Photon energy and band gap from laser wavelength — GaAs laser at 867 nm A GaAs laser emits light of wavelength 8670 × 10^−10 m. Given h = 6.626 × 10^−34 J·s, c = 3 × 10^8 m/s, and 1 eV = 1.602 × 10^−19 J, the energy gap (in eV) of GaAs is:

Introduction / Context:
This numerical problem checks the relationship between photon wavelength and energy, and how a semiconductor laser's emission wavelength reflects its band gap (for near direct transitions). Converting wavelength to photon energy in electron-volts is a staple calculation in electronics and photonics.

Given Data / Assumptions:

• Laser wavelength λ = 8670 × 10^−10 m = 8.67 × 10^−7 m = 867 nm.
• Planck constant h = 6.626 × 10^−34 J·s and speed of light c = 3 × 10^8 m/s.
• Energy conversion 1 eV = 1.602 × 10^−19 J.
• Neglect temperature-dependent shifts and refractive index dispersion for this estimate.

Concept / Approach:
Photon energy E is E = h * c / λ. For direct-gap semiconductors like GaAs, the emitted photon energy approximately equals the band gap E_g under lasing conditions (ignoring small losses). Converting E from joules to eV yields the numerical band gap.

Step-by-Step Solution:
Compute E (J): E = (6.626 × 10^−34 * 3 × 10^8) / (8.67 × 10^−7).Multiply numerator: 6.626 × 3 = 19.878 → 19.878 × 10^−26 J·m / 8.67 × 10^−7 m.Divide: 19.878/8.67 ≈ 2.293 → E ≈ 2.293 × 10^−19 J.Convert to eV: E_eV = (2.293 × 10^−19) / (1.602 × 10^−19) ≈ 1.43 eV.

Verification / Alternative check:
Use the handy relation E(eV) ≈ 1240 / λ(nm). With λ = 867 nm → E ≈ 1240/867 ≈ 1.43 eV, confirming the result.

Why Other Options Are Wrong:
0.18 eV and 0.70 eV are far too small for GaAs; 2.39 eV is too large and would correspond to blue-green light, not 867 nm. 1.00 eV is also inconsistent with the given wavelength.

Common Pitfalls:

• Forgetting to convert wavelength units (meters vs nanometers).
• Rounding too early; keep significant figures until the end.

Final Answer:
1.43 eV