Geometric Optics: Mirrors, Refraction, and Lenses

Foundations of Geometric Optics and Plane Mirror Reflection

• The Ray Approximation in Geometric Optics:

  • Geometric optics models light propagation using straight-line rays that travel in the direction of energy flow perpendicular to the wave fronts.

  • Optical components, including mirrors and lenses, redirect these light rays via reflection and refraction to form images of objects.

  • Point source representation: An object can be modeled as a single point source $O$ or an extended collection of point sources emitting light rays in all forward directions.

  • Object distance ($p$): The physical distance measured from the object to the reflecting or refracting surface.

  • Image distance ($q$): The distance measured from the reflecting or refracting surface to the resulting image.

• Mechanism of Image Formation in Plane Mirrors:

  • Light rays diverge outward from an object point $O$ and strike the flat surface of the plane mirror.

  • Each ray reflects from the boundary obeying the Law of Reflection, which states that the angle of incidence equals the angle of reflection ($\theta_i = \theta_r$).

  • The reflected rays continue to diverge into the space in front of the mirror; to an observer, these rays appear to intersect at a common point $X$ situated behind the mirror.

• Image Classification:

  • Real Image: Formed when light rays physically converge and intersect at a common spatial point. Real images can be displayed or projected onto a physical screen placed at the image plane.

  • Virtual Image: Formed when light rays do not physically pass through or intersect at the image point, but rather diverge away from the optical element such that they appear to originate from that point when traced backward. Virtual images cannot be projected onto a screen.

  • Images formed by plane mirrors for real objects are always virtual.

• Geometric Proof of Plane Mirror Image Distance:

  • Ray tracing requires following at least two distinct rays originating from an object point $A$:

  • Ray 1: Directed perpendicularly toward the mirror along path $AP$. It strikes the mirror at normal incidence ($\theta_i = 0^{\circ}$) and reflects directly back upon itself along line $PA$.

  • Ray 2: Directed along an oblique path $AO$, striking the mirror at point $O$ at an angle $\theta$ relative to the surface normal, reflecting away along ray $OC$.

  • Tracing the extensions of both reflected rays behind the mirror locates the virtual image point $A'$

• Triangles $\triangle APO$ and $\triangle A'PO$ share the side $PO$, each have a right angle at point $P$, and have identical interior angles $\theta$ at point $O$.

• Because the two right triangles share a common side and identical acute angles, they are congruent ($\triangle APO \cong \triangle A'PO$):     $AP = A'P$

• Therefore, the image formed by an object placed in front of a plane mirror lies as far behind the mirror as the object is in front of the mirror:     $p = |q|$

  • Lateral Magnification ($M$) in Plane Mirrors:

• Lateral magnification is defined for any optical element as the ratio of image height to object height:     $M = \frac{\text{image height}}{\text{object height}} = \frac{h_i}{h_o} = \frac{h'}{h}$

• Congruent geometry demonstrates that the image height $h_i$ equals the object height $h_o$:     $h_i = h_o$

• Consequently, the lateral magnification of a plane mirror is always unity:     $M = 1$

  • Summary of Plane Mirror Image Properties:

• Distance: The image is located behind the mirror at a distance equal to the object distance in front ($|q| = p$).

• Orientation & Size: The image is erect (upright), unmagnified ($M = 1$), and virtual.

• Lateral Inversion: The image exhibits right-left reversal (lateral inversion). When an observer in front of a plane mirror raises their right hand, the perceived image raises its left hand.

Spherical Mirrors: Geometry and Mathematical Formulations

• Anatomy of Spherical Mirrors:

  • A spherical mirror is formed from a curved segment of a sphere of radius $R$.

  • Concave Mirror: Formed when the inner, concave surface of the sphere is polished to reflect light.

  • Convex Mirror: Formed when the outer, convex surface of the sphere is polished to reflect light (diverging mirror).

  • Center of Curvature ($C$): The physical center of the sphere from which the mirror surface is cut.

  • Radius of Curvature ($R$): The distance from the center of curvature $C$ to any point on the mirror surface.

  • Pole or Vertex ($V$ or $A$): The geometric center of the curved spherical mirror segment.

  • Principal Axis: The straight line passing through both the center of curvature $C$ and the vertex $V$.

