Derivation of the Thin Lens Equation

In the realm of geometric optics, the ability to predict how light propagates through optical elements is fundamental to designing everything from simple magnifying glasses to complex astronomical telescopes. At the heart of this predictive power lies the thin lens equation, a deceptively simple mathematical relationship that connects object distance, image distance, and focal length. While often memorized as a standalone formula, its true utility is best appreciated when one understands the physical principles and geometric logic that underpin it.

This article explores the rigorous derivation of the thin lens equation, specifically the Gaussian form. We will move from the fundamental laws of refraction at a single spherical surface to the composite behavior of a thin lens, highlighting the critical role of the paraxial approximation in simplifying the mathematics without sacrificing accuracy for standard optical systems.

Foundational Concepts and Assumptions

Before diving into the algebra, it is essential to define the variables and the physical constraints that allow for a tractable derivation.

Core Variables

  • Object Distance ($s$ or $u$): The distance from the object to the optical center of the lens. By convention, a real object placed on the incoming side of the light is assigned a positive value.
  • Image Distance ($s'$ or $v$): The distance from the optical center to the formed image. A positive value indicates a real image formed on the outgoing side of the light, while a negative value denotes a virtual image on the same side as the object.
  • Focal Length ($f$): The distance from the optical center to the focal point, where parallel rays converge (for a converging lens) or appear to diverge from (for a diverging lens).

The Paraxial Approximation

The derivation relies heavily on the paraxial approximation (or Gaussian approximation). This assumption states that we only consider light rays that travel close to the optical axis and make small angles with it. Mathematically, for small angles $\theta$ (in radians):
$$ \sin \theta \approx \tan \theta \approx \theta $$
This simplification is crucial. It allows us to replace complex trigonometric functions with linear ratios of lengths, transforming the problem from one involving circles and arcs into one involving similar triangles and linear algebra.

The Thin Lens Assumption

A "thin" lens is defined as one whose thickness is negligible compared to its radii of curvature and the object/image distances. Consequently, we assume that refraction occurs at a single plane—the optical center—rather than at two distinct surfaces separated by a finite distance. This allows us to treat the lens as a single refractive element rather than a compound system with internal propagation.

Step 1: Refraction at a Single Spherical Surface

To derive the lens equation, we first must understand how a single curved interface bends light. Consider a spherical surface separating two media with refractive indices $n_1$ (object side) and $n_2$ (image side). Let $R$ be the radius of curvature of this surface.

Imagine a point object $P$ on the optical axis. A ray from $P$ strikes the surface at point $M$ and refracts to meet the axis at the image point $P'$. Let $s$ be the object distance and $s'$ be the image distance.

According to Snell’s Law:
$$ n_1 \sin i = n_2 \sin r $$
where $i$ is the angle of incidence and $r$ is the angle of refraction.

Under the paraxial approximation, $\sin i \approx i$ and $\sin r \approx r$. Thus, the law simplifies to:
$$ n_1 i = n_2 r $$

Through geometric analysis of the triangle formed by the object, the center of curvature, and the point of incidence, we can express the angles $i$ and $r$ in terms of the ray height $y$ and the distances $s$, $s'$, and $R$. Specifically, the angles are related to the slopes of the lines connecting these points. After substituting these geometric relationships into the simplified Snell’s Law and eliminating the intermediate variable $y$ (the height of the ray), we arrive at the single-surface refraction equation:

$$ \frac{n_2}{s'} - \frac{n_1}{s} = \frac{n_2 - n_1}{R} $$

This equation is powerful because it shows that for a given object distance and surface geometry, the image distance is uniquely determined, independent of the specific ray height (within the paraxial limit). This property is known as stigmatic imaging.

Step 2: Deriving the Thin Lens Equation

A thin lens can be modeled as two spherical surfaces in close proximity. Let the lens have a refractive index $n$ and be surrounded by air (refractive index $1$). Let the radius of curvature of the first surface (facing the object) be $R_1$, and the second surface be $R_2$.

We treat the lens as a two-step process:

  1. Refraction at the first surface.
  2. Refraction at the second surface.

The First Surface

Light travels from air ($n_1 = 1$) into the lens material ($n_2 = n$). Let $s$ be the object distance and $s_1'$ be the intermediate image distance formed by the first surface. Applying the single-surface equation:

$$ \frac{n}{s_1'} - \frac{1}{s} = \frac{n - 1}{R_1} \quad \text{--- (Equation 1)} $$

The Second Surface

The image formed by the first surface acts as the object for the second surface. Because the lens is thin, the distance between the two surfaces is negligible. Therefore, the object distance for the second surface, $s_2$, is effectively equal to the intermediate image distance $s_1'$. However, we must be careful with signs. If the intermediate image is real (on the right side of the first surface), it lies to the right of the second surface as well, meaning it is a virtual object for the second surface. Thus, $s_2 = -s_1'$.

