Mathematical Explanation

Refraction and Dispersion in Diamonds

The optical properties of diamonds are heavily influenced by their high refractive index and dispersion:

• Snell’s Law: Refraction at an interface between two media with different IORs is given by:

  $$n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$$

  Where $n_1$ and $n_2$ are the indices of refraction, and $\theta_1$ and $\theta_2$ are the angles of incidence and refraction.

• Dispersion: The change in IOR with wavelength $\lambda$ leads to chromatic dispersion, given by:

  $$n(\lambda) = n_0 + k(\lambda – \lambda_0)$$

  Where $n_0$ is the base refractive index, $k$ is a dispersion constant, and $\lambda_0$ is a reference wavelength.

Combining Refraction and Dispersion

To simulate a diamond’s sparkle, multiple Glass BSDF nodes with slightly different IORs are used to mimic the dispersion effect. Each color channel (red, green, blue) has its own IOR value:

• Red:

  $$n_r = n + \Delta n_{\text{adjusted}}$$

• Green:

  $$n_g = n$$

• Blue:

  $$n_b = n – \Delta n_{\text{adjusted}}$$

These values are fed into separate Glass BSDF nodes, and their outputs are combined using Add Shader nodes to achieve the dispersion effect.

Reflection and Refraction Combination

The final step combines the dispersed shader with an additional Glass BSDF to ensure accurate rendering of both reflection and refraction. The Mix Shader node blends these effects based on a specified factor, ensuring the diamond looks realistic under various lighting conditions.

Application in Blender

By using this node setup, you can create highly realistic simulations of diamonds and other gemstones. The key aspects include:

1. Accurate IOR Values: Setting the correct IOR for diamonds.
2. Enhanced Dispersion: Simulating the splitting of light into its constituent colors.
3. Combining Effects: Using Mix Shader to balance reflection and refraction.

This setup ensures that the diamond shader captures the brilliant sparkle and chromatic effects typical of real-world diamonds.