• Paraxial Rays versus Spherical Aberration:

  • Paraxial Rays: Light rays that diverge from an object at very small angles relative to the principal axis.

  • Paraxial rays that depart from a point source on the principal axis reflect from a spherical concave mirror and converge at a single distinct image point $I$.

• Spherical Aberration: Rays that strike the spherical mirror far from the principal axis do not cross at the same focal point, instead intersecting the axis at various distances.

• This geometric imperfection causes blurring of the image and is inherently present to some degree in all spherical optical mirrors.

• Derivation of the Mirror Equation and Lateral Magnification:

• Consider an object of height $h$ placed at distance $p$ from vertex $V$, producing an inverted image of height $h'$ at distance $q$.

• Ray 1 departs the tip of the object, travels through the center of curvature $C$, strikes the mirror surface normally, and reflects directly back along its incoming path.

• Ray 2 departs the tip of the object, strikes the mirror vertex $V$ at angle $\theta$, and reflects away at angle $\theta$ below the principal axis.

• From the right triangles formed with vertex $V$:     $\tan(\theta) = \frac{h}{p}$     $\tan(\theta) = \frac{-h'}{q}$

• Equating these expressions yields the lateral magnification formula:     $M = \frac{h'}{h} = -\frac{q}{p}$     (The negative sign indicates that when $h'$ is negative, the image is inverted relative to the object.)

• From the right triangles sharing vertex angles $\alpha$ at the center of curvature $C$:     $\tan(\alpha) = \frac{h}{p - R}$     $\tan(\alpha) = \frac{-h'}{R - q}$

• Equating the expressions for $\tan(\alpha)$:     $\frac{h'}{h} = -\frac{R - q}{p - R}$

• Equating both expressions for $\frac{h'}{h}$:     $\frac{q}{p} = \frac{R - q}{p - R}$     $q(p - R) = p(R - q)$     $qp - qR = pR - pq$     $2pq = pR + qR$

• Dividing every term by the common factor $pqR$:     $\frac{2pq}{pqR} = \frac{pR}{pqR} + \frac{qR}{pqR}$     $\frac{1}{p} + \frac{1}{q} = \frac{2}{R}$

  • Focal Point and Focal Length:

• When an object is located at an infinite distance ($p \rightarrow \infty$), incident light rays are parallel to the principal axis.

• Substituting $p \rightarrow \infty$ into the mirror relation yields $\frac{1}{p} = 0$:     $\frac{1}{q} = \frac{2}{R} \implies q = \frac{R}{2}$

• The point where parallel incident rays converge is called the focal point ($F$), and the distance from the vertex to $F$ is the focal length ($f$):     $f = \frac{R}{2}$

• Expressing the mirror equation in terms of focal length gives:     $\frac{1}{p} + \frac{1}{q} = \frac{1}{f}$

  • Convex Mirrors:

• Silvered such that light reflects from the outer convex curvature.

• Reflected rays diverge outward upon reflection; their backward extensions meet at a virtual focal point located behind the mirror.

• For all real object positions in front of a convex mirror, the image is virtual, erect (upright), and reduced in size ($M < 1$).

• Sign Conventions for Spherical Mirrors:

  • Front Side: The region in front of the mirror where incident and reflected light rays propagate.

  • Back Side: The region behind the reflecting surface where light does not enter and virtual images form.

Ray Tracing and Image Formation in Spherical Mirrors

• Standard Ray Tracing Construction Rules:

  • Graphical determination of image position and size requires constructing three principal rays from the top of the object:

  • Ray 1: Drawn parallel to the principal axis; reflects through the focal point $F$ (or diverges along a line passing through $F$ for convex mirrors).

  • Ray 2: Drawn through the focal point $F$ (or directed toward $F$ for convex mirrors); reflects parallel to the principal axis.

  • Ray 3: Drawn through the center of curvature $C$ (or directed toward $C$ for convex mirrors); strikes the mirror surface normally and reflects back on itself.

  • The intersection of any two rays locates the image point; the third ray serves as an independent graphical check.

• Ray Tracing Scenarios for Mirrors:

  • Concave Mirror ($p > 2f$): Object positioned beyond the center of curvature $C$.

