[Optics] Find maximum order number, Fraunhofer diffraction

[Decimal]

Homework Statement
This problem concerns a single slit diffraction experiment where Fraunhofer diffraction is observed on an observing screen via a positive converging lens with focal distance f. The diffraction pattern has a central maximum of width dy, and the vacuum wavelength of the light is lambda.The attempt at a solution

The first problem was to find the slit width of this experiment of this experiment with the above data. I was able to solve this fine, by finding the position of the spots where the irradiance of the light becomes zero and then solving for the slit width.

However the second part of the question asks me to assume the the slit width as a given quantity b, and then calculate the "maximum order number for irradiance nulls". I don't really understand the question here. Is there a maximum order number? Don't the irradiance nulls just repeat forever (theoretically)? Because I don't really understand the question I also don't know where to start. Could somebody give me some pointers?

Thanks!

 

[jtbell]

"maximum order number for irradiance nulls"

Is this for an experiment that you actually performed? If yes, then I suspect that this means "visible irradiance nulls". That is, what is the maximum value of n (order number) for which your observed pattern is actually visible to your eyes?

 

[Decimal]

This is just a problem from a problem sheet, not an actual experiment I performed. It doesn't say explicitly whether this concerns visible nulls. The equation I used to explore irradiance nulls in the first part of the exercise and what we derived in the lecture is the following $$ I(\beta) = I_0 * (\frac {sin(\beta)} {\beta})^2 $$

Here beta equals

$$ \beta = \frac {kby} {2f}$$

I suspect these equations will have to be used in some form in this part of the problem as well, but I am not sure. As I said I don't fully understand the question.

 

[jtbell]

Do the viewing screen or the lens have a given width?

 

[Decimal]

No, sadly they don't. All given data is stated in the post. I am starting to think it has something to do with the way the sinc function behaves at larger order numbers but I would actually expect the irradiance nulls to become more frequent at higher order numbers, not disappear.

 

[jtbell]

In Fraunhofer diffraction the maxima become weaker and weaker with increasing angle or distance from the center, but never disappear completely, at least in principle.

I think this needs some clarification from your instructor. I suspect that some information is missing, or you’re expected to assume something that isn’t stated explicitly in the problem.

 

[Decimal]

Alright, I guess I will save this problem for next week. Thanks for the help!

 

[Decimal]

In case anyone else has this question, I was able to find the solution in a textbook somewhere.

Apparently what was meant by a maximum order number was using the equation $$\beta = \frac {kb} {2} * sin(\theta)$$ Here theta is the angle between the light ray and the normal vector of the screen. This equation formed an intermediate step for the derivation with the focal length I used in the first part of the exercise. The maximum order number for irradiance nulls meant setting theta so that sin(theta) would equal 1 and rewriting the wave number to include the known wavelength in the equation.

I still don't really understands what this physically means though. Say I set theta equal to Pi/2, then apparently I will find a maximum order number? But this means the light will never hit the screen, so how can you even speak of order numbers in this case?