Calculating visibility of interference fringe

[Bacat]

Homework Statement

A photomultiplier (PMT) is arranged to detect the photons from a double-slit experiment. It is placed at a point P in the detection plane and makes an angle $$\theta$$ with the horizontal of one of the slits. Assume that the two slits have different widths and that the widths are much less than the wavelength of light, $$\lambda$$.

The probability amplitude for a single photon of wavelength $$\lambda$$ to strike the PMT from one of the slits is $$\sqrt{2}$$ more than for the other slit. Calculate the visibility of the interference fringes:

$$V=\frac{P_{max} - P_{min}}{P_{max}+P_{min}}$$

Where $$P_{max}$$ is the maximum probability and $$P_{min}$$ is the minimum probability that a photon is detected.

Homework Equations

$$A = A_1 + A_2$$ (amplitudes are summed)

$$P = (Abs(A_1 + A_2))^2$$ (probability is sum of amplitudes squared)

The Attempt at a Solution

I let $$A_1$$ be the greater amplitude, so it is equal to $$A_2 + \sqrt{2}$$. Then I use the equation for probability above to solve for $$A_2$$.

First problem:

I get different results if I solve it by hand or if I use Mathematica. Solving by hand I find that $$A_2 = A_1 = \frac{1-\sqrt{2}}{2}$$, but Mathematica tells me that $$A_2 = -1.20711 < 0$$ or $$A_2 = -0.207107 < 0$$

The answer Mathematica gives is in decimal format...but the fact that it comes up negative is really confusing me...I feel like the amplitude must be positive...?

Second problem:

If I take the answer I calculate by hand and add the amplitudes and square to find the probability I get:

$$P = A_1 + A_2 = (Abs(\frac{1-\sqrt{2}}{2} + \frac{1-\sqrt{2}}{2}))^2 = 3 - 2\sqrt{2} = 0.171573 < 1$$

The probability should sum to one, but it is much less than one.

What am I doing wrong?

 

[Bacat]

Should I post this somewhere else?

 

[Visibility]

This work is intended as a help to show, using very simple geometrical models, how http://www.visibilityacceleration.com is built into the Young's interference patterns produced by incoherent sources.