Additional: Lensmaker's Formula and Aberrations

Lensmaker's Formula ($ \frac{1}{f} = (n-1)(\frac{1}{R_1} - \frac{1}{R_2}) $)

The Lensmaker's Formula is a crucial formula for designing lenses. It directly relates the focal length of a thin lens to the properties of the lens material (refractive index) and the geometry of its surfaces (radii of curvature).

Derivation Outline

The formula can be derived by applying the formula for refraction at a single spherical surface twice, once for each surface of the lens. Consider a thin lens made of a material with refractive index $n_2$ and placed in a medium with refractive index $n_1$. Let the radii of curvature of the first and second surfaces be $R_1$ and $R_2$, respectively. Assume light travels from left to right.

Let a point object be placed on the principal axis at O, at a distance $u$ from the first surface (near the optical center P1). Refraction at the first surface forms an image I1 at distance $v_1$ from P1. Applying the refraction formula at a single spherical surface for the first surface:

$ \frac{n_2}{v_1} - \frac{n_1}{u} = \frac{n_2 - n_1}{R_1} $

This image I1 then acts as the object for the second surface (P2, near the optical center). For a thin lens, P1 and P2 are very close, so the distance of I1 from the second surface P2 is approximately $v_1$. The second surface refracts the rays from I1 to form the final image I at distance $v$ from P2.

When applying the refraction formula to the second surface, light is travelling from medium 2 (lens material, $n_2$) to medium 1 (surrounding medium, $n_1$). The object distance for the second surface is $u_2 = v_1$. Note that the direction of light is now from medium 2 to medium 1. The sign convention needs careful application here. If I1 is real (formed after the first surface in the direction of light), $v_1$ is positive in the first formula. For the second surface, the incident rays come from the left, converging towards I1 on the right. I1 acts as a virtual object for the second surface, located at distance $v_1$. So, the object distance for the second surface should be taken as $u_2 = +v_1$ (positive because it's on the right side of the second surface). The radius of curvature $R_2$ is positive if its center of curvature is on the right, negative if on the left (consistent with the standard convention). The formula for refraction at the second surface (from $n_2$ to $n_1$) is:

$ \frac{n_1}{v} - \frac{n_2}{u_2} = \frac{n_1 - n_2}{R_2} $

$ \frac{n_1}{v} - \frac{n_2}{v_1} = \frac{n_1 - n_2}{R_2} = -\frac{n_2 - n_1}{R_2} $

Add the two equations (for first and second surfaces):

$ \left(\frac{n_2}{v_1} - \frac{n_1}{u}\right) + \left(\frac{n_1}{v} - \frac{n_2}{v_1}\right) = \frac{n_2 - n_1}{R_1} - \frac{n_2 - n_1}{R_2} $

$ \frac{n_1}{v} - \frac{n_1}{u} = (n_2 - n_1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) $

$ n_1 \left(\frac{1}{v} - \frac{1}{u}\right) = (n_2 - n_1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) $

We know that for an object at infinity ($u \to \infty$), the image is formed at the focal length ($v = f$). Substituting this into the equation:

$ n_1 \left(\frac{1}{f} - \frac{1}{\infty}\right) = (n_2 - n_1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) $

$ \frac{n_1}{f} = (n_2 - n_1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) $

$ \frac{1}{f} = \left(\frac{n_2 - n_1}{n_1}\right) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) = \left(\frac{n_2}{n_1} - 1\right) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) $

Let $n = n_2/n_1$ be the refractive index of the lens material relative to the surrounding medium. The Lensmaker's Formula is:

$ \frac{1}{f} = (n-1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) $

where the signs of $R_1$ and $R_2$ are determined by the convention where the radius of curvature is positive if the center of curvature is on the right side of the surface, negative if on the left, assuming light comes from the left. For a convex surface viewed from the left, $R > 0$. For a concave surface viewed from the left, $R < 0$. $R_1$ is the radius of the first surface encountered by light, $R_2$ for the second.

Applications

The Lensmaker's Formula is used:

To calculate the focal length of a lens given its material and surface curvatures.
To design lenses with a desired focal length.
To understand how the focal length changes if the lens is placed in a different medium.

