Maybe this article helps. It speaks of R2 high order gravity theory.

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@Dick Vestdijk, thanx for the reference, but the authors don't actually touch on this point. 

Rogier,

This is a very good question and the answer depends on the model of photons being used.  I am going to start with a thought experiment in which we assume that we are using electrons with a well-defined velocity rather than photons to measure distance using the time of flight.  With particles such as electrons, we do not need to be concerned about polarization of photons and the effect this might have on time of flight measurements. (This will be explained later.) We will also assume that the experiment includes some mechanism for reversing the direction of the electrons equivalent to the massive mirrors used in the laser interferometer experiments.  The first point is that the mirrors do not physically move when a gravitational wave passes.  This can be proven by imagining multiple mirrors placed at different separation distances generally along the same optical path.  If the mirrors actually moved, this would create impossible results when the distance between various mirror combinations was measured.  Therefore, the length change occurs because the properties of space are affected by the gravitational wave.  The distance measurement made by particles propagating at a known velocity would agree with a hypothetical resonant bar gravitational wave detector. 

Now we get to the hard part and assume photons.  Most scientists visualize photons as packets of energy which propagate in the empty void of spacetime. Therefore, if you adopt this model, the waves are just a property of the photon that does not affect the propagation speed.  For example, the current record for the shortest pulse of laser light is about 8×10-17 seconds.  Even though this would not achieve the accuracy required, in principle the wavelength would not affect the arrival time if a light pulse (photon pulse) propagates like a particle in an empty void. 

However, I have written two papers (attached below) which in part deduce properties of spacetime from gravitational wave equations.   For example, gravitational waves propagate in the medium of spacetime and it is possible to determine the impedance encountered by gravitational waves as they propagate through the medium of spacetime.  The impedance of spacetime was first determined by Blair to be Zs = c3/G ≈ 4×1035 kg/s.  In the papers below, I have determined that photons encounter the same impedance.  This has profound implications because it implies that photons are not packets of energy propagating through the empty void of spacetime, instead, photons are waves possessing quantized angular momentum propagating in the medium of spacetime.  This and other implications are discussed in the papers. 

So, what does this have to do with your question about the proportional wavelength change canceling the gravitational wave signal?  If you visualize waves actually propagating in the medium of spacetime and if you visualize the vacuum having energy density as implied by QM, QED and QCD, then it is necessary to consider the actual mechanics of wave propagation.  Light is a transverse wave.  Its propagation speed can be polarization dependent if there is a birefringence in the medium through which the light propagates.  Normally spacetime is very homogeneous, but the presence of a gravitational wave effects the transverse dimensions differently.  A spherical volume of spacetime becomes an oscillating ellipsoid as a gravitational wave passes.  Light is also a transverse wave and the speed of propagation depends on the transverse properties to the propagation direction.  Therefore, if you accept the previous model, it makes predictions about the detection of gravitational waves. The two most important predictions are: 1) Since the arms of the interferometer are in the local horizontal plane, it is impossible to detect gravitational waves if linearly polarized light is used with vertical polarization.  Both beams would experience the same effects because they both have the same vertical polarization.  The fact that they are propagating at 90° angles does not matter because both beams are encountering the same transverse property.  2) The gravitational wave should produce a polarization effect which would make it much easier to detect gravitational waves if the experiment is designed to monitor changes in polarization.  For example, a single circularly polarized beam should exhibit a polarization effect which should be easier to detect because many of the sources of noise present in interferometers are greatly reduced. 

Therefore, to answer your question, a strong gravitational wave would be detectable if the gravitational wave makes the medium of spacetime into an oscillating birefringent medium and if photons are quantized waves propagating in the medium of spacetime.  Also, gravitational waves would be detectable if photons are packets of energy which propagate through the empty void of space without any wavelength dependence.

Article Spacetime-based model of EM radiation

Chapter Spacetime Based Foundation of Quantum Mechanics and General Relativity

 Dear @John Macken, thanks for you paper and explanation.  I am going to read if I have a little more time.

I am not sure that bringing in quantum mechanics into the picture is really helpful though. The point I am making is purely based on classical General Relativistic electrodynamics and so should be explainable purely in terms of  classical concepts. But I will see what your papers say! 

Please read the paper, may be helpful for you.  A new approach to the correction of

Galilean transformation.

Answering this interesting and legit question in Huge words does not help. Some are bringing quantum mechanics and some with some complicated mathematics that a common man who knows a little bit of special relativity does not understand at all.

Here is my personal view of the experiment.

