What is the formula for the angle of deviation in a thin prism?

[Misr]

I finally found something about thin prisms on the web

a thin prism, which is generally a prism with an apical angle of less than 15°. For simplicity, we will assume that the initial object ray is striking the first surface normally (or perpendicularly). Since the ray is normal—or at a 90° angle—to the first surface, no refraction occurs. However, once the ray strikes the second surface, it reaches the surface at an angle i and is therefore refracted (or bent) in the direction of the base.
For thin prisms, which include most prisms in ophthalmic optics, the refraction at the second surface depends primarily on the apical angle (a) of the prism and the refractive index (n) of the material. In practice, the angle of incidence at the first surface will affect the extent to which light is deviated by the prism. The amount of deviation, in degrees, is given by the angle (d), while d = i' - i.
[PLAIN]http://img6.imageshack.us/img6/1474/slide16v.gif
Moreover, the angles of incidence (i) and refraction (i') are related by the refractive index (n) of the prism material according to Snell's law. For a thin prism, with a relatively small apical angle (a), it can be shown that the approximate deviation (d), in degrees, is given by:
d = (n - 1) × a
For small amounts of deviation, 1 degree of deviation is roughly equal to 1.75 prism diopters. For prisms made from hard resin, the refractive index (n) is 1.500. This simplifies our formula even further, since 1.500 - 1 = 1/2. Consequently, for prisms made from hard resin, the deviation in degrees is roughly equal to half the apical angle.
For example, consider a ray of light from an object point passing through a hard resin prism (n = 1.500) with an apical angle of 10°. The deviation (d) of this ray is equal to (n - 1) × a = (1.500 - 1) × 10 = 5°. This is approximately 8.75 prism diopters.

I don't understand whether the angle of deviation depends on a the angle of incidence in a thin prism or not?
Is the thin prism always in the position of minimum deviation?

 

[xts]

If you want to get understanding (rather than get ready answer to simplified yes/no question) - compute it yourself.
Compute what is deviation angle as a function of angle of incidence and apical angle, then make several plots (for several apical angles, e.g. 10°, 20°, 30°, 45°): deviation angle as a function of angle of incidence. And compare the plots.
That's pretty easy calculation. You know Snell's law. You can do it yourself! Yes, you can!

 

[Misr]

You are right,I already tried something like that,but I failed .I always get different angles of deviation for different angles of incidence,and this should not happen,because we study that the angle of deviation in a thin prism isn't affected by the angle of incidence.

 

[xts]

Please, post the plots you made and formulae you got - I'll try to point out what you did wrong.

 

[Misr]

That's my work-very sorry for the bad resolution
I drew two diagrams
At A=45 degrees
[PLAIN]http://img190.imageshack.us/img190/9889/dsc00745nq.jpg
[PLAIN]http://img90.imageshack.us/img90/2920/dsc00748yk.jpg
-----
At A=10 degrees

[PLAIN]http://img831.imageshack.us/img831/7805/dsc00735xl.jpg
[PLAIN]http://img854.imageshack.us/img854/1451/dsc00738vc.jpg
The first graph looks logical but there's a math error
the second one looks very strange
I don't know any laws about thin prism ,so I drew the angles manually,and at every angle of incidence,the angle of deviation has a different value.It's not supposed to be so according to the relation
alpha=A(n-1)
so the angle of deviation is only dependent on the angle of the prism and the refractive index
Hope you could find the problem with the second graph
Thanks

 

[Misr]

I've put my work as you told me,so could you tell me what's wrong with it?

 

[xts]

First graph looks reasonably, the second is totally wrong - check calculations (the calculator may help)

You should find first the formula describing the deflection angle as a function of incidence angle and apical angle, then make several graphs of this formula for various fixed values of apical angle.
You may want to use any computer program for drawing graphs of functions (you'd probably been taught one of such programs at school - but even excel can do this), rather than computing all that trigonometry by hand...

 

[Misr]

You should find first the formula describing the deflection angle as a function of incidence angle and apical angle, then make several graphs of this formula for various fixed values of apical angle.
You may want to use any computer program for drawing graphs of functions (you'd probably been taught one of such programs at school - but even excel can do this), rather than computing all that trigonometry by hand...

which formula?I don't know how to do all of this :(

 

[xts]

Derive a general formula, describing the deflection angle as the function of two parameters: apical angle and incidence angle.

Derivation is pretty easy. You must just draw a picture, give names (symbols, rather than actual values in degrees) to all angles, then combine and reduce several simple trigonometrical formulae.