  • Image is real, inverted, reduced in size ($|M| < 1$), and positioned between $C$ and $F$.

• Concave Mirror ($p < f$): Object positioned between the focal point $F$ and the mirror vertex.

  • Image is virtual, upright, magnified ($M > 1$), and located behind the mirror ($q < 0$).

• Convex Mirror: Real object positioned anywhere in front of the convex mirror.

  • Image is virtual, upright, reduced in size ($0 < M < 1$), and located behind the mirror between the vertex and $F$.

  • As the object distance $p$ increases toward infinity, the image shrinks and approaches the focal point $F$.

• Worked Examples: Concave Mirror ($f = 10\,\text{cm}$):

  • Case (a) $p = 25\,\text{cm}$:     $\frac{1}{f} = \frac{1}{p} + \frac{1}{q} \implies \frac{1}{10\,\text{cm}} = \frac{1}{25\,\text{cm}} + \frac{1}{q}$     $\frac{1}{q} = \frac{1}{10\,\text{cm}} - \frac{1}{25\,\text{cm}} = \frac{5 - 2}{50\,\text{cm}} = \frac{3}{50\,\text{cm}}$     $q = \frac{50\,\text{cm}}{3} \approx 16.7\,\text{cm}$     $M = -\frac{q}{p} = -\frac{16.7\,\text{cm}}{25\,\text{cm}} = -0.668$     Description: The image is real ($q > 0$, in front of mirror), inverted ($M < 0$), and smaller than the object ($|M| < 1$).

  • Case (b) $p = 10\,\text{cm}$:     $\frac{1}{q} = \frac{1}{10\,\text{cm}} - \frac{1}{10\,\text{cm}} = 0 \implies q = \infty$     Description: The image is formed at infinity; reflected rays emerge parallel to each other.

  • Case (c) $p = 5\,\text{cm}$:     $\frac{1}{q} = \frac{1}{10\,\text{cm}} - \frac{1}{5\,\text{cm}} = \frac{1 - 2}{10\,\text{cm}} = -\frac{1}{10\,\text{cm}}$     $q = -10\,\text{cm}$     $M = -\frac{q}{p} = -\frac{-10\,\text{cm}}{5\,\text{cm}} = +2$     Description: The image is virtual ($q < 0$, behind mirror), erect/upright ($M > 0$), and enlarged by a factor of $2$.

  • Case (d) $p = 20\,\text{cm}$ ($p = 2f = R$):     $\frac{1}{q} = \frac{1}{10\,\text{cm}} - \frac{1}{20\,\text{cm}} = \frac{2 - 1}{20\,\text{cm}} = \frac{1}{20\,\text{cm}}$     $q = 20\,\text{cm}$     $M = -\frac{q}{p} = -\frac{20\,\text{cm}}{20\,\text{cm}} = -1$     Description: The image is real ($q > 0$), inverted ($M < 0$), and identical in size to the object ($|M| = 1$).

• Worked Example: Convex Mirror ($h = 3\,\text{cm}$, $p = 20\,\text{cm}$, $f = -8\,\text{cm}$):

  • (a) Position of the image ($q$):     $\frac{1}{f} = \frac{1}{p} + \frac{1}{q} \implies -\frac{1}{8\,\text{cm}} = \frac{1}{20\,\text{cm}} + \frac{1}{q}$     $\frac{1}{q} = -\frac{1}{8\,\text{cm}} - \frac{1}{20\,\text{cm}} = \frac{-5 - 2}{40\,\text{cm}} = -\frac{7}{40\,\text{cm}}$     $q = -\frac{40\,\text{cm}}{7} \approx -5.71\,\text{cm}$     Description: The image is virtual and located $5.71\,\text{cm}$ behind the mirror.

  • (b) Magnification of the mirror ($M$):     $M = -\frac{q}{p} = -\frac{-5.71\,\text{cm}}{20\,\text{cm}} = +0.286$     Description: The image is erect/upright ($M > 0$) and reduced in size.

  • (c) Height of the image ($h_i$):     $M = \frac{h_i}{h_o} \implies 0.286 = \frac{h_i}{3\,\text{cm}}$     $h_i = 3\,\text{cm} \times 0.286 = 0.857\,\text{cm}$

Images Formed by Refraction at Spherical and Plane Surfaces

• Refraction at a Single Spherical Boundary:

  • Consider two transparent optical media with indices of refraction $n_1$ and $n_2$ separated by a spherical interface of radius $R$.

