**2.1. Symmetric 3 × 3 optical fiber coupler**

**Figure 1** shows the schematic diagram of the proposed 120° phase difference technology. As we know, the interferometric signals are 180° out of phase because of 2 × 2 optical fiber coupler. In an ideal 3 × 3 optical fiber coupler, there is a 120° phase difference between any two of the three output ports. A symmetric 3 × 3 optical fiber coupler is described by the matrix.

120° Phase Difference Interference Technology Based on 3 × 3 Coupler and Its Application... http://dx.doi.org/10.5772/66129 235

**Figure 1.** Schematic diagram of the proposed 120° phase difference technology.

and photonics field. The phase and frequency noise of such lasers can be conveniently described either in terms of linewidth or in terms of the power spectral density (PSD) of their phase or frequency noise. The linewidth gives a basic and concise parameter for characterizing laser coherence but lacks detailed information on frequency noise and its Fourier frequency spectrum, which is needed for understanding the noise origins and improving laser performances. Therefore, the measurement of frequency noise PSD is a focus of attention in the field, especially for lasers of very high coherence, whose linewidth is not easy to be measured.

To measure the phase and frequency noise, many methods have been proposed, such as beat note method [10], recirculating delayed self-heterodyne (DSH) method [11], DSH technique based on Mach-Zehnder interferometer with 2 × 2 coupler [12, 13], or Michelson interferometer (MI) with 2 × 2 coupler [14]. These methods can obtain good measurement results but need some strict conditions. The beat note method needs a high coherent source as a reference. The recirculating DSH method needs very long fiber delay lines. The DSH interferometers with 2 × 2 coupler need to control the quadrature point by some active feedback

To overcome these difficulties, we introduce a robust technique that can demodulate directly the laser differential phase accumulated in a delay time and then derive strict mathematical relations between the laser differential phase and the laser phase noise or frequency noise that can describe the complete information on laser phase and frequency noise. Because 3 × 3 optical fiber coupler acts as a 120° optical hybrid, it can demodulate the differential phase of the input light and has been used for DxPSK signal demodulation [15], optical sensors [16], optical field reconstruction, and dynamical spectrum measurement [17]. In this chapter, 120° phase difference interference technology based on an unbalanced Michelson interferometer, which is composed of a 3 × 3 optical fiber coupler and two Faraday rotator mirrors, is utilized to demodulate the differential phase of a laser. The structure has the advantage of being polarization insensitive and adjust-free. Especially, it does not need any active controlling operation that is used in the DSH methods with 2 × 2 coupler. Furthermore, based on the differential phase and strict physical and mathematical derivation, the PSD of the differential phase fluctuation and frequency fluctuation, the PSD of the instantaneous phase fluctuation and frequency fluctuation, laser phase noise, and linewidth are completely calculated and dis-

**Figure 1** shows the schematic diagram of the proposed 120° phase difference technology. As we know, the interferometric signals are 180° out of phase because of 2 × 2 optical fiber coupler. In an ideal 3 × 3 optical fiber coupler, there is a 120° phase difference between any two of the three output ports. A symmetric 3 × 3 optical fiber coupler is described by the

methods and accurate calibration.

234 Optical Interferometry

**2. 120° phase difference interference technology**

**2.1. Symmetric 3 × 3 optical fiber coupler**

cussed.

matrix.

$$T\_3 = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 & \exp\frac{i2\pi}{3} & \exp\frac{i2\pi}{3} \\ \exp\frac{i2\pi}{3} & 1 & \exp\frac{i2\pi}{3} \\ \exp\frac{i2\pi}{3} & \exp\frac{i2\pi}{3} & 1 \end{pmatrix} \tag{1}$$

The matrix of the two arms of the Michelson interferometer is given by,

$$P = \begin{pmatrix} \exp(i\phi\_1) & 0 & 0 \\ 0 & \exp(i\phi\_2) & 0 \\ 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \exp(i\Lambda\phi) & 0 \\ 0 & 0 & 0 \end{pmatrix} \exp(i\phi\_l) \tag{2}$$

Where Δ*ϕ* = *ϕ*2 – *ϕ*1 = 2π*n*Δ*L*/*λ* is the phase difference of the two arms of the Michelson interferometer, *n* is the refractive index of fiber, *λ* is the wavelength of transmit light, Δ*L* is the length difference of the two arms. Setting 2π/3 = *θ*, the operation of the whole interferometer is then described by the system matrix.

$$M = T\_{\mathfrak{z}} P T\_{\mathfrak{z}} = \frac{e^{i\boldsymbol{\eta}}}{3} \begin{pmatrix} 1 + e^{i(\boldsymbol{\Lambda}\boldsymbol{\eta} + 2\boldsymbol{\vartheta})} & (\mathbf{l} + e^{i\boldsymbol{\Lambda}\boldsymbol{\eta}}) e^{i\boldsymbol{\theta}} & e^{i\boldsymbol{\theta}} (\mathbf{l} + e^{i(\boldsymbol{\Lambda}\boldsymbol{\eta} + \boldsymbol{\vartheta})}) \\\ e^{i\boldsymbol{\theta}} (\mathbf{l} + e^{i\boldsymbol{\Lambda}\boldsymbol{\eta}}) & e^{i2\boldsymbol{\theta}} + e^{i\boldsymbol{\Lambda}\boldsymbol{\eta}} & e^{i\boldsymbol{\theta}} (e^{i\boldsymbol{\theta}} + e^{i\boldsymbol{\Lambda}\boldsymbol{\eta}}) \\\ e^{i\boldsymbol{\theta}} (\mathbf{l} + e^{i(\boldsymbol{\Lambda}\boldsymbol{\eta} + \boldsymbol{\vartheta})}) & e^{i\boldsymbol{\theta}} (e^{i\boldsymbol{\theta}} + e^{i\boldsymbol{\Lambda}\boldsymbol{\eta}}) & e^{i2\boldsymbol{\theta}} (\mathbf{l} + e^{i\boldsymbol{\Lambda}\boldsymbol{\eta}}) \end{pmatrix} \tag{3}$$

The electric field of input laser is expressed by,

$$E(t) = \left| E(t) \right| \exp[i\alpha\_0 t + i\varphi(t)] \tag{4}$$

with amplitude |*E*(*t*)|, center frequency *ω*0, and instantaneous phase fluctuation *ϕ*(*t*). The input field *K*in = [*E*(*t*), 0, 0]T is a vector whose elements are the amplitudes of the input modes to the interferometer. *K*in is transformed by the interferometer into an output vector.

$$\left[\left[E\_1^{\rm out}, E\_2^{\rm out}, E\_3^{\rm out}\right]^T = MK^{\rm in} = \frac{e^{i\boldsymbol{\rho}\_1}}{3} \cdot \begin{pmatrix} 1 + e^{i(\boldsymbol{\Delta}\boldsymbol{\rho} + 2\boldsymbol{\theta})} \\ e^{i\boldsymbol{\theta}} (1 + e^{i\boldsymbol{\Delta}\boldsymbol{\rho}}) \\ e^{i\boldsymbol{\theta}} (1 + e^{i(\boldsymbol{\Delta}\boldsymbol{\rho} + \boldsymbol{\theta})}) \end{pmatrix} \cdot E\left(t\right) \tag{5}$$

The measured intensities *In* = | *En* out |2 (*n*=1, 2, 3) of the three outputs are,

$$
\begin{pmatrix} I\_1 \\ I\_2 \\ I\_3 \end{pmatrix} = \frac{2I\_0}{9} \begin{pmatrix} 1 + \cos(\Lambda \varphi - 2\pi/3) \\ 1 + \cos(\Lambda \varphi) \\ 1 + \cos(\Lambda \varphi + 2\pi/3) \end{pmatrix} \tag{6}
$$

where *I*0 = |*E*(t)|2 is the input intensity. Then the differential phase can be obtained.

$$
\Delta \varphi = \arctan \left[ \frac{\sqrt{3} \left( I\_3 - I\_1 \right)}{I\_3 + I\_1 - 2I\_2} \right] \tag{7}
$$

#### **2.2. Asymmetric 3 × 3 optical fiber coupler**

However, the commercially available 3 × 3 optical fiber coupler is usually asymmetric and lossy. So the transmissivity of 3 × 3 coupler from port *m* to port *n* (*m*, *n* = 1, 2, 3) is *b*mnexp(*iθ*mn), where *b*mn and *θ*mn are the splitting ratio and phase delay of coupler, respectively. The forward transmission matrix of 3 × 3 coupler is then given by.

$$T\_{3\mu m} = b\_{m\nu} \exp(\mathrm{i}\,\theta\_{m\nu})\tag{8}$$

Similarly, the backward transmission matrix of 3 × 3 coupler is then given by,

$$T'\_{\rm 3uu} = b'\_{\rm nu} \exp(\rm i \, \theta'\_{\rm nu}) \tag{9}$$

where *b*ʹ*mn* is the splitting ratio and *θ*ʹ*mn* is the phase delay of 3 × 3 coupler from port *n* to port *m*. The matrix of the two arms of the Michelson interferometer is given by,

