The lens shown in FIGURE CP35.49 is called an achromatic doubl...

Question

The lens shown in FIGURE CP35.49 is called an achromatic doublet, meaning that it has no chromatic aberration. The left side is flat, and all other surfaces have radii of curvature R. Because of dispersion, either lens alone would focus red rays and blue rays at different points. Define ∆n₁ and ∆n₂ as n_blue - n_red for the two lenses. What value of the ratio ∆n₁ / ∆n₂ makes f_blue = f_red for the two-lens system? That is, the two-lens system does not exhibit chromatic aberration.

Accepted Answer

Hello, fellow physicists today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. An optical engineer is designing an achromatic lens for a new microscope. The engineer combines two lenses of different materials and shapes, plano concave and biconvex. As shown in the image below. As a result, all the wavelengths of light will focus at the same distance resulting in a sharper image. The engineer starts by selecting two materials for her lenses. The plano concave lens has refractive indices of N subscript, one comma B and N subscript, one comma R for blue and red wavelengths respectively. The biconvex lens has refractive indices of N subscript, two comma B and N subscript, two comma R for blue and red wavelengths respectively. The curved side of the plano concave lens and the curve side of the biconvex lens have equal radius of curvature R determine the relation between N subscript two comma B minus N subscript, two comma R and N subscript, one comma B minus N subscript, one comma R in order that the different wavelengths focus on the same point and eliminate chromatic abrasion. So that's our end goals. We're trying to figure out the relationship between N two comma B minus N two comma R and N one comma B minus N one comma R. So we're trying to determine the relationship between those two mathematical statements in order that different wavelengths will focus on the same point and eliminate chromatic abrasion. So we're trying to figure out when the chromatic abrasion will be eliminated. So that's our end goal. Awesome. So quickly, let's look at our little diagram or the image that's provided to us in the prom itself on the left of our lens here and the lens is our lenses I should say are represented in blue. So on the left, we have that's more boxy shaped is our Planno con lens. And the one that's more circular shaped is our biconvex lens and then cutting both of these lenses in half is our optical axis. And it's also our image is telling us to note that the medium surrounding our lenses is air. I'll circle that in blue to show that that's what our medium that's surrounding the lenses are is air. And for our final concave lens, we have N subscript one and for our biconvex lens, we have N subscript two. Awesome. Awesome. So now let's look at our multiple choice answers to see what our final answer might be. So for a, it's N subscript two comma B minus N subscript two comma R is equal to 0.5 multiplied by N subscript one comma B minus N subscript one comma R for B it's N subscript two comma B minus N subscript two comma R is equal to N subscript one comma B minus N subscript one comma R C is N subscript two B minus N subscript two R is equal to two multiplied by N subscript one B minus N subscript one R. And finally, for D it's N subscript two comma B minus N subscript two comma R is equal to four multiplied by N subscript one comma B minus N subscript one comma R. So to keep things simple, so I don't have to keep saying subscript a bunch of times. I'm just gonna say N two B and N two R and then you'll know that I'm saying and to, you know, comma B et cetera, but just to keep it simple, I'm gonna say that we're gonna denote it like that. So with that in mind, let's start solving for this problem. So first off, let us note that we are looking for the focal point on the achromatic lens as described to us in the problem text, this lens is made using a plano concave lens which we call lens one and a biconvex lens which will call lens two to find the focal points. We will consider a ray that is parallel to the optical axis. So where S equals infinity. Awesome. So the parallel rays do not experience refraction at the interface at the lens, the lens is left interface. So at the lenses left interface, it does not experience refraction at that particular interface. So as a result, no changes will occur until these rays encounter the first curved surface which is located between lens one and lens two. For the first curved surface, the boundary between the Plano concave and bio convex lens, we have the position of the image S prime which we can write as and this is the equation that we need to use. So N one divided by S so N one divided by S plus N two divided by S prime is equal to N two minus N one divided by R which we now need to start isolating and solving for S prime. So let's do that. So we will get that N one divided by infinity plus N two divided by S prime is equal to N two minus N one all divided by capital R which finally we can isolate and solve for S prime. So S prime is equal to N two multiplied by capital R all divided by N two minus N one. OK. So the Plano concave lens produces an image at S prime is equal to N two multiplied by R divided by N two minus N one. Now we will consider the image formed by the biconvex lens, the distance between the plano concave and biconvex lens is zero. Thus, the object of this biconvex lens is S prime is equal to N two multiplied by R all divided by N two minus N one. The image found in this case of the mono concave lens. So the focal point of an optical system for an object at infinity S equals infinity is the point where parallel rays of light converge after passing through the system. So when the light exits the biconvex lens, we obtain the focal lens of this achromatic lens which we can write as N two divided by N two multiplied by capital R divided by N two minus N one which this is all added to an air divided by negative F which is equal to N air minus N two divided by capital R. So now we need to take this equation and rearrange it and isolate and sulfur F which is our focal length. So let's do that. So N two multiplied by N two minus N one all divided by N two multiplied by R plus one divided by negative F is equal to one minus N two divided by capital R which we can simplify and write as negative one divided by F is equal to one minus N two divided by R plus N one minus N two all divided by R or capital R I should say which is equal to one minus N two plus N one minus N two, all divided by R which is equal to one minus two, N two plus N one all divided by R. Oh That was a mouthful. OK. So now that we have our equation a little bit more simplified, we can easily go on and just isolate and solve for F all by itself the focal length. So when we do that, we get that F is equal to negative R negative capital are divided by one minus two, multiplied by N two plus N one. So now if we consider all of the visible wavelengths focus on the same point, we can state that F blue is equal to F red. So the focal point for blue light is equal to the focal point of red light. So we can therefore go on to say that negative capital R divided by one minus two N two B plus N one B is equal to negative capital R one minus two and two are plus N one R. Awesome. Awesome. So now we can simplify this equation and write that one minus two N two B plus N one B is equal to one minus two N two R plus N one R. So we need to keep simplifying here. So let's do that. You will get that two N two R minus N two B is equal to N one R minus N one B which we can simplify once more to state that N two, B minus N two, R is equal to 0.5 multiplied by N one B minus N one or hooray. We did it and this will be our final answer. Awesome. Awesome. So thus to obtain an achromatic lens, we should have N subscript two comma B minus N subscript two comma R is equal to 0.5 multiplied by N subscript one comma B minus N subscript one comma R. So looking at our multiple choice answers, the correct answer has to be the letter A. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

Key Concepts

Chromatic Aberration

Refractive Index and Dispersion

Focal Length and Lens Combination

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