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When is the image produced by a thin lens sharp?

  1. Nov 25, 2016 #1
    1. The problem statement, all variables and given/known data

    Hi everybody! Being given three identical thin lenses with the same focal length, I have to determine ##a## (distance object-screen), ##d_a## (distance lens 1-lens2) and ##d_b## (distance lens 2-lens 3) so that a sharp image of the object appears on the screen regardless of the position of the optical system between the object and the screen (see picture).

    2. Relevant equations

    Lens equation: ##\frac{1}{f} = \frac{1}{s_i} + \frac{1}{s_o}## with ##s_i##: distance from lens to screen and ##s_0##: distance from lens to object.

    3. The attempt at a solution

    Well I didn't get very far because I don't really know what is the condition for an image on the screen to be sharp. Is it the case only when the lens equation is fulfilled, that is when I have a certain ##s_i## and ##s_o## so that the sum of their inverses is equal to ##1/f##?

    And if so, is the following thinking correct? Say a ray of light is going from ##S## to ##L_1##. I want it to be sharp when meeting ##L_2##, so the following equation has to hold:

    ##\frac{1}{x} + \frac{1}{d_a} = \frac{1}{f}##.

    Is that correct? If so I can set up an equality with three equations, but I am afraid it remains dependent of ##x## then. Any clue about how to tackle such problems?


    Thanks a lot in advance for your answers.
     

    Attached Files:

  2. jcsd
  3. Nov 25, 2016 #2
    What about the matrix method for example? If I put one after the other translation and refraction matrices I would say that a ray reaching the screen has for incident angle and height:

    ##\begin{bmatrix} \theta_r \\ r_r \end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} \theta_i \\ r_i \end{bmatrix}##
    ##= \underbrace{\begin{bmatrix} 1 & 0 \\ x & 1 \end{bmatrix}}_{\mbox{object} \to L_1} \underbrace{\begin{bmatrix} 1 & -1/f \\ 0 & 1 \end{bmatrix}}_{\mbox{refraction } L_1} \underbrace{\begin{bmatrix} 1 & 0 \\ d_a & 1 \end{bmatrix}}_{L_1 \to L_2} \underbrace{\begin{bmatrix} 1 & -1/f \\ 0 & 1 \end{bmatrix}}_{...} \begin{bmatrix} 1 & 0 \\ d_b & 1 \end{bmatrix} \begin{bmatrix} 1 & -1/f \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ k_3 & 1 \end{bmatrix} \begin{bmatrix} \theta_i \\ r_i \end{bmatrix}##

    where ##k_3## is the distance between the third lens and the screen. Is that expression correct? And more importantly, could that bring me somewhere? I just started using the matrix notation for lenses today, so I am very unexperienced with that method and unsure about what I can and can't do with it.

    I think the image will be sharp if all rays intersect at some point on the screen. I read somewhere that ##C## has to be zero for the rays to intersect at the same point independently of ##\theta##, does that make sense? If so I get a crazy equation that doesn't simplify so easily, so I'd rather wait for an answer before diving into it. :)

    Thank you in advance.


    Julien.
     
    Last edited: Nov 25, 2016
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