120° Phase Difference Interference Technology Based on 3 × 3 Coupler and Its Application... http://dx.doi.org/10.5772/66129 237

$$P = \begin{pmatrix} p\_{11} & 0 & 0 \\ 0 & p\_{22} \exp(-i o \tau - i \delta) & 0 \\ 0 & 0 & 0 \end{pmatrix} \tag{10}$$

where *p*11 and *p*22 are the transmissivity of the two arms, respectively; *τ* and *δ* are the differential delay time and the differential phase delay between the signals in two arms of the Michelson interferometer. The operation of the whole interferometer is then described by the system matrix.

$$\begin{split} M\_{\boldsymbol{nn}} &= \left( T\_3' P T\_3 \right)\_{\boldsymbol{nn}} \\ &= b\_{n1}' p\_{11} b\_{1n} \exp\left[ i \left( \theta\_{n1}' + \theta\_{1n} \right) \right] + b\_{n2}' p\_{22} b\_{2n} \exp\left[ i \left( \theta\_{n1}' + \theta\_{2n} \right) \right] \exp(-i\alpha \tau - i\delta) \end{split} \tag{11}$$

The output light from the output port *n* is detected by a photodetector having a responsivity of *rn*. The detected intensities from the output port *n*, *In*, can be expressed as.

$$\begin{split} I\_{\boldsymbol{n}}(t) &= r\_{\boldsymbol{n}} \left| E\_{\boldsymbol{n}}^{\text{out}} \right|^{2} \\ &= r\_{\boldsymbol{n}} c\_{\boldsymbol{n}1}^{2} \left| E\left(t\right) \right|^{2} + r\_{\boldsymbol{n}} c\_{\boldsymbol{n}2}^{2} \left| E\left(t-\tau\right) \right|^{2} + \eta\_{\boldsymbol{n}} \cos\left(\delta + \Delta\phi\right) \left| E\left(t\right)E\left(t-\tau\right) \right| + \\ &\quad \left. \zeta\_{\boldsymbol{n}} \sin\left(\delta + \Delta\phi\right) \right| E\left(t\right)E\left(t-\tau\right) \right| \end{split} \tag{12}$$

Define an intermediate matrix,

$$X' = \begin{pmatrix} X\_1'(t) & X\_2'(t) & X\_3'(t) \end{pmatrix}^\top = \begin{pmatrix} \cos\left(\delta + \Delta\phi\right) \left| E\begin{pmatrix} t \end{pmatrix} E\begin{pmatrix} t - \tau \end{pmatrix} \right| \\ \sin\left(\delta + \Delta\phi\right) \left| E\begin{pmatrix} t \end{pmatrix} E\begin{pmatrix} t - \tau \end{pmatrix} \right| \\ \left( \left| E\begin{pmatrix} t \end{pmatrix} \right|^2 + \left| E\begin{pmatrix} t - \tau \end{pmatrix} \right|^2 \right) \end{pmatrix} \tag{13}$$

Then,

<sup>0</sup> *Et Et i t i t* ( ) ( ) exp[ ( )] = + w

to the interferometer. *K*in is transformed by the interferometer into an output vector.

, , (1 ) <sup>3</sup>

out |2

0

arctan

j

The forward transmission matrix of 3 × 3 coupler is then given by.

æ ö <sup>+</sup> ç ÷ é ù = =× + × ç ÷ ë û ç ÷ <sup>+</sup> è ø

*<sup>i</sup> <sup>T</sup> i i*

j

1 cos( 2 / 3) <sup>2</sup> 1 cos( ) <sup>9</sup> 1 cos( 2 / 3)

3

é ù - D = ê ú

*T b* <sup>3</sup>*nm nm* = exp i( )

*T b* <sup>3</sup>*nm nm* exp i( )

Similarly, the backward transmission matrix of 3 × 3 coupler is then given by,

*m*. The matrix of the two arms of the Michelson interferometer is given by,

æö æ + D -p <sup>ö</sup> ç÷ ç <sup>÷</sup> = +D ç÷ ç <sup>÷</sup> ç÷ ç <sup>÷</sup> èø è + D +p <sup>ø</sup>

j

j

j

is the input intensity. Then the differential phase can be obtained.

( ) 3 1 31 2

ê ú + - ë û

However, the commercially available 3 × 3 optical fiber coupler is usually asymmetric and lossy. So the transmissivity of 3 × 3 coupler from port *m* to port *n* (*m*, *n* = 1, 2, 3) is *b*mnexp(*iθ*mn), where *b*mn and *θ*mn are the splitting ratio and phase delay of coupler, respectively.

q

q

where *b*ʹ*mn* is the splitting ratio and *θ*ʹ*mn* is the phase delay of 3 × 3 coupler from port *n* to port