  • An object is located at point $O$ within the medium of refractive index $n_1$.

• Derivation of the Single Surface Refraction Equation:

• Applying Snell's Law to an arbitrary paraxial ray leaving point $O$ and refracting toward point $I$:     $n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$

• In the paraxial limit (small angles measured in radians), $\sin(\theta) \approx \theta$:     $n_1 \theta_1 = n_2 \theta_2$

• Utilizing exterior angle properties of triangles $\triangle OPC$ and $\triangle PIC$:     $\theta_1 = \alpha + \beta$     $\beta = \theta_2 + \gamma \implies \theta_2 = \beta - \gamma$

• Substituting these geometric angle relations into Snell's Law:     $n_1(\alpha + \beta) = n_2(\beta - \gamma)$     $n_1 \alpha + n_2 \gamma = (n_2 - n_1)\beta$

• Using small-angle approximations for the tangents ($\tan(\theta) \approx \theta$) with arc height $d$:     $\alpha \approx \frac{d}{p}, \quad \beta \approx \frac{d}{R}, \quad \gamma \approx \frac{d}{q}$

• Substituting these values and dividing through by $d$ yields the fundamental equation for refraction at a spherical surface:     $\frac{n_1}{p} + \frac{n_2}{q} = \frac{n_2 - n_1}{R}$

• Because this equation does not depend on the ray angle $\alpha$, all paraxial rays leaving point $O$ focus at the exact same image point $I$.

  • Sign Conventions for Spherical Refracting Surfaces:

• Front Side: The side of the refracting surface from which incident light approaches.

• Back Side: The side opposite incident light into which refracted rays enter.

• Real images form on the back side of the refracting surface ($q > 0$), whereas virtual images form on the front side ($q < 0$).

• Refraction at a Flat (Plane) Surface:

  • For a flat refracting interface, the radius of curvature approaches infinity ($R \rightarrow \infty$).

  • Substituting $R \rightarrow \infty$ into the spherical refraction formula:     $\frac{n_1}{p} + \frac{n_2}{q} = 0 \implies \frac{n_1}{p} = -\frac{n_2}{q}$     $q = -\frac{n_2}{n_1} p$

  • The sign of $q$ is always opposite to $p$. For a real object ($p > 0$), $q$ is negative, meaning the image formed by a plane refracting boundary is always virtual and lies on the same side of the surface as the object.

  • When light travels from an optically denser to rarer medium ($n_1 > n_2$), the virtual image forms closer to the boundary than the object ($|q| < p$).

• When light travels from a rarer to denser medium ($n_1 < n_2$), the virtual image forms farther from the boundary than the object ($|q| > p$).

  • Worked Example: Coin Embedded in a Spherical Paperweight:

• A coin is embedded inside a spherical plastic paperweight with radius $R = 3.0\,\text{cm}$, index of refraction $n_1 = 1.50$, and outer medium air ($n_2 = 1.00$).

• The coin is placed $p = 2.0\,\text{cm}$ from the front curved surface. The center of curvature is on the object side (in front of the surface relative to exiting light), so $R = -3.0\,\text{cm}$.

• Applying the spherical refraction equation:     $\frac{n_1}{p} + \frac{n_2}{q} = \frac{n_2 - n_1}{R}$     $\frac{n_2}{q} = \frac{n_2 - n_1}{R} - \frac{n_1}{p}$     $\frac{1.00}{q} = \frac{1.00 - 1.50}{-3.0\,\text{cm}} - \frac{1.50}{2.0\,\text{cm}} = \frac{-0.50}{-3.0\,\text{cm}} - 0.75\,\text{cm}^{-1} = \frac{1}{6.0\,\text{cm}} - \frac{3}{4.0\,\text{cm}}$     $\frac{1}{q} = \frac{2 - 9}{12.0\,\text{cm}} = -\frac{7}{12.0\,\text{cm}}$     $q = -\frac{12.0\,\text{cm}}{7} \approx -1.7\,\text{cm}$

• The negative sign indicates that the virtual image appears $1.7\,\text{cm}$ behind the front surface inside the plastic.

