Manley-Rowe Relations

In lossless linear systems, a harmonic input at a frequency ω remains harmonic, the system's steady-state response to a linear combination of individual harmonic inputs is a linear combination of invididual responses, and an average input power is zero at each frequencies. In lossless nonlinear systems, harmonic inputs at some frequencies generate also inputs at multiple and combination frequencies. The average input powers at each frequency may not be zero but they are predicted from the general relations found by Manley and Rowe in 1956.

Suppose the input applied to a nonlinear system consists of two harmonic signals of frequencies ω₁ and ω₂. The nonlinear system produces an output at all frequencies m ω₁ + n ω₂, where m and n are integers. For example, m = 2, n = 0 and m = 0, n = 2 are signals at the double frequencies 2 ω₁ and 2 ω₂, while m = n = 1 is the signal on the combinational frequency ω₁ + ω₂. Denote P_m,n the total input power at frequency m ω₁ + n ω₂. In lossless systems, the total power is zero; thus ∑ P_m,n = 0. In other words, the power P_m,n is negative if the signal at m ω₁ + n ω₂ is generated due to nonlinearity, and the power is positive if the signal is applied from input to the system.

The Manley-Rowe relations are relations between powers P_m,n and frequencies m ω₁ + n ω₂:

As an elementary example, we consider a heterodyne (an oscillator with frequency ω₁), which is mixed with the carrier incident frequency ω, such that a combination frequency ω₂ = ω - ω₁ occurs in the spectrum of the nonlinear system. We neglect signals at other frequencies and denote powers at frequencies ω₁, ω₂ and ω = ω₁ + ω₂ as P₁, P₂, and P = - P₁ - P₂. The Manley-Rowe relations between the powers and frequencies of the three signals are:

We note that these relations preserve conservation of powers P₁ + P₂ + P = 0 at the three resonant frequencies ω₁ + ω₂ - ω = 0. If ω > ω₁,ω₂, the Manley-Rowe relations show that the power P₂ at the combinational frequency ω₂ is smaller than the input power P by a factor of ω₂/ω, while the power at the heterodyne frequency ω₁ is smaller than P by a factor of ω₁/ω. Therefore, when an input signal of larger frequency transforms into output signals of smaller frequencies, the powers of output signals get smaller.

As another application, we consider an electro-optical modulator that takes an input signal at frequency 1 GHz and multiplies with another signal at frequency 100 MHz to produce a modulated output signal at combinational frequency 1.1 GHz. If the power of the output signal has to be 1mW, the Manley--Rowe relations require powers 0.0909 mW at 100 MHz and 0.9090 mW at 1 GHz. The 1 GHz signal is sometimes called the pump since it provides most of the power needed in the modulation process.

Manley--Rowe relations naturally appear as constants of motion in the time evolution of the resonant nonlinear wave interactions. If three waves with frequencies ω₁, ω₂ and ω = ω₁ + ω₂ and wavevectors k₁, k₂ and k = k₁ + k₂ satisfy the phase matching conditions:

then their interaction is resonant in time and is described by the system of three-wave interaction equations:

where d = ω(k) - ω(k₁) - ω(k₂) is the frequency detuning from the exact resonance, v, v₁, v₂ are group velocities of the three waves, and γ is the real-valued coupling coefficient in nonlinear systems with quadratic nonlinearities. The system of amplitude equations has integrals of motions:

The integrals of motions are the Manley--Rowe relations for the powers:

The resonant interaction of three waves result in decay of the wave with larger frequency ω > ω₁,ω₂ into waves of smaller frequencies ω₁, ω₂, which alternates with annihilation of the two waves of smaller frequencies ω₁, ω₂ into the wave of larger frequency ω. If the wave of smaller frequency ω₁ is initially large, compared to the other two waves ω and ω₂, it can not decay into the other two waves. On the other hand, if the wave of larger frequency ω is initially large, it can decay into two wave particles of smaller frequencies ω₁ and ω₂.

Manley-Rowe invariants play an important role in studies of properties of three-wave interactions. In particular, optical solitons are supported by dispersive and diffraction effects in nonlinear three-wave interactions. Stability of optical solitons is determined by the Vakhitov-Kolokolov criterion, which involves derivatives of the Manley--Rowe invariants with respect to parameters of optical solitons.

When the frequencies ω₁ and ω₂ of the two resonant waves coincide, the three-wave interactions degenerate into the resonant second-harmonic generation, when the wave a₁ = a₂ = a₀ with the fundamental frequency ω₁ = ω₂ = ω₀ generates the wave a at the double frequency ω = 2 ω₀. The system of three-wave equations simplifies then to the form:

The system of two-wave equations has only one Manley-Rowe invariant: