Thick Lens Formula
1/f = (n-1)[1/R₁ - 1/R₂ + (n-1)d/(nR₁R₂)]
Center thickness of the lens (e.g., 0.01 m = 1 cm)
Thin Lens vs Thick Lens Formula
Understanding when to use the thin lens formula versus the thick lens formula is crucial for accurate optical calculations. Here's a detailed comparison:
Thin Lens Formula
1/f = (n-1)(1/R₁ - 1/R₂)
- Assumption: Lens thickness is negligible (d ≈ 0)
- Accuracy: Good for d < 5% of radii
- Use cases: Classroom problems, thin spectacle lenses, simple optical systems
- Advantage: Simpler calculation
Thick Lens Formula
1/f = (n-1)[1/R₁ - 1/R₂ + (n-1)d/(nR₁R₂)]
- Assumption: Lens thickness is significant
- Accuracy: Required when d > 5% of radii
- Use cases: Camera lenses, microscope objectives, high-power lenses
- Advantage: More accurate for real lenses
Rule of Thumb
If the lens thickness is greater than 10% of either radius of curvature, use the thick lens formula. The correction term (n-1)d/(nR₁R₂) becomes significant and ignoring it can lead to errors of several percent in focal length calculations.
Thick Lens Formula Derivation
The thick lens formula is derived by treating the lens as two refracting surfaces separated by a distance d (the lens thickness). Here's the step-by-step derivation:
Step 1: First Surface Refraction
n₁/s₁ + n₂/s₁' = (n₂ - n₁)/R₁
Light refracts at the first surface from air (n₁=1) into the lens (n₂=n)
Step 2: Second Surface Refraction
n₂/s₂ + n₁/s₂' = (n₁ - n₂)/R₂
Light refracts at the second surface from the lens (n₂=n) back into air (n₁=1)
Step 3: Account for Thickness
s₂ = s₁' - d
The image from the first surface becomes the object for the second, shifted by thickness d
Step 4: Combine for Focal Length
1/f = (n-1)[1/R₁ - 1/R₂ + (n-1)d/(nR₁R₂)]
The final thick lens formula with thickness correction term
The additional term (n-1)d/(nR₁R₂) is the thickness correction. When d approaches zero, this term vanishes and we recover the thin lens formula.
When to Use Thick Lens Formula
The standard lens maker formula assumes a "thin lens" where thickness is negligible. Use the thick lens formula when:
- Lens thickness is comparable to the radii of curvature
- High precision is required in optical design
- Working with thick meniscus or high-power lenses
- Designing camera or microscope objectives
Real-World Applications
Camera Lenses
Professional camera lenses use multiple thick lens elements. The thick lens formula ensures accurate focal length calculations for each element in the optical stack.
Microscope Objectives
High-magnification microscope objectives require precise thick lens calculations to achieve diffraction-limited performance and minimize aberrations.
High-Prescription Eyeglasses
Strong corrective lenses (±6 diopters or more) have significant thickness, requiring thick lens formula for accurate prescription calculations.
Telescope Eyepieces
Wide-field eyepieces often use thick meniscus elements where the thickness correction significantly affects the optical performance.
Frequently Asked Questions
What is the thick lens formula?
The thick lens formula is 1/f = (n-1)[1/R₁ - 1/R₂ + (n-1)d/(nR₁R₂)], where d is the center thickness of the lens. This formula accounts for the separation between the two refracting surfaces that is ignored in the thin lens approximation.
How much does thickness affect focal length?
The effect depends on the lens geometry. For a typical biconvex lens with d = 10% of the radii, the focal length can differ by 1-3% from the thin lens calculation. For high-power lenses or thick meniscus designs, the difference can be 5-10% or more.
What are principal planes in thick lens optics?
Principal planes are hypothetical planes where all refraction can be considered to occur. For thick lenses, there are two principal planes (H and H') that don't coincide. Distances for image calculations should be measured from these planes, not the lens surfaces.
Can I use the thin lens formula for preliminary calculations?
Yes, the thin lens formula is useful for initial estimates and understanding the basic behavior of an optical system. For final design verification and precision applications, always use the thick lens formula to account for real lens geometry.