Both the arm length and the wavelength stretch/compress because of the gravitational wave. But no one is talking about the constant speed of light. Of course, the returning light from both arms come to the beam splitter covering exactly equal number of waves in both arms. If they return at same time, they interfere destructively for sure. But, the fact is, light from compressed arm comes faster than than that from the stretched arm because speed of light in both arm is equal, c.

We can ask if the time dilation by GW compensates the interference. 

My answer is: There is only one beam splitter/detector so it can not show two types of time dilations for two arms at the same time. So the time gap between the arrival of two beams is not affected by the time dilation caused by the gravitational wave on the detector. 

Even in simple words: this experiment is detecting optical beats: the phase difference between two light beams, not because of the path difference but because of the frequency difference. The light in compressed arm is blue-shifted and that in stretched arm is red-shifted. But don't expect that they are truly waiting for a complete beat to come. That requires the laser to travel more than billion light years and the gravitational wave does not wait that long to create a single beat! 

Let me suppose two sine curves  Y1 =  sin (2*pi*f*x) and Y2 = sin (2*pi*f+df*x) drawn in paper. They differ in frequency by df but have initial phase difference 0.  (Light in two arms start at same time from the splitter but are of different frequency because of red-shift/blue-shift). As I keep drawing the curves, the phase difference between the two sine curves keep increasing with x (phase difference = 2*pi*df*x). That is what they are doing by increasing the arm length and using resonant cavity to increase the distance covered by light and hence to obtain detectable phase shift!

Your doubts are the same as mine. If gravitational waves are an oscillation of gravitational field, then light is influenced in the same way as bodies - that is why Michelson experiment is independent of orientation and of motion. There is a tiny possibility of detection of gravitational waves in this way which arise from the fact that matter has inertia and so the consequences of gravitational waves on matter are delayed by it; in this case, the interferometer is a high pass filter for gravitational waves.

Usually, I can obtain whatever result I wish by manipulating equations; therefore, for me, results from equations without an explanation I can understand are meaningless.

I think also that this is a wonderful experiment to detect minimum mechanical vibrations at Earth scale, due to internal phenomena in Earth or extraordinary phenomena at surface; which seem to be a much better explanation for the result then the one presented. I don't understand how can the possibility of a mechanical vibration too small to be detectable by conventional devices be ruled out, although I recognize that they have a theory in support of the signal detected, although a rather speculative one.

I've approached this question from the field equation point of view a number of years ago with similar questions in mind. The exercise was confusing to say the least and while correct, I think is somewhat "wrong minded" in some sense. The approach is to look at the modifications to the EM field equations to first order. In this approach the antisymmetric tensors F = (E,B) and M = (D,H) are related by the metric. Expanding this relation yields an effective gravitationally induced current and charge. Here we are working in a lab frame a TT gauge for the wave. This effective source gives rise to it's own EM field in addition to the one being used to probe geometry. This ends up being a very complex way to get the very same result as a length change. The standard way of viewing LIGO is much much cleaner to the point and quite correct.

  Now, having said this, the field equation approach does provide a means of analyzing approaches which are not based on interferometery where computing the effect due to length changes is not at all apparent. 

 @Paul Colby.

dear Paul, thank you for you answer. I would be interested to see your computations. Your point of view (an effective charge/current) is interesting, and rather surprising to me.

@Rogier Brussee

I've written a set of slides on the work I did in 2011 and posted them on my RG profile some time ago. My understanding has improved after some reflection and some help from a kindly expert mostly in regard to the response of matter well above resonance. 

So, an anonymous down vote, while always up lifting, adds no real information to the discussion. The slides I posted are based on exceedingly standard stuff. What's not to like? 

Glasses are suspended in such a way that they are not affected by variation in lengths of arms.  However, it seems that the laser production unit and interference detecting unit are attached with that the arms in ends.  They are affected by variations of lengths in arms and so variations in lengths of laser beams happen, which produce interferences.  I like to post this answer for a different reason.  An explanation for detection of gravitational waves can be applied even for detection of usual electromagnetic waves.  Details can be found in the article attached with this answer. 

Article LIGOs Detected Magnetic Field Waves; not Gravitational Waves

Rogier: Your mental picture is just fine because it can also be made into a real picture. The LIGO folks and the theorists who stand by their claims, meanwhile, have been revising their explanations (reworded LIGO FAQ, reworded article about LIGO on Wikipedia, etc.), making it ever more foggy, muddy, and impossible to picture at all.

They never provide a sensible picture of their "very slightly" deformed wavelength. They never provide a picture of how the "newly emitted" laser light is SOMEHOW affected differently from the light that had been in the tube milliseconds earlier. If their claim made sense, they'd draw it out. They never do.

Instead they wave their hands over their Riemann tensors expecting their audience to be taken in by their schtick. In my humble opinion, something is very fishy with the whole G-Wave industry.