# Coherent Detection

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It is clear from the discussion of optical receiver noise tutorial that, even though shot noise sets the fundamental limit, it is the thermal noise that limits a photodetector in practice. The use of APDs helps to reduce the impact of thermal noise to some extent, but it also enhances the shot noise. One may ask if it is possible to design a detection scheme that is limited by shot-noise alone. The answer is provided by a technique known as *coherent detection*, so called because it combines the incoming optical signal coherently with a CW optical field before it falls on the detector. An added benefit is that such a technique can also be used for systems that encode information in the optical phase (such as FSK and PSK modulation formats) because it converts phase variations into amplitude variations.

#### 1. Local Oscillator

The basic idea behind coherent detection is shown schematically in the figure below.

A coherent field is generated locally at the receiver using a narrow-linewidth laser, called the *local oscillator* (LO), a term borrowed from the radio and microwave literature. It is combined with the incoming optical field using a beam splitter, typically a fiber coupler in practice. To see how such a mixing can improve the receiver performance, let us write the optical signal using complex notation as

E_{s} = A_{s}exp[-i(ω_{0}t + φ_{s})]

where ω_{0} is the carrier frequency, A_{s} is the amplitude, and φ_{s} is the phase. The optical field associated with the local oscillator is given by a similar expression,

E_{LO} = A_{LO}exp[-i(ω_{LO}t + φ_{LO})]

where A_{LO}, ω_{LO}, and φ_{LO} represent the amplitude, frequency, and phase of the local oscillator, respectively. The scalar notion is used for both E_{s} and E_{LO} after assuming that the two fields are identically polarized. The optical power incident at the photodetector is given by P = |E_{s} + E_{LO}|^{2}. Using the two equations above,

where

P_{s} = A_{s}^{2}, P_{LO} = A_{LO}^{2}, ω_{IF} = ω_{0} - ω_{LO}

The frequency ν_{IF} ≡ ω_{IF}/2π is known as the intermediate frequency (IF). When ω_{0} ≠ ω_{LO}, the optical signal is demodulated in two stages. Its carrier frequency is first converted to an intermediate frequency ν_{IF} (typically 0.1-5 GHz). The resulting radio-frequency (RF) signal is then processed electronically to recover the bit stream. It is not always necessary to use an intermediate frequency. In fact, there are two different coherent detection techniques to choose from, depending on whether or not ω_{IF} equals zero. They are known as *homodyne* and *heterodyne* detection techniques.

#### 2. Homodyne Detection

In this coherent-detection technique, the local-oscillator frequency ω_{LO} is selected to coincide with the signal-carrier frequency ω_{0} so that ω_{IF} = 0. From the equation above, the photocurrent (I = R_{d}P, where R_{d} is the detector responsivity) is given by

Typically, P_{LO} >> P_{s}, and P_{s} + P_{LO} ≈ P_{LO}. The last term in this equation contains the information transmitted and is used by the decision circuit. Consider the case in which the local-oscillator phase is locked to the signal phase so that φ_{s} = φ_{LO}. The homodyne signal is then given by

The main advantage of homodyne detection is evident from this equation if we note that the signal current in the direct-detection case is given by I_{dd}(t) = R_{d}P_{s}(t). Denoting the average signal power by , the average electrical power is increased by a factor of with the use of homodyne detection. Since P_{LO} can be made much larger than , the power enhancement can exceed 20 dB. Although shot noise is also enhanced, it is shown later that homodyne detection improves the SNR by a large factor.

Another advantage of coherent detection is evident from the I(t) equation above. Because the last term in this equation contains the signal phase explicitly, it is possible to recover data transmitted using the phase or frequency of the optical carrier. Direct detection does not allow this because all information in the signal phase is lost. Several modulation formats requiring phase encoding are discussed in another tutorial.

A disadvantage of homodyne detection also results from its phase sensitivity. Since the last term in I(t) equation above contains the local-oscillator phase φ_{LO} explicitly, clearly φ_{LO} should be controlled. Ideally, φ_{s} and φ_{LO} should stay constant except for the intentional modulation of φ_{s}. In practice, both φ_{s} and φ_{LO} fluctuate with time in a random manner. However, their difference φ_{s} - φ_{LO} can be forced to remain nearly constant through an optical phase-locked loop. The implementation of such a loop is not simple and makes the design of optical homodyne receivers quite complicated. In addition, matching of the transmitter and local-oscillator frequencies puts stringent requirements on the two optical sources.