  • Worked Example: Apparent Depth of a Swimming Fish:

• A fish swims at actual depth $d$ below the flat surface of a pond ($p = d$, $n_1 = 1.33$ for water, $n_2 = 1.00$ for air).

• Applying the plane surface refraction equation:   $q = -\frac{n_2}{n_1} p = -\frac{1.00}{1.33} d = -0.750 d$

• The apparent depth seen by an observer directly above the pond is three-quarters of the actual depth ($0.750 d$).

Thin Lens Theory and the Lens Maker's Equation

• Two-Surface Lens Model:

  • A lens comprises two refracting surfaces. Light passing through the lens undergoes two sequential refractions.

  • The analytical method models image formation such that the image formed by the first surface serves as the object for the second surface.

• Derivation of the Thin Lens Equation:

  • Let the lens have index of refraction $n$, thickness $t$, and surface radii of curvature $R_1$ and $R_2$, surrounded by air ($n_1 = 1$).

  • Refraction at Surface 1: An object is placed at distance $p_1$. Applying the single-surface equation with $n_1 = 1$ and $n_2 = n$:     $\frac{1}{p_1} + \frac{n}{q_1} = \frac{n - 1}{R_1} \quad \text{--- (1)}$     (where $q_1$ is the position of intermediate image $I_1$; $q_1 < 0$ if virtual, $q_1 > 0$ if real).

  • Refraction at Surface 2: Light passes from the lens medium ($n_1 = n$) back into air ($n_2 = 1$):     $\frac{n}{p_2} + \frac{1}{q_2} = \frac{1 - n}{R_2} \quad \text{--- (2)}$

  • Boundary Coupling Condition: The object distance for surface 2 is $p_2 = -q_1 + t$. For a thin lens, thickness $t$ is negligible ($t \approx 0$), giving:     $p_2 = -q_1$

  • Substituting $p_2 = -q_1$ into Equation (2):     $-\frac{n}{q_1} + \frac{1}{q_2} = -\frac{n - 1}{R_2} \quad \text{--- (3)}$

  • Adding Equation (1) and Equation (3) eliminates the intermediate image distance term $\frac{n}{q_1}$:     $\frac{1}{p_1} + \frac{1}{q_2} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$

  • Setting $p = p_1$ (original object distance) and $q = q_2$ (final image distance):     $\frac{1}{p} + \frac{1}{q} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$

• The Lens Maker's Equation:

  • Letting object distance $p \rightarrow \infty$, the corresponding image distance equals the focal length $f$ ($q = f$):     $\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$

  • This relation defines the focal length of any thin lens as a function of the refractive index of the lens material and the radii of curvature of both faces.

• The Thin Lens Equation:

  • Combining the thin lens formula with the Lens Maker's equation gives the Gaussian Thin Lens Equation:     $\frac{1}{p} + \frac{1}{q} = \frac{1}{f}$

• Focal Points of Thin Lenses:

  • Because light can travel in either direction through a lens, every thin lens possesses two symmetric focal points ($F_1$ and $F_2$) located equidistant ($f$) on either side of the optical center.

• Lens Geometries and Classifications:

  • Converging Lenses ($f > 0$): Thicker in the middle than at the edges. Types include biconvex, convex-concave (meniscus), and plano-convex.

• Diverging Lenses ($f < 0$): Thinner in the middle than at the edges. Types include biconcave, convex-concave, and plano-concave.

• Sign Conventions for Thin Lenses:

  • Front Side: Side from which incident light arrives (virtual side for image formation).

  • Back Side: Side into which refracted light departs (real side for image formation).

• Lateral Magnification of Thin Lenses:   $M = \frac{h'}{h} = \frac{h_i}{h_o} = -\frac{q}{p}$

  • When $M > 0$, the image is upright and on the front (same) side of the lens as the object.

  • When $M < 0$, the image is inverted and on the back (opposite) side of the lens.

Ray Tracing and Image Formation in Thin Lenses

• Principal Rays for Converging Lenses:

  • Ray 1: Enters parallel to the principal axis; refracts through the focal point $F_2$ on the back side of the lens.

  • Ray 2: Enters through the focal point $F_1$ on the front side (or as if originating from $F_1$ if $p < f$); refracts parallel to the principal axis.

