Refraction at Curved Optical Surfaces and Paraxial Optics

Vergence and Wavefront Optics

• Definition of Wavefront: A wavefront represents the locus of points having the same phase of oscillation. The curvature of a wavefront as it propagates through space defines its vergence.

• Definition of Vergence: Vergence is the quantitative measure of the extent of convergence or divergence of incident or emergent light rays.

• Mathematical Formula for Vergence: An inverse relationship exists between vergence and the distance from the wavefront to the point source or point focus:

$\text{Vergence} = \frac{1}{\text{distance-to-source}}$

• Determinants of Wavefront Vergence:

  • Distance from the point of interest on the wavefront to the source (or focal point).

  • Refractive index ($n$) of the medium in which the light rays are traveling.

• Units of Vergence:

  • Measured in reciprocal meters ($\text{m}^{-1}$), universally referred to as Diopters ($\text{D}$).

• Diverging Rays:

  • Light rays originating from a point object spread outward.

  • Represented by negative vergence ($-\text{ve}$ vergence).

• Parallel Rays / Light from Infinity:

  • When light originates from a source at infinity, the wavefronts are flat (plano).

  • Because the distance to the source is infinite, the curvature is zero:

$\text{Vergence} = \frac{1}{\infty} = 0$

• Converging Rays:

  • Light rays traveling toward a point focus ($F'$) contract inward.

  • Represented by positive vergence ($+\text{ve}$ vergence).

  • The wavefront curvature becomes steeper (higher vergence) as it gets closer to the point focus.

Cartesian Optical Sign Convention

• Purpose: Optical sign conventions use positive ($+\text{ve}$) and negative ($-\text{ve}$) mathematical signs to establish spatial orientation, indicating whether images are real or virtual, erect or inverted, and positioned to the left or right of refracting elements.

• Origin / Vertex ($A$):

  • The vertex ($A$) is the geometric center point of the refracting surface where the principal optical axis intersects the surface.

  • $A$ coincides directly with the origin $(0,0)$ of the Cartesian coordinate system.

• Direction of Light Propagation:

  • Light rays are drawn traveling strictly from left to right.

  • Objects are located on the left side, and emergent rays/images form toward the right.

• Horizontal Axis Rules ($x$ -axis):

  • Distances measured to the left of vertex $A$ are negative ($-$-direction).

  • Distances measured to the right of vertex $A$ are positive ($+$-direction).

• Vertical Axis Rules ($y$ -axis):

  • Distances measured above the principal optical axis are positive ($+$-direction).

  • Distances measured below the principal optical axis are negative ($-$-direction).

• Angles:

  • Angles are measured relative to the principal horizontal optical axis ($x$ -axis) or the surface normal.

Refraction Principles at Flat and Curved Surfaces

• Snell's Law of Refraction:

  • Applies universally to all optical boundaries.

  • When a light ray passes from a medium with refractive index $n$ into a medium with higher refractive index $n'$, the ratio of the sine of the angle of incidence ($i$) to the sine of the angle of refraction ($i'$ or $r$) equals the inverse ratio of the refractive indices:

$\frac{n'}{n} = \frac{\sin(i)}{\sin(i')}$

$\text{Re-expressed as: } n \sin(i) = n' \sin(i')$

• Refraction at Curved Interfaces:

  • At any point of incidence on a curved surface, a tangent line is defined.

  • The surface normal is drawn perpendicular to the tangent at the point of entry and passes through the center of curvature ($C$).

• Refraction at Convex Surfaces ($n' > n$):

  • Light traveling from a lower index medium ($n$) into a higher index medium ($n'$) undergoes a reduction in phase velocity.

  • Wavefront spacing/wavelength shortens inside the higher index medium.

  • Rays refract toward the surface normal.

  • Convex refracting surfaces cause light rays to converge.

• Refraction at Concave Surfaces ($n' > n$):

  • Light entering a medium of higher refractive index slows down, shortening wavefronts.

  • Rays bend toward the normal, directed away from the principal axis.

  • Concave refracting surfaces cause light rays to diverge.

