Quartz Wave Plate

How do you find the "x-th" order thickness of a lambda/4 waveplate?

I have no idea how you solve for the order of a waveplate, I have two questions that talk about it:

1. What is the thickness of a zeroth-order quarter wave plate made of quartz?

2. If quartz plate 0.735mm thick acts as a quarter wave plate, what is its order?

n_fast = 1.5462
n_slow = 1.5553
lambda = 546 nm (incident)

For an "x-th" order lambda/4 waveplate:
(delta n)*(thickness) = (lambda)*(x + 1/4)

1. zeroth-order quarterwave plate:
(delta n)*(thickness) = (lambda)/4
so thickness = (546 x 10^-9 m)/4/(1.5553-1.5462)
thickness = 1.5 x 10^-5 m = .015 mm

2. (delta n)*(thickness) = (lambda)*(x+1/4)
(x + 1/4) = (1.5553-1.5462)*(0.735 x 10^-3 m)/(546 x 10^-9 m)
(x + 1/4) = 12.25
so x = 12

By the way, the .015 mm thickness is pretty fragile, so zero-order quarterwave plates are sometimes made by the combination of two waveplates in series, with one rotated 90 degrees relative to the other. For example, a 5.25 waveplate could be crossed with a 5.00 waveplate, for a net retardance of 0.25 waves.

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