  • Ray 3: Directed straight through the optical center of the lens; proceeds without angular deviation.

• Principal Rays for Diverging Lenses:

  • Ray 1: Enters parallel to the principal axis; refracts outward along a line projected from the front focal point $F_1$.

  • Ray 2: Directed toward the back focal point $F_2$; refracts parallel to the principal axis.

  • Ray 3: Directed straight through the optical center of the lens; proceeds without angular deviation.

• Image Formation Regimes:

  • Converging Lens ($p > f$): Produces a real, inverted image on the back side of the lens.

• Converging Lens ($p < f$): Produces a virtual, upright, magnified ($M > 1$) image on the front side of the lens. In this regime, the lens functions as a magnifying glass.

• Diverging Lens (Arbitrary $p$): Produces a virtual, upright, reduced ($|M| < 1$) image on the front side of the lens for all real object locations.

• Worked Examples: Converging Lens ($f = 10.0\,\text{cm}$):

  • Case 1: $p = 30.0\,\text{cm}$:     $\frac{1}{f} = \frac{1}{p} + \frac{1}{q} \implies q = \frac{pf}{p - f} = \frac{30.0\,\text{cm} \times 10.0\,\text{cm}}{30.0\,\text{cm} - 10.0\,\text{cm}} = +15.0\,\text{cm}$     $M = -\frac{q}{p} = -\frac{15.0\,\text{cm}}{30.0\,\text{cm}} = -0.5$     Description: Real ($q > 0$, back side), inverted ($M < 0$), reduced in size ($|M| = 0.5 < 1$).

• Case 2: $p = 10.0\,\text{cm}$ ($p = f$):   $q = \frac{10.0\,\text{cm} \times 10.0\,\text{cm}}{10.0\,\text{cm} - 10.0\,\text{cm}} = \infty$   Description: Image forms at an infinite distance; rays exit parallel to each other.

• Case 3: $p = 5.00\,\text{cm}$:   $q = \frac{pf}{p - f} = \frac{5.00\,\text{cm} \times 10.0\,\text{cm}}{5.00\,\text{cm} - 10.0\,\text{cm}} = -10.0\,\text{cm}$   $M = -\frac{q}{p} = -\frac{-10.0\,\text{cm}}{5.00\,\text{cm}} = +2.0$   Description: Virtual ($q < 0$, front side), upright ($M > 0$), enlarged ($M = 2.0 > 1$).

• Worked Example: Diverging Lens ($f = -20.0\,\text{cm}$, $h = 2.0\,\text{cm}$, $p = 30.0\,\text{cm}$):

  • Image distance calculation:     $q = \frac{pf}{p - f} = \frac{30.0\,\text{cm} \times (-20.0\,\text{cm})}{30.0\,\text{cm} - (-20.0\,\text{cm})} = \frac{-600\,\text{cm}^2}{50.0\,\text{cm}} = -12.0\,\text{cm}$

  • Lateral magnification:     $M = -\frac{q}{p} = -\frac{-12.0\,\text{cm}}{30.0\,\text{cm}} = +0.40$

  • Image height:     $h_i = M h_o = 0.40 \times 2.0\,\text{cm} = 0.80\,\text{cm}$

  • Description: Virtual ($q < 0$, front side), upright ($M > 0$), reduced in size ($M = 0.40 < 1$).

• Diverging Lens Variations Across Relative Distances ($f = -10.0\,\text{cm}$):

  • Object farther than focal length ($p = 30.0\,\text{cm}$):     $q = \frac{30.0\,\text{cm} \times (-10.0\,\text{cm})}{30.0\,\text{cm} - (-10.0\,\text{cm})} = -7.50\,\text{cm}$

• Object at the focal distance ($p = 10.0\,\text{cm}$):   $q = \frac{10.0\,\text{cm} \times (-10.0\,\text{cm})}{10.0\,\text{cm} - (-10.0\,\text{cm})} = -5.00\,\text{cm}$

• Object closer than focal distance ($p = 5.00\,\text{cm}$):   $q = \frac{5.00\,\text{cm} \times (-10.0\,\text{cm})}{5.00\,\text{cm} - (-10.0\,\text{cm})} = -3.33\,\text{cm}$

Thin Lens Combinations and Optical Systems

• System Treatment Methodology:

  • Analyze sequential optical elements by treating the image produced by the first element as the object for the second element.