Light now travels from the lens material ($n_1 = n$) into air ($n_2 = 1$). Let $s'$ be the final image distance. Applying the single-surface equation again:

$$ \frac{1}{s'} - \frac{n}{s_2} = \frac{1 - n}{R_2} \quad \text{--- (Equation 2)} $$

Substituting $s_2 = -s_1'$ into Equation 2:

$$ \frac{1}{s'} - \frac{n}{-s_1'} = \frac{1 - n}{R_2} $$
$$ \frac{1}{s'} + \frac{n}{s_1'} = \frac{1 - n}{R_2} \quad \text{--- (Equation 3)} $$

Combining the Steps

Now we have a system of two equations. We want to eliminate the intermediate variable $s_1'$. Let's rearrange Equation 1 to isolate the term $\frac{n}{s_1'}$:

$$ \frac{n}{s_1'} = \frac{1}{s} + \frac{n - 1}{R_1} $$

Substitute this expression into Equation 3:

$$ \frac{1}{s'} + \left( \frac{1}{s} + \frac{n - 1}{R_1} \right) = \frac{1 - n}{R_2} $$

Rearranging the terms to group the object and image distances on the left side:

$$ \frac{1}{s'} + \frac{1}{s} = \frac{1 - n}{R_2} - \frac{n - 1}{R_1} $$

Notice that $\frac{1 - n}{R_2}$ is equivalent to $-(n - 1)\frac{1}{R_2}$. Factoring out $(n - 1)$:

$$ \frac{1}{s'} + \frac{1}{s} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) $$

Introducing the Focal Length

The term on the right-hand side depends only on the physical properties of the lens (its refractive index and curvature radii). This constant is defined as the reciprocal of the focal length, $\frac{1}{f}$. This relationship is known as the Lensmaker’s Formula:

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

By substituting $\frac{1}{f}$ back into our derived equation, we arrive at the classic Gaussian Thin Lens Equation:

$$ \frac{1}{s} + \frac{1}{s'} = \frac{1}{f} $$

Or, using the common notation where $u$ is the object distance and $v$ is the image distance:

$$ \frac{1}{u} + \frac{1}{v} = \frac{1}{f} $$

Sign Conventions and Practical Application

The power of this equation lies in its universality, provided one adheres to a consistent sign convention. The most common convention in physics is:

  • Real is Positive: Real objects and real images have positive distances.
  • Virtual is Negative: Virtual objects and virtual images have negative distances.
  • Converging vs. Diverging: Convex lenses (converging) have positive focal lengths; concave lenses (diverging) have negative focal lengths.

A Worked Example

Consider a convex lens with a focal length of $f = 10 \text{ cm}$. An object of height $h_o = 2 \text{ cm}$ is placed $u = 15 \text{ cm}$ in front of the lens. We want to determine the image distance and its characteristics.

  1. Find Image Distance ($v$):
    $$ \frac{1}{15} + \frac{1}{v} = \frac{1}{10} $$
    $$ \frac{1}{v} = \frac{1}{10} - \frac{1}{15} $$
    Finding a common denominator (30):
    $$ \frac{1}{v} = \frac{3}{30} - \frac{2}{30} = \frac{1}{30} $$
    $$ v = 30 \text{ cm} $$

  2. Analyze the Image:

    • Since $v$ is positive, the image is real and located $30 \text{ cm}$ on the opposite side of the lens.
    • To find the magnification ($m$), we use the formula $m = -\frac{v}{u}$:
      $$ m = -\frac{30}{15} = -2 $$
    • The negative sign indicates the image is inverted.
    • The magnitude of 2 indicates the image is twice the size of the object.
    • Image height $h_i = m \times h_o = -2 \times 2 \text{ cm} = -4 \text{ cm}$.

Conclusion

The derivation of the thin lens equation is a testament to the elegance of geometric optics. By decomposing the complex refraction of light through a lens into two manageable spherical surface interactions and applying the paraxial approximation, we derive a linear relationship between the reciprocals of the distances. This formula is not merely a textbook exercise; it is the foundational tool for optical engineering. Whether calculating the focus of a camera lens, designing a microscope objective, or analyzing the vision correction provided by eyeglasses, the thin lens equation provides the quantitative backbone for understanding how light forms images. Understanding its derivation ensures that practitioners can apply it correctly, respecting the sign conventions and the physical limits imposed by the paraxial assumption.