Light: interference - two slit

[sightempest]

Homework Statement

Two isotropic point sources of light (s1 and s2) are separated by 2.7micrometers along a y-axis and emit in phase at wavelength 900 nm and at the same amplitude. A point detector is located at point P at coordinate Xp on the x-axis (s1 is also on the x axis, at (0,0).) What is the greatest value of Xp at which the detected light is minimum due to destructive interference?

I don't know how to make a diagram for this on the computer but say there is an x,y plane: S1 is on the origin, P is at distance Xp to the right of S1 (on the X axis) and S2 is at 2.7um from S2 (lower, on the Y axis.)

Homework Equations

Y = (λL (m+½)) / d
Where Y is the position of the interference minima
L is the distance Xp
D is the slit space (2.7 um)
and λ is the wavelength.
m= mode.

1.d\sin\theta=m\lambda for constructive interference
2.d\sin\theta=(m+\frac{1}{2})\lambda for destructive interference
3.\phi=\frac{2\pi}{\lambda}(r_2-r_1) phase difference related to path difference

The Attempt at a Solution

I figure since the center of interference is always between the two slits (or in this case point sources), y = 1/2 of 2.7um, i.e. 1.35 micrometers, and thus

Y = (λL (m+½)) / d
1.35 x 10 ^- 6 = [ (900 * 10^-9) * L (m+½) / 2.7 * 10 ^ 0.6 ]
or
4.05 x 10 ^ -6 = L (m+½).

And I'm stuck there, I don't know if I'm to suppose that m is 0 seeing as how that would give the biggest L.

The answer is 7.88 x 10^-6 m, but I can't get it...

 

[Doc Al]

Rather than try to force fit the formula for two-slit interference to your problem, why not attack the problem directly? Imagine your detector on the x-axis, starting at the origin and moving outward. Now consider the path length difference between the light reaching the detector from both sources. As you move the dectector further along the x-axis, does the path length difference--and thus the phase difference--increase or decrease? What's the minimum path length difference that will lead to destructive interference? Write an expression for the path difference as a function of x, and use the insights from answering the above questions to solve for the maximum value of x.

 

[sightempest]

Thank you, kind sir!