant phase difference in time ensures stationary fringes.Recognize spatial coherence: while important for extended sources, it is not strictly required to answer the basic two-wave coherence condition at one point.Hence, select 'constant in time' as the core condition in many introductory contexts.

Verification / Alternative check:
Fringe visibility V deteriorates when phase fluctuates in time faster than the detector integration time, demonstrating the centrality of temporal coherence.

Why Other Options Are Wrong:

• 'Constant in space' alone does not guarantee stable interference over time.
• 'Constant in time and space' is overly restrictive for the basic two-wave condition assessed at a point.
• 'None of these' contradicts wave optics fundamentals.

Common Pitfalls:

• Confusing requirements for extended-source interferometry (which needs spatial coherence) with the minimal definition for two beams.

Final Answer:
constant in time