Chromatic Aberration

Ideal lenses, designed using simple formulas, suffer from certain imperfections or defects in image formation, known as aberrations. Chromatic aberration is one such aberration, related to the dispersion of light.

Cause of Chromatic Aberration

Chromatic aberration occurs because the refractive index ($n$) of the lens material is different for different wavelengths (colours) of light. This means the focal length ($f$) of the lens is also different for different colours, as given by the Lensmaker's Formula: $1/f = (n-1)(1/R_1 - 1/R_2)$. Since $n$ is higher for violet light than for red light ($n_V > n_R$), the focal length for violet light ($f_V$) is shorter than the focal length for red light ($f_R$) for a convex lens.

When white light from a source passes through a lens, the different colours are focused at slightly different points along the principal axis. Violet light, being deviated more, focuses closer to the lens, while red light focuses farther away. This results in the image being blurred and appearing coloured, especially around the edges.

Diagram illustrating chromatic aberration in a convex lens.

(Image Placeholder: A convex lens. A beam of white light parallel to the axis is incident on the lens. Show the beam splitting into colours inside the lens. Show the refracted rays for red and violet focusing at different points on the axis after the lens (violet closer to the lens than red). Label the focal points for violet (Fv) and red (Fr).)

Consequences

Chromatic aberration degrades the quality of the image, causing lack of sharpness and coloured fringes. It is particularly noticeable in lenses with large apertures and short focal lengths.

Correction (Achromatic Doublet)

Chromatic aberration can be reduced or eliminated by using a combination of lenses called an achromatic doublet. This typically consists of a convex lens made of one type of glass (e.g., Crown glass) cemented together with a concave lens made of a different type of glass (e.g., Flint glass). The materials are chosen such that the dispersion produced by the convex lens is cancelled out by the dispersion produced by the concave lens, while still achieving a net convergence or divergence of light.

Spherical Aberration

Spherical aberration is another type of aberration that affects the image quality of spherical lenses (and mirrors). It is caused by the geometry of the spherical surface itself and occurs even with monochromatic light.

Cause of Spherical Aberration

Spherical aberration occurs because rays of light that are farther from the principal axis (marginal rays) are focused at a different point compared to rays that are closer to the principal axis (paraxial rays) after refraction by a spherical lens or reflection by a spherical mirror. For a convex lens, paraxial rays focus at the principal focus, but marginal rays are bent more and focus closer to the lens. For a concave lens, marginal rays diverge more and appear to come from a point closer to the lens than the paraxial rays.

This means that instead of forming a sharp point image of a point object, a spherical lens or mirror forms a blurred patch of light. The deviation from ideal focusing is due to the spherical shape; ideally, a parabolic shape would perfectly focus parallel rays to a single point, but spherical shapes are easier to manufacture.

Diagram illustrating spherical aberration in a convex lens.

(Image Placeholder: A convex lens. A beam of parallel light rays incident on the lens. Show rays near the center (paraxial) focusing at the principal focus F. Show rays near the edges (marginal) focusing at a point closer to the lens than F. Indicate the region where the beam is narrowest (circle of least confusion), representing the best focus.)

Consequences

Spherical aberration results in a blurred image, especially in lenses with large apertures. The blurring increases as the aperture increases because more marginal rays are involved. For a given aperture, lenses with shorter focal lengths exhibit more spherical aberration.

Correction and Minimization

Spherical aberration cannot be completely eliminated in spherical lenses, but it can be minimized by:

Using smaller apertures: By using a diaphragm to block the marginal rays, only the paraxial rays pass through, reducing aberration but also reducing the amount of light and increasing diffraction.
Using combinations of lenses: Combining convex and concave lenses with different radii of curvature can help cancel out spherical aberration.
Using aspheric lenses: Lenses with non-spherical surfaces (parabolic, hyperbolic) can be designed to eliminate spherical aberration for certain object positions, but they are more expensive and difficult to manufacture.
Choosing optimal curvatures: For a single lens, spherical aberration can be minimized by distributing the total deviation over both surfaces rather than just one.

Spherical aberration, along with chromatic aberration and other aberrations like coma, astigmatism, field curvature, and distortion, limits the performance of simple optical systems. Complex optical instruments use multiple lens elements with carefully designed shapes and materials to minimize these aberrations and achieve high image quality.