Spherical Aberration, Hyperbolic Surfaces, and Paraxial Optics Assumptions

• Spherical Aberration in Spherical Surfaces:

  • Marginal rays (rays striking the outer edge of a spherical surface) undergo greater refraction than paraxial rays (rays passing near the vertex).

  • This uneven focusing forms a envelope of focused rays called a caustic curve or caustic surface, preventing a single sharp focus point.

• Hyperbolic Surfaces:

  • Hyperbolic optical surfaces refract all incident parallel rays directly into a single sharp point focus, eliminating spherical aberration natively.

• Paraxial Optics Model:

  • Paraxial optics restricts analysis to light rays traveling very close to the principal optical axis (chief ray) passing through the vertex ($A$).

• Core Paraxial Assumptions:

  1. Spherical wavefronts refracted by spherical surfaces are assumed to emerge as perfect spherical wavefronts focusing at a single, precise point focus (ignoring differences between spherical and hyperbolic shapes).

  2. Refraction is assumed to occur along a single plane perpendicular to the optical axis (the distance between the actual curved surface and a flat reference plane, known as the sagitta, is treated as negligible).

  3. Variations in refractive index due to light frequency/color are neglected (monochromatic light is assumed, ignoring chromatic aberration).

Derivation of the Fundamental Paraxial Equation

• Small Angle Approximations:

  • For points $P$ close to the vertex $A$, angles of incidence ($i$), refraction ($i'$), object slope ($\alpha$), surface slope ($\beta$), and image slope ($\gamma$) are extremely small.

  • Under the small angle approximation in radians:

$\sin(i) \approx i \approx \tan(i)$

• Simplifying Snell's Law:

$n \sin(i) = n' \sin(i') \implies n i \approx n' i'$

• Geometric Relationships from Triangles:

  • From triangle $\Delta BPC$ (exterior angle theorem):

$i = \alpha + \beta$

*   From triangle $\Delta B'PC$:

\beta = i' + \gamma \implies i' = \tbeta - \gamma

• Substituting into Snell's Law:

$n(\alpha + \beta) = n'(\beta - \gamma)$

• Expressing Angles in Terms of Heights and Distances:

  • Let $l = AB$ be the object distance (measured negatively from vertex $A$).

  • Let $l' = AB'$ be the image distance (measured positively from vertex $A$).

  • Let $r = AC$ be the surface radius of curvature.

  • For small angles, arc length $AP$ approximates a straight vertical segment perpendicular to the axis:

$\tan(\alpha) \approx \frac{PA}{-l}$

$\tan(\beta) \approx \frac{PA}{r}$

$\tan(\gamma) \approx \frac{PA}{l'}$

• Algebraic Expansion:

$n\left(\frac{PA}{-l} + \frac{PA}{r}\right) = n'\left(\frac{PA}{r} - \frac{PA}{l'}\right)$

• Dividing both sides by segment $PA$:

$\frac{n}{-l} + \frac{n}{r} = \frac{n'}{r} - \frac{n'}{l'}$

• Rearranging terms puts object and image quantities on the left side and surface curvature on the right side:

$\frac{n'}{l'} - \frac{n}{l} = \frac{n' - n}{r}$

Refracting Surface Power and Reduced Vergence

• Surface Power ($F$):

  • Defined as the quantitative change in reduced vergence imparted by a refracting interface:

$F = \frac{n' - n}{r}$

*   Where radius of curvature $r$ is measured in meters ($\text{m}$), surface power $F$ is expressed in Diopters ($\text{D}$, or $\text{m}^{-1}$).

