定义：
振荡相位随时间的微分除以2π。

在描述非单色信号时需要用到瞬时频率，它的定义式为： 

也就是振荡相位 φ对时间的微分。（除去因子 1/2π，就得到瞬时角频率。）与傅里叶频率不同的是，瞬时频率通常与时间有关。正弦信号的瞬时频率是常数等于振荡的频率。 

图1：强上啁啾脉冲的电场，其中瞬时频率随时间增加而增大。 
在考虑频率噪声和相位噪声时，瞬时频率非常重要，它也经常用于啁啾光脉冲（图1），其瞬时频率与时间有关。基本思想比傅里叶频率要更加直观。在音乐中也有相同的概念：乐谱中将音符表示为时间间隔，每一个间隔内的瞬时频率为一个定值（对应于音调）。然而，这一概念对于复杂的信号，例如白噪声，就出现问题了。在激光器中，很容易对单频激光器定义瞬时频率，而对于多模激光器，则首先需要将各个不同频率组分分离开（采用滤波技术），然后才能得到瞬时频率。在啁啾光脉冲中瞬时频率概念也是很有用处的，其中不同脉冲的瞬时频率是不同的。 

振荡信号的傅里叶光谱并不代表瞬时频率的概率分布，因此采用这一光谱测量的线宽也不是瞬时频率的均方根值。瞬时频率域傅里叶频率之间的关系更加复杂和微秒。 

瞬时频率随时间的变化可以根据光谱图的出来。但是，仅仅一个瞬时频率随时间变化的曲线并不能给出全部变化信息。 

测量瞬时频率 
电子信号（例如，拍音）的瞬时频率可以由锁相环路（PLL）得出，环路包含一个电压控制振荡器（VCO）和反馈系统的鉴相器使VCO与入射信号同步。VCO的入射信号可以测量瞬时频率。 这一方案的原理也可以运用到相位跟踪器软件中，来测量记录信号的瞬时频率。这一方案很简单但是也有一些缺点，尤其是其有限的带宽，存在延迟响应。快速傅里叶变换方法更加强大，但是也会更加复杂。
 

Definition: temporal derivative of the oscillation phase divided by 2π

More general term: optical frequency

Formula symbol: ν, ν_i

Units: Hz

The instantaneous frequency is a useful concept for describing non-monochromatic (polychromatic) signals – not only in optics. It is defined as

i.e. essentially as the temporal derivative of the oscillation phase φ. (Without the factor 1/2π, one would have the instantaneous angular frequency.) In contrast to a Fourier frequency, the instantaneous frequency is generally a time-dependent frequency. The instantaneous frequency of a sinusoidal signal is constant and equals the oscillation frequency, as expected.

Figure 1: Electric field of a strongly up-chirped pulse, where the instantaneous frequency increases with time.

The concept of the instantaneous frequency is particularly important in the context of frequency noise and phase noise, but it is also applied to chirped optical pulses (Figure 1), which have a time-dependent instantaneous frequency. The basic idea is intuitive, actually more than that of Fourier frequencies. The same concept is used e.g. in music: music scores essentially specify notes as time intervals for which the instantaneous frequency has a certain value (corresponding to the pitch of one voice, and disregarding overtones). However, the concept can become problematic for complicated signals, e.g. for white noise.

In the context of lasers, the instantaneous frequency can be easily defined for single-frequency lasers, whereas for multimode lasers one would first have to separate the different frequency components (with some filtering technique) before retrieving their instantaneous frequencies. The instantaneous frequency is also useful in the context of chirped optical pulses, where the instantaneous frequency varies during the pulse.

Note that the Fourier spectrum of an oscillating signal (in optics, the optical spectrum) does not represent the probability distribution of instantaneous frequencies, and that the linewidth measured from such a spectrum is not an r.m.s. (root mean squared) value of the instantaneous frequency. The relation between instantaneous frequency and Fourier frequency is substantially more subtle than that.

The time dependence of the instantaneous frequency can sometimes be estimated from a spectrogram. However, a plot of instantaneous frequency versus time generally does not carry the same information.

Measuring the Instantaneous Frequency

The instantaneous frequency of an electronic signal (e.g. a beat note) can be obtained using a phase-locked loop (PLL), containing a voltage-controlled oscillator (VCO) and phase discriminator in a feedback system which forces the VCO to follow the oscillation of the input signal. The input signal of the VCO can be used as a measure of the instantaneous frequency.

The basic idea of this approach can also be utilized in the form of a software phase tracker, which evaluates the instantaneous frequency of an already recorded signal. This approach is simple to implement but has some flaws, particularly the limited bandwidth and delayed response. Significantly more powerful, but also more complicated techniques utilize fast Fourier transforms.

The instantaneous frequency of a single-frequency laser may be measured using a stable reference laser at a similar optical frequency, a fast photodetector for recording a beat note, an electronic sampling card and suitable software.

Bibliography