This is a classic **two-lens system** (compound lens system), often used in optics to analyze image formation by two thin lenses separated by a distance `d`. Let's break down the step-by-step solution to finding the final image position \( s_{o2} \), magnification, and effective focal length. --- ## **Step-by-Step Solution** ### **Step 1: Image Formation by the First Lens (L1)** #### **Given:** - Object distance from first lens: \( s_{o1} \) - Focal length of first lens: \( f_1 \) #### **Lens Formula:** \[ \frac{1}{f_1} = \frac{1}{s_{o1}} + \frac{1}{s_{i1}} \] Solve for image distance \( s_{i1} \): \[ s_{i1} = \left( \frac{1}{f_1} - \frac{1}{s_{o1}} \right)^{-1} \] --- ### **Step 2: Object Distance for the Second Lens (L2)** The image formed by L1 acts as the object for L2. The distance between the lenses is \( d \): \[ s_{o2} = d - s_{i1} \] *Be careful with sign conventions (real is positive to the right of the lens).* --- ### **Step 3: Image Formation by the Second Lens (L2)** #### **Given:** - Object distance for L2: \( s_{o2} \) - Focal length of L2: \( f_2 \) #### **Lens Formula:** \[ \frac{1}{f_2} = \frac{1}{s_{o2}} + \frac{1}{s_{i2}} \] Solve for image distance \( s_{i2} \): \[ s_{i2} = \left( \frac{1}{f_2} - \frac{1}{s_{o2}} \right)^{-1} \] --- ### **Step 4: Overall Magnification** \[ M = M_1 \cdot M_2 \] Where, - \( M_1 = -\frac{s_{i1}}{s_{o1}} \) (magnification by L1) - \( M_2 = -\frac{s_{i2}}{s_{o2}} \) (magnification by L2) --- ### **Step 5: Effective Focal Length of the System** For two thin lenses separated by distance \( d \): \[ \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2} \] Where: - \( f \) = effective focal length of the two-lens system --- ## **Summary Table** | Step | Equation | What it Finds | |-------------|---------------------------------------------------------------------------------------|------------------------------| | 1 | \( \frac{1}{f_1} = \frac{1}{s_{o1}} + \frac{1}{s_{i1}} \) | Image by L1 (\( s_{i1} \)) | | 2 | \( s_{o2} = d - s_{i1} \) | Object for L2 | | 3 | \( \frac{1}{f_2} = \frac{1}{s_{o2}} + \frac{1}{s_{i2}} \) | Image by L2 (\( s_{i2} \)) | | 4 | \( M = M_1 \cdot M_2 \) | Total magnification | | 5 | \( \frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2} \) | Effective focal length | --- **If you have actual numerical values, provide them and I can solve it with numbers!**

Let's analyze how to write the ray transfer matrix (also known as the ABCD matrix) for two thick lenses, \(L_1\) and \(L_2\). --- ## **Ray Transfer Matrix (ABCD Matrix) for a Thick Lens** A thick lens can be modeled as: 1. **Propagation through the lens thickness \( t \)** (free space propagation) 2. **Refraction at the front surface** 3. **Propagation through the lens material (assuming the lens is homogeneous)** 4. **Refraction at the back surface** However, in paraxial optics, a **thick lens** is often represented as a **single matrix** combining these effects, characterized by its **focal length** \(f\) and **lens thickness** \(t\). --- ## **Step 1: Transfer matrices for individual components** ### **a) Propagation through free space over distance \(d\):** \[ \mathbf{M}_{\text{free}}(d) = \begin{bmatrix} 1 & d \\ 0 & 1 \end{bmatrix} \] ### **b) Refraction at a spherical interface** For a spherical interface with radius \( R \), the matrix is: \[ \mathbf{M}_{\text{interface}} = \begin{bmatrix} 1 & 0 \\ -\frac{(n_2 - n_1)}{n_2 R} & \frac{n_1}{n_2} \end{bmatrix} \] where: - \( n_1 \) is the refractive index of the incident medium, - \( n_2 \) is that of the transmitted medium. --- ## **Step 2: Matrix for a thick lens** A **thick lens** with focal length \(f\) and thickness \(t\) can be represented as: \[ \mathbf{M}_{\text{thick lens}} = \mathbf{M}_{\text{propagation}}(t) \times \mathbf{M}_{\text{lens surfaces}} \times \mathbf{M}_{\text{propagation}}(t) \] Alternatively, for thin lens approximation, the matrix simplifies to: \[ \mathbf{M}_{\text{thin lens}} = \begin{bmatrix} 1 & 0 \\ -\frac{1}{f} & 1 \end{bmatrix} \] But since the question asks explicitly for **thick lenses**, the general approach is: \[ \boxed{ \mathbf{M}_{L} = \mathbf{M}_{\text{propagation}}(t) \times \mathbf{M}_{\text{surface front}} \times \mathbf{M}_{\text{surface back}} \times \mathbf{M}_{\text{propagation}}(t) } \] --- ## **Step 3: Matrix for the two lenses** Suppose: - \(L_1\) has matrix \( \mathbf{M}_1 \), - \(L_2\) has matrix \( \mathbf{M}_2 \), - Distance between lenses: \(d\). The **overall transfer matrix** from object to image through both lenses is: \[ \boxed{ \mathbf{M}_{\text{system}} = \mathbf{M}_2 \times \mathbf{M}_{\text{free}}(d) \times \mathbf{M}_1 } \] where: \[ \mathbf{M}_{\text{free}}(d) = \begin{bmatrix} 1 & d \\ 0 & 1 \end{bmatrix} \] --- ## **Summary** - Each **thick lens** can be represented by a **4x4 matrix** or, more commonly in paraxial optics, by a **2x2 ABCD matrix** incorporating the effects of thickness and curvature. - The **composite system** matrix is obtained by multiplying the individual matrices, with free space propagations in between. --- ## **Final note** The specific matrices depend on the detailed parameters of the lenses: radii of curvature, thickness, refractive indices, etc. If you provide these parameters, I can help derive explicit matrices for \(L_1\) and \(L_2\).