• Object Vergence ($L$):

  • The reduced vergence of incident wavefronts at the vertex:

$L = \frac{n}{l}$

• Image Vergence ($L'$):

  • The reduced vergence of emergent wavefronts at the vertex:

$L' = \frac{n'}{l'}$

• Fundamental Paraxial Equation in Reduced Vergence:

$L' - L = F \implies L' = L + F$

Linear Magnification at a Single Refracting Surface

• Definition of Linear Magnification ($m$):

  • The ratio of lateral image height ($h'$) to lateral object height ($h$):

$m = \frac{h'}{h}$

• Derivation Using Small Angles and Snell's Law:

  • For small angles of incidence and refraction at the surface:

$\tan(i) = \frac{h}{-l}$

$\tan(i') = \frac{-h'}{l'}$

*   Applying Snell's Law in tangent form:

$\frac{n'}{n} = \frac{\tan(i)}{\tan(i')} = \frac{\frac{h}{-l}}{\frac{-h'}{l'}} = \frac{h \times l'}{h' \times l}$

• Rearranged Linear Magnification Equation:

$m = \frac{h'}{h} = \frac{l' \times n}{n' \times l}$

• Magnification expressed via Reduced Vergence:

  • Recognizing that reduced vergence $L = \frac{n}{l}$ and $L' = \frac{n'}{l'}$, the formula simplifies directly to:

$m = \frac{h'}{h} = \frac{L}{L'}$

First and Second Focal Lengths of Refracting Surfaces

• Second Principal Focus ($F'$) and Second Focal Length ($f'$):

  • The point where incident parallel light rays ($L = 0$, object at infinity $l = -\infty$) focus after refraction.

  • Given $L = 0$, reduced image vergence becomes $L' = F$:

$L' = \frac{n'}{f'} = F \implies f' = \frac{n'}{F}$

*   $f'$ is the second focal length measured from the surface vertex.

• First Principal Focus ($F$) and First Focal Length ($f$):

  • The object position that produces emergent parallel rays ($L' = 0$, image at infinity $l' = +\infty$).

  • Given $L' = 0$, reduced object vergence becomes $-L = F$:

$-L = -\frac{n}{f} = F \implies f = -\frac{n}{F}$

*   $f$ is the first focal length measured from the surface vertex.

• Relationship Between Focal Lengths and Refractive Indices:

$\frac{f}{f'} = -\frac{n}{n'} \implies f = -\frac{n}{n'} \times f'$

*   The first and second focal lengths are unequal in magnitude whenever the media on either side of the surface differ in refractive index ($n \neq n'$).

• Sign Conventions for Focal Lengths:

  • Convex Surface ($F > 0$):

    • Second focal length $f'$ is positive (real focus inside medium $n'$).

    • First focal length $f$ is negative (located in front of vertex in medium $n$).

  • Concave Surface ($F < 0$):

    • Second focal length $f'$ is negative (virtual focus in front of vertex in medium $n$).

    • First focal length $f$ is positive (virtual object focus inside medium $n'$).

Ray Tracing and Graphical Construction of Images

• Three Principal Rays for Refracting Surfaces:

  1. Parallel Ray: Departs top of object parallel to principal axis, refracts through second focal point $F'$.

  2. Undeviated Ray: Passes through center of curvature $C$, striking surface normal to local curvature ($i = 0^{\circ}$, $i' = 0^{\circ}$), passing straight through without bending.

  3. Focal Ray: Passes through (or directs toward) first focal point $F$, emerging from refractive interface parallel to principal axis.

• Image Properties:

  • Convex Surfaces: Produce real inverted images or virtual erect images depending on whether object distance is greater or less than focal length.

  • Concave Surfaces: Consistently form virtual, upright, and diminished images ($m > 0$, $|m| < 1$) located in front of the refracting interface.

Newton's Extrafocal Equations

• Extrafocal Distances:

  • $x$: Extrafocal object distance, measured from first focal point $F$ to object point ($x = l - f$).

  • $x'$: Extrafocal image distance, measured from second focal point $F'$ to image point ($x' = l' - f'$).

