Cardinal points of an optical system

Cardinal points of an optical system:

               A  combination  of  lenses  having common  principal axis is  called a coaxial system of lens. In such  systems, a  method for finding out the distance and size of  the image is very tedious. Gauss showed that a coaxial system  can  be  considered  as  one system. For any system of lenses, there are six points known as cardinal points.

  They are:

  i) two focal points F₁ and F₂
  ii) two principal points P₁ and P₂
  III) two nodal points N₁ and N₂.

             Consider a coaxial optical system of two convex lenses of focal lengths f₁ and f₂ placed at a  distance ‘a' apart  as  shown  in figure.

Focal points :

 
             To explain the first focal point on the object  side  refer  figure. A.  Consider  an incident ray of light AB passing through axial point F₁. After refraction through lens system this ray becomes parallel to the principal axis ( in a direction CD ). Then this axial point F₁ through which rays are incident on the lens system is known as first focal point. A plane passing through F₁ and perpendicular to the principal axis is known as first focal plane.

Focal point

              To explain second focal point on image side refer figure.B. consider incident rays parallel to the principal axis. After refraction through the lense system, these rays focus on the axial point F₂, then this axial point F₂ is known as second focal point. A plane passing through F₂ and perpendicular to the principal axis is known as second focal plane.

Principal points:

                 In  figure .A.  incident  ray  AB produced forward and emergent ray CD produced backward to meet at E. A vertical plane passing through this point E and perpendicular to the principal axis is known as first principal plane. The point where the first principal plane intersect the axis is known as first principal point. Here P₁ is the first principal point. The distance P₁F₁ between first principal point and first focal point is known as the first principal focal length of the system.

                In  figure  B.  Incident  ray  GH produced forward and emergent ray KJ produced backward to meet at M. A vertical plane passing through this point M and perpendicular to the principal axis is known as second principal plane. The point P₂ where the second principal plane intersect the is known as principal point.

                 Principal  points  are  the  points having unit positive magnification for lens system. The distance between first focal point and the first principal point is known as first principal focal length ( P₁F₁ ) of lens system. Similarly the distance between second focal point and second principal point is known as a second principal focal length ( P₂F₂ ) of lens system.

Nodal points : 

                 Nodal points are the two conjugate points on the axis of lens such that the relative angular magnification is unity. When the light ray is incident at an angle  Rita , on the first nodal point N₁, emerges from the optical system at the same angle Rita from the another nodal point N₂. It means these are the points on the axis having unit positive angular magnification. So, the nodal points have a property that the rays , through these points are parallel to each other. The plane which passes through nodal points and perpendicular to the axis XX' are known as nodal planes.

                When the medium on both sides of optical system in same, then the nodal points and principal points coincide.