 

[Misr]

do u mean that
Apical angle=second Angle of incidence-Angle of refraction?
This is just for thin prism I guess.
How about math errors in both graphs

 

[xts]

do u mean that
Apical angle=second Angle of incidence-Angle of refraction?

Not quite that, but that is an idea: to express the deviation angle as a function of apical and incidence angles. You must apply Snell's law twice, and express the incidence/refracted ray angles as sums of other angles.

How about math errors in both graphs

First one seems ok (although I did not check it very carefully), the second is wrong.

 

[Misr]

Not quite that, but that is an idea: to express the deviation angle as a function of apical and incidence angles. You must apply Snell's law twice, and express the incidence/refracted ray angles as sums of other angles.

first:Is the equation I wrote correct?
Could you explain more:(
I still can't imagine what I have to do

 

[Misr]

I checked the numbers in the second graph,there's nothing wrong with them
Could you give me some laws about thin prism to use them instead of drawing the prism manually?

 

[xts]

Could you give me some laws about thin prism to use them instead of drawing the prism manually?

There are no more laws applicable here than Snell's law and trigonometric laws for sinus of sum of angles, nor there is no need for special laws so closely related to Snell's one.
You may easily derve Misr's law: the law describing deflection angle of the prism...

first:Is the equation I wrote correct? - you haven't wrote the equation, except of sime side-notesyou used in calculations, but as you haven't presented it as a part of ordered reasoning, I can't judge them.

I still can't imagine what I have to do
Make a drawing how the ray passes through a prism. Something like in your first post. Mark all angles and give them symbols. Write down Snell's relations between incidence/refraction angles on both surfaces, using n as refraction index. Express all angles as sums/differences of: apical angle, incident angle on first surface, deflection angle.Transform those equations to have it in form:
deflection_angle = some_function_of(incident_angle, apical_angle, n)
Make graphs of that function for several fixed values of apical_angle and single example value of n
Compare those graphs visually and find some regularity of the graph shapes - it will be obvious as you look at the graphs.

 

[Misr]

you mean by "deflection angle" the angle of deviation?

 

[xts]

you mean by "deflection angle" the angle of deviation?

Yes. Sorry for wrong terminology. The angle between final ray and incoming one.

 

[Misr]

I've been working on this for hours
I derived two equations but there's something wrong with them
deviation angle=angle of emergence- angle of incidence+Apical angle
This works on some values of angle of incidence
but while making some calculations I found that's not true-it gives me a negative value for the angle

And it is the same for
Apical angle=Angle of refraction-second angle of incidence
Could you help me with signs because I'm getting very bored

 

[xts]

two equations but there's something wrong with them
deviation angle=angle of emergence- angle of incidence+Apical angle

You may chose any convention of signs you like.
I see you got lost in it. A little help from my side - I spent 10mins to make a readable drawing:

In this convention your equation is:$$\delta=\theta-\beta-\alpha$$

but while making some calculations I found that's not true-it gives me a negative value for the angle

Please stay with naming and sign convention as on this drawing (angles are positive if they are like on the drawing, negative if they lay on opposide side of the normal to the surface), if you prefer other names and other sign convention - it is ok, but post a readable drawing.

 

[Misr]

but according to this drawing
deviation=emergence-incidence+apical

not deviation=emergence-incidence-apical

 

[Misr]

oh sorry you are right

 

[Misr]

Ok
that's the graph
http://imageshack.us/photo/my-images/546/graphp.jpg/
At apical angle=10 degrees,and n=1.5
It is still wrong I guess

 

[Misr]

[PLAIN]http://img546.imageshack.us/img546/2479/graphp.jpg

 

[xts]

Still wrong.
Check if you derived your formula for deviation from Snell's law correctly.
Show me your formula of defiation_angle(incidence_angle, apical_angle, n) - if you want me to check it.
Make such drawings not only for angles 0-40 deg, but for a range -40 - 40 deg
and make such drawings for several apical angles (e.g. 5, 10, 15, 20, 25, 30 deg)

 

[Misr]

I just used the formula you provided in post 18
and I used the formula
Apical angle=second angle of incidence-Angle of refraction

Using Snell's Law I find the angle of refraction "fie" ,then I find the second angle of incidence using this formula :Apical angle=second angle of incidence-Angle of refraction
then I find the angle emergence using Snell's Law again
then I use the formula in post 18 to find deviation
That's the way I work,so what's wrong?