• Derivation via Similar Triangles:

  • Comparing similar triangles formed by object, image, and focal points:

$\frac{h}{-h'} = \frac{x}{f} \quad \text{and} \quad \frac{h}{-h'} = \frac{-f'}{x'}$

• Newton's Product Formula:

$x \times x' = f \times f'$

• Newton's Magnification Equations:

$m = \frac{h'}{h} = -\frac{f}{x} = -\frac{x'}{f'}$

Step-by-Step Worked Problems and Calculations

• Sample Calculation 1: Convex Glass Interface

  • Problem: An object of height $h = 5\,\text{cm}$ in air ($n = 1.0$) is placed $l = -0.75\,\text{m}$ from a glass block ($n' = 1.5$) having a convex spherical surface with radius of curvature $r = +0.10\,\text{m}$. Calculate surface power, focal lengths, image distance, image vergence, magnification, and image height.

  • Step 1: Surface Power ($F$):

$F = \frac{n' - n}{r} = \frac{1.5 - 1.0}{+0.10\,\text{m}} = +5.00\,\text{D}$

*   *Step 2: Focal Lengths ($f$ and $f'$)*:

$f = -\frac{n}{F} = -\frac{1.0}{+5.00\,\text{D}} = -0.20\,\text{m}$

$f' = \frac{n'}{F} = \frac{1.5}{+5.00\,\text{D}} = +0.30\,\text{m}$

*   *Step 3: Object Vergence ($L$)*:

$L = \frac{n}{l} = \frac{1.0}{-0.75\,\text{m}} = -1.333\,\text{D}$

*   *Step 4: Image Vergence ($L'$)*:

$L' = L + F = -1.333\,\text{D} + 5.00\,\text{D} = +3.667\,\text{D}$

*   *Step 5: Image Distance ($l'$)*:

$l' = \frac{n'}{L'} = \frac{1.5}{+3.667\,\text{D}} = +0.408\,\text{m} = +40.8\,\text{cm}$

*   *Step 6: Linear Magnification ($m$)*:

$m = \frac{L}{L'} = \frac{-1.333\,\text{D}}{+3.667\,\text{D}} = -0.3635$

*   *Step 7: Image Height ($h'$)*:

$h' = h \times m = 5\,\text{cm} \times (-0.3635) = -1.82\,\text{cm}$

*   *Interpretation*: The image forms $40.8\,\text{cm}$ inside the glass block to the right of the vertex; it is real, inverted, and reduced in height to $1.82\,\text{cm}$.

• Sample Calculation 2: Determining Radius of Curvature

  • Problem: An object of height $h = 5\,\text{cm}$ in air ($n = 1.0$) is located $50\,\text{cm}$ ($l = -0.50\,\text{m}$) in front of a glass block ($n' = 1.5$). If a real image forms inside the glass at $80\,\text{cm}$ ($l' = +0.80\,\text{m}$) behind the surface, find the surface radius of curvature ($r$), magnification ($m$), and image height ($h'$).

  • Step 1: Calculate Object Vergence ($L$):

$L = \frac{n}{l} = \frac{1.0}{-0.50\,\text{m}} = -2.00\,\text{D}$

*   *Step 2: Calculate Image Vergence ($L'$)*:

$L' = \frac{n'}{l'} = \frac{1.5}{+0.80\,\text{m}} = +1.875\,\text{D}$

*   *Step 3: Calculate Surface Power ($F$)*:

$F = L' - L = +1.875\,\text{D} - (-2.00\,\text{D}) = +3.875\,\text{D} \approx +3.88\,\text{D}$

*   *Step 4: Calculate Radius of Curvature ($r$)*:

$F = \frac{n' - n}{r} \implies +3.875\,\text{D} = \frac{1.5 - 1.0}{r}$

$r = \frac{0.5}{+3.875\,\text{D}} = +0.129\,\text{m} = +12.9\,\text{cm}$

*   *Step 5: Calculate Magnification ($m$)*:

$m = \frac{L}{L'} = \frac{-2.00\,\text{D}}{+1.875\,\text{D}} = -1.0667$

*   *Step 6: Calculate Image Height ($h'$)*:

$h' = h \times m = 5\,\text{cm} \times (-1.0667) = -5.33\,\text{cm}$

*   *Interpretation*: The surface is convex with a radius of curvature $r = +12.9\,\text{cm}$, creating an inverted real image of height $-5.33\,\text{cm}$.