How to get to this Surface slope formula?

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Discussion Overview

The discussion revolves around deriving a formula for the maximum slope of a wavefront as presented in a textbook on optical system design. Participants explore various assumptions and calculations related to wavefront distortion, specifically focusing on the relationship between amplitude, wavelength, and slope. The scope includes theoretical derivation and mathematical reasoning.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant expresses difficulty in deriving the textbook formula for maximum slope and suspects an assumption may be incorrect.
  • Another participant clarifies that the maximum slope refers to the maximum value of the derivative of the wavefront.
  • A participant questions the definition of "maximum slope" and how it relates to the spatial dimension.
  • There is a discussion about the role of sine waves versus saw-tooth approximations in the derivation, with one participant noting that the sine wave introduces a factor of π.
  • Participants discuss the scaling of peak-to-valley amplitude and its relationship to the sine wave function, with one participant deriving a formula that includes π but questions the factor of 2 in the textbook's equation.
  • Another participant raises a question about the relationship between electric field intensity and wavelength, seeking clarification on how these concepts are connected.
  • There is mention of different scalings for wavefront deviation, with references to λ/2, λ/4, and λ/8 irregularities.
  • Participants discuss the common practice of expressing wavefront distortion in terms of fractions of wavelength.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the derivation of the maximum slope formula. Multiple competing views and uncertainties about the assumptions and calculations remain evident throughout the discussion.

Contextual Notes

Participants express uncertainty regarding specific factors in their derivations, particularly the factor of 2 in the textbook's formula and the scaling of amplitude in relation to wavelength. There are also unresolved questions about the definitions and relationships between various parameters involved in the discussion.

Ornit
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This is no homework; but I cannot work it out so I assume one of my assumption is off... no matter how much I looked at it I can't get to the textbook answer.
I tried searching (Google is my friend); I got to Prof. Burge's wonderful lectures slides (Arizona.edu) but he follows a different path so it didn't help.

The image below is taken from Optical System Design by Fischer & Tadic-Galeb. It can be found in the link below. The author describes a simple way of calculating surface slope but I cannot follow his steps and would appreciate help in deriving his simple formula for Maximum Slope from page 346:
upload_2015-7-28_16-29-41.png

http://kundoku.free.fr/O/007/Optical System Design/Optical System Design.pdf

Assuming A is in waves, I tried working out the maximum slope as the magnitude of the disturbance (A*lambda, in length units) divided by the typical length of half a bump (D/(2*n)). This gives a slope of 2*n*lambda*A/D.
My answer is 4/pi off the textbook. What might have went wrong...?

Thanks
Ornit
 
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Ornit said:
(...) I tried working out the maximum slope as the magnitude of the disturbance (A*lambda, in length units) divided by the typical length of half a bump (D/(2*n)).

The maximum slope stands for "the maximum value of the derivative" of the wave front.
 
Thanks soarce, but I still can't understand the textbook version.
"the maximum value of the derivative"
Derivative with respect to what? I assumed with respect to unit length, this is why I tried dividing by the typical length of half a bump.
I completely don't understand how did pi got into the numerator of the textbook equation. What am I missing?
Thanks
Ornit
 
What is your definition of "the slope"? What do you understand by "the maximum slope of the wavefront"?

Following the picture you posted I reffered to the slope of the magnitude of the wavefront with respect to the spatial dimension. Then the derivative can be taken with respect to the spatial dimension (represented by vertical axis in your picture).
 
Thanks soarce,

Thanks for your help. You actually made me find the reason for the pi... At first I approximated the wavefront by saw-tooth, missing the (clearly noted) sine wave mention, hence no π in my initial derivation. But I still can't prove the book is right.

I think the author meant that "maximum slope" would be the max derivative of the curvy wavefront with respect to distance along the Y axis in the picture (I call it x below). If w(x) is the wavefront function, I interpret maximum slope = max(w'(x)). I scaled the PV of the sine to be A*λ.

When done with a sine wave, the π comes from the normalization to one diameter, and d(sin(u)/dx=cos(u)*du/dx.

After checking and rechecking I still end up with max(w'(x))=πAλn/D instead of the πAλn/2D .
Can't find a justification for the factor of 2. :frown:.

Thanks again for your help,
Ornit
 
The peak-to-valley amplitude is twice the amplitude of the sine?
 
True, but it doesn't help.
Let me write it down fully and properly (well, I hope not - if you find a mistake it will actually solve the problem :smile:):

w(x)=0.5*A*λ*(sin(π+2*π*x*n/D)+1)
(0.5 and +1 are for the wavefront to swing from 0 to +1 PV as you mentioned; π is just for phase shift to have the look like the 1.5 bumps illustration; sin(2*π*x*n/D) is what I think to be the critical tern for the missing factor of 2).​

Therefore w'(x)=0.5*A*λ*cos(π+2*π*x*n/D)*(2*π*n/D)

max(w'(x))=0.5*A*λ*1*2*π*n/D=A*λ*π*n/D :eek:
 
How did you scale PV amplitude to be A*lambda (or A*lambda/2) ?
 
Looking at the picture. I interpret A as PV (peak to valley) in waves therefore multiplied the sine expression by A*λ. The 0.5 factor comes from the (sin(y)+1): this expression will vary between 0 and 2, therefore I multiplied by 0.5. Have I misunderstood your question?
 
  • #10
I was just curious. The amplitude of the wavefront is given in terms of electric field intensity and don't understand how the field amplitude is related to lambda. I was thinking that you normalized in some way the energy of the wave and then lambda appears in the amplitude of the field. The factor 1/2 it's ok.

Anyway, who is lambda ? It is some wavelength ? I was reading the page you indicated to but I didn't see any reference to lambda.
 
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  • #11
Lambda is indeed the wavelength. The industry uses units of "waves" to measure Transmitted Wavefront Distortion (the deviation from a perfect wavefront, either plane or sphere). It is quite common to talk in units of waves (or fringes), therefore it seemed natural to me to assume those are the units (and at least it got lambda into the equation...).
Hard to believe the author had missed a factor of two. It haunts me...
 
  • #12
Maybe there are some different scalings to report the wavefront deviation. I have seen on internet that people speak of lambda/2, lambda/4 and lambda/8 irregularities of the surfaces.

LE: Have a look at Chapter 4, the wavefront deviations are given in terms of differences in optical paths and these differences are scaled to lambda. From here they speak of lambda, lambda/2 or lambda/4 criteria.
 
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  • #13
Yes, people usually speak in term of fraction of lambda (aka "wave") to describe surface figure or TWD (Transmitted Wavefront Distortion). My interpretation is that this is the "A" defined in the picture. For example, if the peak-to-valley in the picture is lambda/4, A=1/4 and my scaling is A*lambda.
 

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