Sign-conventions and Huygen’s principle in Geometric Optics by Chiu (4/10/02)

In this note we discuss

· a self consistent “sign-convention” table in geometric optics, and

· show how Huygen’s principle can help us to solve lens problems which involve a sequence of lenses.

We refer students to class notes for details.

Caution: Unfortunately the word-html code does not display figures properly on the web. The same page will be handed out in the class and also in TA sessions. The reader should refer to the handouts for the figures.

1. Image formed due to reflection on a spherical surface (mirrors). Introduce the x-coordinate in such a manner that the location of a real image will always have a positive x-value. For mirrors, the origin is chosen to be at the mirror. Here the x-axis must then be drawn to the left. The universal relationship which is applicable for both mirrors and lenses is given as: (1/p)+(1/x_image)=1/x_focus.

The object coordinate x_obj = p >0. x_focus = x_c /2.

· For a concave mirror, x_c>0, so x_focus>0. It corresponds to a convergent mirror.

· For a convex mirror, , x_c<0, so x_focus<0. It corresponds to a divergent mirror.

2. Image formed to due to refraction on a spherical surface (transparent media)

		x

Here the x-coordinate is from left to right. The origin is the intersection between the spherical surface and the x-axis. Denote the index of refaction of the medium to the left to be n_L and to the right n_R, the appropriate relationship is:

(n_L/p)+(n_R/x_image)=(n_R-n_L)/x_c.

· Convergent surface occurs when the right hand side: RHS = (n_R-n_L)/x_c >0.

· Divergent surface occurs when the right hand side: RHS = (n_R-n_L)/x_c <0.

3. Lenses: For lenses, the x-axis is again from left to right. The origin is located at the lens. The basic relationship is the above mentioned universal expression:

(1/p)+(1/x_image)=1/x_focus.

· Convergent lens is for the case where x_f>0

· Divergent lens is for the case where x_f<0.

4. Lens maker’s formula: Let the ray be going from left to right. It encounters surface #1 first and then #2. The lens maker’s formula is given by:

(1/f)=(n_lens/n_medium –1)[(1/x₁) – (1/x₂)],

x₁ and x₂ are the x-coordinates of the center of surface #1 and surface #2 respectively.

5. Image position for a sequence of lenses: Transport of a wave front.

Define the curvature of a spherical wave front to be C=(1/x_C). Consider a spherical wave front to be symmetric about the x-axis, where the origin is the intersection between the wave front and the x-axis.

	x

C

C

• Convergent wave front has x_c>0
• Divergent wave front has x_c<0.

A new interpretation: Now we would like to reinterpret the meaning of the usual lens formula.

· First rearrange the expression as follows: (1/q)=(-1/p)+(1/f).

· Notice that (1/q) is the final curvature C_f, of the wave front after the wave front has passed the lens.

· And (-1/p) is the initial curvature C_i, of the wave front before it enters the lens.

· We write C_f=(1/f), which is the modification to the initial curvature due to the passage through the lens.

Transport of a wave front: We can rewrite the about expression as: C_f=C_i+C_lens. This is a “local” interpretation to the usual lens formula. The latter has been derived through a global geometric consideration. Our curvature expression will allow us to determine the image location for an object through a sequence of lenses.

Consider a two lens problem. The setup is depicted in the sketch.

B

d

q₁

· The left lens is referred to as lens “A” and the right lens, lens “B”.

· Denote the curvature of the initial wave front before entering lens A as C₁ and the final curvature after it passes through the lens A as C₂. We have C₂=C₁ + C_A.

· Similarly at lens “B”, we have C₂’=C₁’ +C_B.

· What is C₁’? The wave front propagates freely from A to B. By inspection, C₁’=1/(q₁ –d).

A sequence of lenses. One may now describe the general algorithm for a sequence of lenses.

• First a wave front is generated by the object point, say at P.
• The wave front is then corrected by lens #1.
• The corrected wave front is “propagating” freely as a spherical wave front, until it hits lens #2.
• The wave front will then be corrected by lens #2.
• It then propagates freely to lens #3, etc.