We all know about the gravitational lensing effect. From the analogy of fabric of space time used to explain this concept to laymen like me, I understand that light follows the curvature of spacetime.
Following on that same line of thought process, gravitational waves would cause the spacetime stretch and squeeze. Would it affect the path of light in any way?
Gravitational waves do affect the path followed by light. In fact, that's how LIGO works.
Gravitational waves cause space to stretch and shrink only in the directions transverse (perpendicular) to their motion. Lets say that LIGO's two arms are aligned to the $x$ and $y$ directions. A gravitational wave traveling in the $z$ direction will cause space to distort in the $x$ and $y$ directions only. As the wave passes the detector, the "peak" of the wave will maximally stretch one direction ($x$) while shrinking the other ($y$). The "valley" will do the opposite: shrinking $x$ while stretching $y$.
Because of the stretching and shrinking of space, the laser light in the two arms of the detector travel different distances. This relative path length change causes the interference pattern that LIGO detects.
As @eri points out, if light travels the same direction as the gravitational wave, the two will move at the same speed. The light won't notice any periodic changes in space, because it is always at the same spot on the wave (peak, valley, somewhere in between).
If light travels any non parallel direction relative to the wave propagation it will experience changes in the shape of space. They changes will slightly affect arrival times of photons, but not very much.
Yes, without going into calculation light path should be affected since gravitational wave is a perturbation in the spacetime metric. If metric is perturbed, then geodesic equation is affected (which governs motion of all particles and photons in spacetime).
The only tricky thing is the fact that GW propagates at the speed of light; so if the "wavefront" travels in the same direction as photons, then they never overtake each other and as seen from observer at infinity, GW that is lagging behind photons will always lag behind photons since they travel the same speed in space. If light "rides" along GW, then observer at infinity should see light paths get influenced the same way spacetime is rippled. But locally light probably cannot "feel" GW.
Would be nice to see explicit computation of this - I am not too knowledgeable about GW itself.
Probabily, it would be beneficial to first compare gravity, and gravitational waves (GW) as they pass earth's gravitational field.
To compare gravity, and GW, you can consider gravity as a permanent dip in space, while GW is a moving ripple in space.
Wherever this ripple passes, it minutely, and temporarily, changes the shape of the permanent dip. I.e it changes the gravity somewhat (depending upon strength of GW). Then the shape of dip (gravity) would return to its permanent state after the GW has passed.
Therefore, for the impacted time, it should impact speed of light, time, wavelength etc. Because all these get impacted by strength of gravity.
How much - That must have been predicted by general relativity math, may be someone else can quantify that.