2 *I I II I*

*<sup>e</sup> E E E MK e e E t*

out out out in

1

*I <sup>I</sup> <sup>I</sup>*

2 3

*I*

1 23

The measured intensities *In* = | *En*

**2.2. Asymmetric 3 × 3 optical fiber coupler**

where *I*0 = |*E*(t)|2

236 Optical Interferometry

with amplitude |*E*(*t*)|, center frequency *ω*0, and instantaneous phase fluctuation *ϕ*(*t*). The input field *K*in = [*E*(*t*), 0, 0]T is a vector whose elements are the amplitudes of the input modes

1

q

q

*i*

j q

D + D D +

*e*

*i i*

*e e*

 j

( ) <sup>1</sup> ( 2)

 j

(1 )

(*n*=1, 2, 3) of the three outputs are,

( )

 jq

(4)

*nm* (8)

*nm* ¢¢ ¢ = (9)

(5)

(6)

(7)

$$
\begin{pmatrix} I\_1(t) \\ I\_2(t) \\ I\_3(t) \end{pmatrix} = \begin{pmatrix} \eta\_1 & \zeta\_1 & \xi\_1 \\ \eta\_2 & \zeta\_2 & \xi\_2 \\ \eta\_3 & \zeta\_3 & \xi\_3 \end{pmatrix} \begin{pmatrix} X\_1'(t) \\ X\_2'(t) \\ X\_3'(t) \end{pmatrix} + \begin{pmatrix} r\_1 \left(c\_{11}^2 - c\_{12}^2\right) \\ r\_2 \left(c\_{21}^2 - c\_{22}^2\right) \\ r\_3 \left(c\_{31}^2 - c\_{32}^2\right) \end{pmatrix} \frac{\left\| E\begin{pmatrix} t \\ \end{pmatrix} \right\|^2 - \left| E\begin{pmatrix} t - \tau \end{pmatrix} \right\|^2}{2} \tag{14}
$$

Where

$$\begin{aligned} \eta\_n &= 2r\_n c\_{n1} c\_{n2} \cos \theta\_n \\ \xi\_n &= 2r\_n c\_{n1} c\_{n2} \sin \theta\_n \\ \xi\_n &= r\_n \left( c\_{n1}^2 + c\_{n2}^2 \right) \\ c\_{nn} &= p\_{nm} b\_{nn}' b\_{m1} \\ \theta\_n &= \left( \theta\_{n2}' + \theta\_{21} \right) - \left( \theta\_{n1}' + \theta\_{11} \right) \\ \Delta \varphi &= \varphi(t) - \varphi(t-\tau) \end{aligned} \tag{15}$$

The parameters *ηn*, *ςn*, and *ξn* are constant for the setup, once the devices and structure are determined. They can be obtained by a broadband light source without measuring each parameter one by one as said in Ref. [15]. Eq. (13) can be transformed as.

$$X' = \begin{pmatrix} \eta\_1 & \zeta\_1 & \tilde{\xi}\_1 \\ \eta\_2 & \zeta\_2 & \tilde{\xi}\_2 \\ \eta\_3 & \zeta\_3 & \tilde{\xi}\_3 \end{pmatrix}^{-1} \begin{pmatrix} I\_1(t) \\ I\_2(t) \\ I\_3(t) \end{pmatrix} - \begin{pmatrix} \eta\_1 & \zeta\_1 & \tilde{\xi}\_1 \\ \eta\_2 & \zeta\_2 & \tilde{\xi}\_2 \\ \eta\_3 & \zeta\_3 & \tilde{\xi}\_3 \end{pmatrix}^{-1} \begin{pmatrix} r\_1 \left(c\_{11}^2 - c\_{12}^2\right) \\ r\_2 \left(c\_{21}^2 - c\_{22}^2\right) \\ r\_3 \left(c\_{31}^2 - c\_{32}^2\right) \end{pmatrix} \frac{\left| E\left(t\right) \right|^2 - \left| E\left(t - \tau\right) \right|^2}{2} \tag{16}$$

Eq. (16) shows the relation between the differential phase ∆*ϕ* and the detectors outputs (*I*1, *I*2, *I*3). We note that the second term on the right-hind side of Eq. (16) becomes zero or can be omitted under the following conditions: (a) the splitting ratios of the 3 × 3 coupler are uniform (i.e., *c*n1 = *c*n2 for all *n*), or (b) the intensity of the laser under test is constant or periodical (i.e., |*E*(*t*)|2 – |*E*(*t* – *τ*)|2 =0), or (c) the extinction ratio of the Michelson interferometer. As a result, the differential phase Δ*ϕ*(*t*) accumulated in delay time *τ* can be obtained in the following simple form

$$
\Delta\varphi(t) = \varphi(t) - \varphi(t-\tau) = \arctan\left(\frac{X\_z'(t)}{X\_1'(t)}\right) - \arctan\left(\frac{X\_z'(t)}{X\_1'(t)}\right) \tag{17}
$$
