Derivation of the Relationship Between Light Intensity and Complex Amplitude

In the framework of wave optics, describing the spatial propagation and temporal evolution of light waves relies heavily on the mathematical formalism of the complex amplitude. This construct elegantly encapsulates both the amplitude and the phase of an optical field, serving as an essential bridge between classical electromagnetism and observable phenomena. Establishing a rigorous quantitative relationship between light intensity and complex amplitude is therefore a prerequisite for analyzing any interference, diffraction, or scattering problem.

Strictly speaking, light intensity is defined as the time-averaged energy flux density—the amount of electromagnetic energy passing through a unit area perpendicular to the propagation direction per unit time. For a monochromatic plane wave, the electric field vector $\mathbf{E}$ can be expressed in a complex representation as:
$$ \mathbf{E}(\mathbf{r}, t) = \text{Re}[\mathbf{A}(\mathbf{r}) e^{-i\omega t}] $$
where $\mathbf{A}(\mathbf{r})$ denotes the spatial complex amplitude, $\omega$ is the angular frequency, and $\text{Re}{\cdot}$ extracts the real part.
According to classical electrodynamics, the instantaneous energy flux density of an electromagnetic wave is governed by the Poynting vector $\mathbf{S}$:
$$ \mathbf{S}(t) = \frac{1}{\mu_0} \mathbf{E}(t) \times \mathbf{B}(t) $$
In a standard linear, isotropic dielectric medium, the magnetic field is in phase with the electric field and proportional to it, meaning the instantaneous power flow scales quadratically with the electric field magnitude.

Because optical frequencies are exceptionally high (on the order of $10^{14}$ Hz for visible light), no conventional photodetector or optical sensor can resolve these rapid oscillations. Instead, detectors register a time-averaged value. The measurable intensity $I$ is defined as the time average of the Poynting vector's projection along the direction of propagation:
$$ I = \langle \mathbf{S} \cdot \hat{\mathbf{n}} \rangle_t $$
where $\hat{\mathbf{n}}$ is the unit propagation vector, and $\langle \cdot \rangle_t$ signifies a temporal average over an optical cycle $T = 2\pi/\omega$.

Mathematical Derivation via Complex Representation

To evaluate this average explicitly, let us adopt a scalar formulation for a fixed polarization state. The real-valued electric field can be written as:
$$ E(t) = \text{Re}[A e^{-i\omega t}] = \frac{1}{2} \left( A e^{-i\omega t} + A^* e^{i\omega t} \right) $$

Since the instantaneous intensity is proportional to $E^2(t)$, we expand the squared expression:
$$ E^2(t) = \left[ \frac{1}{2} \left( A e^{-i\omega t} + A^* e^{i\omega t} \right) \right]^2 = \frac{1}{4} \left( A^2 e^{-2i\omega t} + 2|A|^2 + (A^*)^2 e^{2i\omega t} \right) $$

Next, we take the time average over the period $T$:

  1. The cross-terms $A^2 e^{-2i\omega t}$ and $(A^*)^2 e^{2i\omega t}$ oscillate at twice the optical frequency ($2\omega$), and their integrals over a full cycle vanish to zero.
  2. The middle term $2|A|^2$ is time-independent, and its average is simply itself.

Consequently, the time-averaged squared electric field reduces to:
$$ \langle E^2(t) \rangle_t = \frac{1}{4} \cdot 2 |A|^2 = \frac{1}{2} |A|^2 $$

Incorporating the fundamental electromagnetic constants of free space, the definitive relationship between light intensity $I$ and the complex amplitude $\mathbf{A}$ emerges as:
$$ I = \frac{1}{2} c \epsilon_0 |\mathbf{A}|^2 $$
where $c$ is the speed of light and $\epsilon_0$ is the vacuum permittivity. By grouping the material constants into a single proportionality factor $K = \frac{1}{2} c \epsilon_0$, the equation simplifies to the familiar proportional form:
$$ I = K |\mathbf{A}|^2 $$

Physical Implications and Analytical Significance

This fundamental derivation highlights two profound physical consequences that govern optical experiments:

  • Loss of Phase Information: Intensity depends solely on the squared magnitude of the complex amplitude ($|A|^2 = A A^*$). Consequently, standard direct detection methods record only the envelope of the light wave, discarding the absolute phase information. Overcoming this "phase problem" is the primary motivation behind advanced optical techniques like holography and interferometry, which convert unmeasurable phase shifts into detectable intensity variations through reference beams.
  • Nonlinear Superposition of Fields: When two or more coherent light waves overlap, their complex amplitudes add linearly: $\mathbf{A}_{\text{total}} = \mathbf{A}_1 + \mathbf{A}2$. However, because intensity scales quadratically with the amplitude, the resulting total intensity exhibits cross-terms:
    $$ I    {\text{total}} = K |\mathbf{A}_1 + \mathbf{A}_2|^2 = K \left( |\mathbf{A}_1|^2 + |\mathbf{A}_2|^2 + \mathbf{A}_1 \mathbf{A}_2^* + \mathbf{A}_1^* \mathbf{A}_2 \right) $$
    The final two terms combine to form the interference term $2K \text{Re}(\mathbf{A}_1 \mathbf{A}_2^*)$. This mathematical outcome underscores how the linear superposition of complex fields yields intricate, nonlinear intensity distributions characteristic of wave interference.

Conclusion

The proportional link between light intensity and the squared magnitude of the complex amplitude ($I \propto |A|^2$) forms the bedrock of wave optics. Recognizing that intensity is fundamentally an energy metric derived from time-averaging a complex field allows physicists and optical engineers to accurately model complex phenomena such as diffraction gratings, laser cavities, and wave propagation. Mastering this derivation ensures a seamless transition from abstract electromagnetic wave equations to concrete, verifiable optical measurements.