# Perfect Two-Lens System Characteristics

During investigations of a point source of light that does not lie in the focal plane of a lens, it is often convenient to represent a perfect lens as a system composed of two individual lens elements. This tutorial explores off-axis oblique light rays passing through such a system.

The tutorial initializes with a parallel beam of light passing through the double lens system in coincidence with the optical axis and traveling from left to right. The **Tilt Angle** slider can be employed to tilt the axis of the light beam through ± 25 degrees, and the **Object Side Focal Length** slider adjusts the focal length of the lens nearest the object between a range of 0.4 to 0.8 centimeters. The **Image Side Focal Length** slider can be utilized to change this value between 0.8 and 2.0 centimeters. A checkbox toggles simulations of plane and spherical wavefronts on and off, allowing the visitor to view how spherical waves are produced when a plane wavefront passes through the lens system. The blue **Reset** button is utilized to re-initialize the tutorial.

In the dual-lens system illustrated in the tutorial window, a spherical wavefront emanating from light source point **S(1)**, and located at a distance **δ** from the optical axis of the lens, is converted by **Lens(a)** into a plane wave. As it exits from **Lens(a)**, the plane wave is tilted with respect to the lens axis by an angle **α**. Both **δ** and **α** are related by the sine equation, with the value for **f** being replaced by **f(a):**

**δ = f × sin(α)**

where **f** is the focal length of the perfect lens. After passing through the second lens (**Lens(b)**), the plane wave is converted back into a spherical wave having a center located at **S(2)**. The result is that a perfect lens, which equals **Lens(a)** + **Lens(b)**, focuses light from point **S(1)** onto point **S(2)** and also performs the reverse action by focusing light from point **S(2)** onto point **S(1)**. Focal points having such a relationship in a lens system are commonly referred to as **conjugate points**.

In the nomenclature of classical optics, the space between light source **S(1)** and the entrance surface of the first lens is referred to as the **object space**, while the region between the second lens exit surface and point **S(2)** is known as the **image space**. All points involved with primary or secondary light rays are termed **objects** (or **specimens** in optical microscopy), while the regions containing light rays concentrated by refraction by the lens are called **images**. If the light waves themselves intersect, the image is **real**, whereas if only the projected extensions of refracted light rays intersect, a **virtual** image is formed by the lens system.

If the point **S(1)** is expanded into a series of points spread throughout the same focal plane, then a perfect lens will focus each point in the series onto a conjugate point in the focal plane of **S(2)**. In the case where a point set of **S(1)** lies in a plane perpendicular to the optical axis of the lens, then the corresponding conjugate points in set **S(2)** would also lie in a plane that is perpendicular to the axis. The reverse is also true: the lens will focus every point in the set **S(2)** onto a conjugate point on the plane or surface of point set **S(1)**. Corresponding planes or surfaces of this type are known as **conjugate planes**.