 

[xts]

Using Snell's Law I find the angle of refraction "fie" ,then I find the second angle of incidence using this formula :Apical angle=second angle of incidence-Angle of refraction
then I find the angle emergence using Snell's Law again
then I use the formula in post 18 to find deviation
That's the way I work,so what's wrong?

The sequence you use is ok, but apparently you did something wrong in any of those steps, because the numbers you got are wrong.
Show me the final formula you use for: $\delta$ as a function $\delta(\beta, \alpha, {\rm{n}})$.

 

[Misr]

I didn't use any other formulas I'm not very good at Math
I don't understand what you mean,I checked the numbers well
Could you write this final formula ?

 

[xts]

I'm not very good at Math...Could you write this final formula ?

No. I won't do it for you. If you want to learn something - you must do some calculations yourself. I've already made a sketch for you - which was easy enough that you could do it yourself.

Put a little your own effort to find an answer to your question!
Ready answers from others won't teach you nothing. Do it yourself.

 

[Misr]

Okay,that's right.but do i need this final formula?I already drew this wrong graph without the formula
Could you give me a small hint or tell me what kind of formula you mean

 

[Misr]

which formulae would I need to derive this formula

 

[xts]

do i need this final formula?I already drew this wrong graph without the formula

As you already drew this wrong graph without the formula, you need either:
- simpler formula (derive one yourself, I've already told you its form: δ as a function δ(β,α,n);
- more careful calculations, if you prefer to make it in multiple steps, rather than using single formula.

which formulae would I need to derive this formula
Read previous posts. I answered this question already.

 

[Misr]

I checked my calculations several times and there's nothing wrong with them
for incidece=0,refraction=0
then second angle of incidence is eqivalent to the apical angle=10 degrees
by applying Snell's law,we find the emergence angle = 1.5sin10

using the formula in post 18 ,you will find that deviation=5 degrees

I repeat these steps for different angles of incidence and i get different angles of deviation(misfortunately)

You can check the calculations yourself and you'll find that there's nothing wrong with them

as for the simple formula,I can't derive one formula unless I have other variables
n=Sin(angle of incidence)/sin(angle of refraction)=Sin(angle of emergence)/sin(angle of second incidence)=sin(deviation+incidence+Apical angles)/sin(Apical+refraction angles)
So I still have "refraction" angle in the formula

 

[xts]

for incidece=0,refraction=0
then second angle of incidence is eqivalent to the apical angle=10 degrees
by applying Snell's law,we find the emergence angle = 1.5sin10
using the formula in post 18 ,you will find that deviation=5 degrees

Wrong! You made your plot with 0.1° accuracy - that is ok.
But with such accuracy you should get 5.1°, not 5°.

 

[Misr]

Wrong! You made your plot with 0.1° accuracy - that is ok.
But with such accuracy you should get 5.1°, not 5°.

So what I have to do?
Would it make such a great difference if I get 5.1 or 5?
how can I draw an accurate graph?

 

[xts]

So what I have to do?

Calculate everything again with the accuracy you assumed at start. If you assumed accuracy 0.1° - make all calculations and plots with such accuracy.
And make your plots with points placed much densier, starting not from 0, but in range, let's say -40° to 40°.

Would it make such a great difference if I get 5.1 or 5?

You don't know that.
If you believe the accuracy 0.1° is too high, and lower accuracy (e.g. 1° or 0.5°) would be sufficient - then make a lower accuracy plot and mark points according to this, at the risk that after making the plot you may spot it was not accurate enough and you must do it again with better accuracy. The best practice is to make such plots with best accuracy reacheable with your computing and presentation tools.

But if you assume that the plot is made with 0.1°, and you mark 5° where appropriate value is 5.1°, you are not able to say if the plot you just made results from the formula or from your calculation errors, thus you cannot trust it.

how can I draw an accurate graph?

Up to you: from computing values for multiple points and plotting them manually with pencil and graph paper, through manual computation of points and plotting them in some graphic program (like you did recently) to using any data/function plotting program you like (the one you were taught at school). If you haven't learned any at school - you have opportunity to learn one now - there are lots of such programs available free, very popular one is gnuplot (http://www.gnuplot.info/). As the last resort you may use some spreadsheet program to make plots: openoffice.org:Calc or MS-Excel

Added: Don't ask me to teach you how to use gnuplot! It is pretty well documented, there are examples and good manual.

 

[Misr]

Okay,one more trial
[PLAIN]http://img849.imageshack.us/img849/3051/graphio.jpg
I hope you spend some time checking these calculations
I tried to be more accurate this time