#### 3. Heterodyne Detection

In the case of heterodyne detection the local-oscillator frequency ω_{LO} is chosen to differ from the signal-carrier frequency ω_{0} such that the intermediate frequency ω_{IF} is in the microwave region (ν_{IF} ~ 1 GHz). Using the P(t) equation from above together with I = R_{d}P, the photocurrent is now given by

Since P_{LO} >> P_{s} in practice, the direct-current (dc) term is nearly constant and can be removed easily using bandpass filters. The heterodyne signal is then given by the alternating-current (ac) term in this equation or by

Similar to the case of homodyne detection, information can be transmitted through amplitude, phase, or frequency modulation of the optical carrier. More importantly, the local oscillator still amplifies the received signal by a large factor, thereby improving the SNR. However, the SNR improvement is lower by a factor of 2 (or by 3 dB) compared with the homodyne case. This reduction is referred to as the heterodyne detection penalty. The origin of the 3-dB penalty can be seen by considering the signal power (proportional to the square of the current). Because of the ac nature of I_{ac}, the electrical power is reduced by a factor of 2 when I_{ac}^{2} is averaged over a full cycle at the intermediate frequency (recall that the average of cos^{2}θ over θ is 1/2).

The advantage gained at the expense of the 3-dB penalty is that the receiver design is simplified considerably because an optical phase-locked loop is no longer needed. Fluctuations in both φ_{s} and φ_{LO} still need to be controlled using narrow-linewidth semiconductor lasers for both optical sources. However, the linewidth requirements are relatively moderate when an asynchronous demodulation scheme is employed. This feature makes the heterodyne-detection scheme quite suitable for practical implementation in coherent lightwave systems.

#### 4. Signal-to-Noise Ratio

The advantage of coherent detection for lightwave systems can be made more quantitative by considering the SNR of the receiver current. For this purpose, it is necessary to extend the analysis of optical receiver noise to the case of heterodyne detection. The receiver current fluctuates because of shot noise and thermal noise. The variance σ^{2} of current fluctuations is obtained by adding the two contributions so that

σ^{2} = σ_{s}^{2} + σ_{T}^{2}

where

σ_{s}^{2} = 2q(I + I_{d})Δf, σ_{T}^{2} = (4k_{B}T/R_{L})F_{n}Δf

It is important to note that the I in this equation is the total current generated at the detector and is given by I(t) from the two equations above, depending on whether homodyne or heterodyne detection is employed. In practice, P_{LO} >> P_{s}, and I in this equation can be replaced by the dominant term RP_{LO} for both cases.

The SNR is obtained by dividing the average signal power by the average noise power. In the heterodyne case, it is given by

In the homodyne case, the SNR is larger by a factor of 2 if we assume that φ_{s} = φ_{LO}. The main advantage of coherent detection can be seen from this equation. Since the local-oscillator power P_{LO} can be controlled at the receiver, it can be made large enough that the receiver noise is dominated by shot noise. More specifically, σ_{s}^{2} >> σ_{T}^{2} when

P_{LO} >> σ_{T}^{2} /(2qR_{d}Δf)

Under the same conditions, the dark-current contribution to the shot noise is negligible (I_{d} << RP_{LO}). The SNR is then given by

where R_{d} = ηq/hν was used. The main point to emphasize is that the use of coherent detection allows one to achieve the shot-noise limit even for p-i-n receivers whose performance is generally limited by thermal noise. Moreover, in contrast with the case of APDs, this limit is realized without adding any excess shot noise.

It is useful to express the SNR in terms of the number of photons, N_{p}, received within a single bit. At the bit rate B, the signal power is related to N_{p} as . Typically, Δf ≈ B/2. Using these values in the SNR equation above, the SNR is given by a simple expression

SNR = 2ηN_{p}

In the case of homodyne detection, SNR is larger by a factor of 2 and is given by

SNR = 4ηN_{p}