  • Step 1: Calculate the image distance $q_1$ produced by Lens 1 as if Lens 2 were absent.

  • Step 2: Determine the object distance $p_2$ for Lens 2 based on the physical separation $d$ between the lenses:     $p_2 = d - q_1$

  • If $q_1 < d$, the intermediate image lies in front of Lens 2 and acts as a real object ($p_2 > 0$).

  • If $q_1 > d$, the rays are intercepted before forming the intermediate image; the image lies behind Lens 2 and acts as a virtual object ($p_2 < 0$).

  • Step 3: Calculate the final image position $q_2$ using the thin lens equation for Lens 2.

  • Total System Magnification: Because magnification multiplies cumulatively across elements, the net magnification is the product of the individual magnifications:     $M = M_1 M_2$     (For a system containing $N$ elements, $M = M_1 M_2 \dots M_N$.)

• Thin Lenses in Contact:

  • Consider two thin lenses of focal lengths $f_1$ and $f_2$ placed in direct physical contact ($d \approx 0$).

  • For the first lens:     $\frac{1}{p} + \frac{1}{q_1} = \frac{1}{f_1}$

  • Because the lenses are in contact and infinitesimally thin, $p_2 = -q_1$. For the second lens:     $\frac{1}{p_2} + \frac{1}{q} = \frac{1}{f_2} \implies -\frac{1}{q_1} + \frac{1}{q} = \frac{1}{f_2}$

  • Adding these equations eliminates $q_1$:     $\frac{1}{p} + \frac{1}{q} = \frac{1}{f_1} + \frac{1}{f_2}$

  • Replacing the combination with a single equivalent thin lens gives the effective focal length $f$:     $\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2}$

• Worked Example: Two Separated Converging Lenses:

  • System Parameters:

  • Lens 1: $f_1 = +10.0\,\text{cm}$

  • Lens 2: $f_2 = +20.0\,\text{cm}$

  • Lens separation: $d = 20.0\,\text{cm}$

  • Object position: $p_1 = 30.0\,\text{cm}$ to the left of Lens 1.

• Calculations for Lens 1:   $\frac{1}{q_1} = \frac{1}{f_1} - \frac{1}{p_1} = \frac{1}{10.0\,\text{cm}} - \frac{1}{30.0\,\text{cm}} = \frac{3 - 1}{30.0\,\text{cm}} = \frac{2}{30.0\,\text{cm}}$   $q_1 = +15.0\,\text{cm}$   $M_1 = -\frac{q_1}{p_1} = -\frac{15.0\,\text{cm}}{30.0\,\text{cm}} = -0.5$   (Image $I_1$ is real, inverted, and located $15.0\,\text{cm}$ to the right of Lens 1.)

• Calculations for Lens 2:

  • Object distance for Lens 2:     $p_2 = d - q_1 = 20.0\,\text{cm} - 15.0\,\text{cm} = 5.00\,\text{cm}$

  • Final image distance $q_2$:     $\frac{1}{q_2} = \frac{1}{f_2} - \frac{1}{p_2} = \frac{1}{20.0\,\text{cm}} - \frac{1}{5.00\,\text{cm}} = \frac{1 - 4}{20.0\,\text{cm}} = -\frac{3}{20.0\,\text{cm}}$     $q_2 = -\frac{20.0\,\text{cm}}{3} \approx -6.67\,\text{cm}$

  • Magnification of Lens 2:     $M_2 = -\frac{q_2}{p_2} = -\frac{-6.67\,\text{cm}}{5.00\,\text{cm}} = +1.33$

• Overall System Results:

  • Net lateral magnification:     $M = M_1 M_2 = (-0.5)(+1.33) = -0.665 \approx -0.67$

  • Position and character of final image $I_2$:

  • Located $6.67\,\text{cm}$ in front (to the left) of Lens 2, placing it $13.33\,\text{cm}$ to the right of Lens 1.

  • The final image is virtual with respect to Lens 2 ($q_2 < 0$), inverted with respect to the initial object ($M < 0$), and reduced in size ($|M| = 0.67 < 1$).