When is the image produced by a thin lens sharp?

• JulienB
In summary, the conversation discusses determining the necessary distances between three identical thin lenses with the same focal length in order to produce a sharp image on a screen, regardless of the position of the optical system between the object and the screen. The lens equation is mentioned and it is suggested that the matrix method could also be used. The condition for a sharp image is when all rays intersect at the same point on the screen, and it is noted that this may involve a complicated equation.
JulienB

Homework Statement

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).

Homework 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.

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.

Attachments

• IMG_0953.jpg
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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. :)

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1. How can I determine if an image produced by a thin lens is sharp?

To determine if an image produced by a thin lens is sharp, you can use the thin lens equation, which takes into account the focal length, object distance, and image distance. If the resulting image distance is equal to the object distance, then the image is considered sharp.

2. What factors affect the sharpness of an image produced by a thin lens?

The sharpness of an image produced by a thin lens is affected by several factors, including the focal length of the lens, the size of the aperture, and the distance between the lens and the object being photographed. The quality of the lens also plays a significant role in determining the sharpness of the image.

3. Can the sharpness of an image produced by a thin lens be improved?

Yes, the sharpness of an image produced by a thin lens can be improved by adjusting the focus of the lens, using a smaller aperture, and ensuring that the lens is clean and free from any debris or scratches. Upgrading to a higher quality lens can also improve the sharpness of the image.

4. How does the thickness of a lens affect the sharpness of an image?

The thickness of a lens does not have a significant impact on the sharpness of an image. As long as the lens is within the range of being considered a "thin" lens, the thickness should not affect the sharpness. However, thicker lenses may have more aberrations, which can impact the overall quality of the image.

5. Is it possible for an image produced by a thin lens to be too sharp?

In most cases, it is not possible for an image produced by a thin lens to be too sharp. However, if the lens is of very high quality and the aperture is extremely small, it is possible for the image to appear too sharp, resulting in a loss of detail and overall quality. This is known as diffraction, and it can be avoided by using a larger aperture or reducing the sharpness in post-processing.

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