# Step-by-Step Solution: Field Flattener for Petzval Curvature ## 1. **Background: Petzval Curvature** In an optical system, the **Petzval surface** is generally curved, not flat. The radius of curvature of this surface is called the **Petzval radius** ($R_P$). A **field flattener** is a thin lens inserted near the image plane to make the final image surface flat. ## 2. **Focus Shift by a Thin Slab (from Homework)** When a glass slab of thickness $t$ and refractive index $n$ is inserted perpendicular to the optical axis in a converging beam: - The **optical path length** increases by $(n-1)t$. - The **paraxial focus shift** $\Delta x$ (in the direction of propagation) is: \[ \Delta x = t \left(1 - \frac{1}{n}\right) \] This is the distance by which the image plane must be moved to refocus. ## 3. **Field Flattener Placement and Effect** We desire a plano-convex lens (field flattener) with: - **Plano side** toward the image plane, - **Convex side** toward the objective, - **Refractive index** $n$, - **Unknown radius** $\rho$ of the convex surface. The lens is **thin** and placed close to the image plane. ## 4. **Required Correction** The field flattener must introduce a focus shift that **matches and cancels the Petzval curvature**. - The Petzval surface is a sphere of radius $R_P$ (often negative, i.e., concave toward the image). - The field flattener's power must produce an **equal and opposite curvature** to flatten the field. ## 5. **Lensmaker's Formula for Thin Lens** For a **plano-convex lens** (plano side faces image), the lensmaker's equation: \[ \frac{1}{f} = (n - 1) \left[ \frac{1}{R_1} - \frac{1}{R_2} \right] \] - $R_1 = \rho$ (convex side, facing object, by sign convention), - $R_2 = \infty$ (plano side). So: \[ \frac{1}{f} = (n-1) \frac{1}{\rho} \implies f = \frac{\rho}{n-1} \] ## 6. **Matching the Petzval Curvature** The **paraxial image curvature** introduced by a thin lens at the image plane, of focal length $f$, is a **field curvature** of radius $f$ (for a thin lens in air). - To **flatten** the field, the lens must **produce a focal surface with curvature opposite to the Petzval surface**: \[ \rho_{\text{flattener}} = -R_P \] (since the sign convention: if the Petzval radius is $R_P$, we need a lens of focal length $f = -R_P$). - From above: \[ f = \frac{\rho}{n-1} \] Set $f = -R_P$: \[ -R_P = \frac{\rho}{n-1} \implies \rho = -(n-1) R_P \] ## 7. **Final Answer** ### **The required radius of the plano-convex field flattener lens is:** \[ \boxed{ \rho = -(n-1) R_P } \] - $\rho$: radius of the convex surface (positive if convex toward object). - $n$: refractive index of flattener glass. - $R_P$: Petzval radius of curvature (use the sign convention: negative if concave toward image). **This lens, with its plano side toward the image, will flatten the field.** --- ## **Summary of Steps** 1. **Focus shift by glass**: $\Delta x = t (1-1/n)$. 2. **Lensmaker’s formula**: $f = \rho/(n-1)$ for plano-convex lens. 3. **Set $f = -R_P$** to match and cancel Petzval curvature. 4. **Solve for $\rho$**: $\rho = -(n-1)R_P$. --- **If you know the Petzval radius $R_P$ and glass index $n$, plug in those values to get $\rho$.**