Chapter 3 Sagnac Fiber Interferometers

Chapter 3

Sagnac Fiber Interferometers

Nonlinear effects seen in the previous chapter can be exploited for optical processing by using interferometric structures. Fiber interferometers are based on simple schemes and they are widely used as key-elements in many fiber-based optical devices. The most common fiber interferometers are the fiber-version of Fabry-Perot, Sagnac, Mach-Zehnder and Michelson interferometers. For our purposes the most interesting is the fiber Sagnac Interferometer, as the NOLMs are based on that. In the fibre Sagnac Interferometer a central role is covered by the fiber coupler [3].

3.1 Fiber Couplers

Fig.3.2 Schematic illustration of a fiber coupler.

Fiber couplers cover an essential role in optical fibers technology. They are widely used whenever the splitting of an optical signal (or the coupling of two) is required. Fiber couplers are four-port devices with two input and two output ports, signals incident on an input port are coherently split and the two parts are directed to the output ports. Several different techniques can be used to create a fiber coupler; one way is to realize a fused fiber coupler in which the cores of two single-mode fibers are brought close together in a central region so that the distance between the cores is comparable with their diameters. Another solution can be a dual core fiber, designed to have two cores close to each other throughout its length. In both cases the fundamental modes propagating in each core overlap partially in the cladding region between the two cores. Such evanescent wave coupling between the two modes can lead to the transfer of optical power from one core to the other. Fiber couplers are called symmetric when their cores are identical; in general they need not to be identical and such couplers are called asymmetric. The power coming out of the two output ports depends on the coupler length and on the input power. It can be demonstrated that for a symmetric coupler with continuous waves incident on input ports, the outgoing signals are given by:

1 1 2 2 A (L) cos( L) jsin( L) A (0) A (L) jsin( L) cos( L) A (0) κ κ ⎛ ⎞ ⎛ ⎞⎛ = ⎜ ⎟ ⎜ _{κ κ} ⎟⎜ ⎝ ⎠ ⎝ ⎠ ⎝ ⎞ ⎟ ⎠

The determinant of the 2x2 transfer matrix is unity as it should be for lossless couplers. The coupling coefficient κ depends on the spacing d between the two cores and it is usually computed using the following empirical expression:

2 0 1 2 2 0 0 V exp[ (c c d c d )] 2k n a π κ = − + +

where V = 2πaNA/λ0, NA is the fiber numerical aperture (it represents the light-gathering

capacity of an optical fiber), a is the core radius and d ≡ d/a is the normalized center-to-center spacing between the two cores (d>2). The constants c0,c1 and c2 depend on V as c0 =

5.2789 - 3.663V + 0.3841V2, c1 = -0.7769 + 1.2252V – 0.0152V2 and c2 = -0.0175 – 0.0064V

-0.0009V2. This equation is accurate to within 1% for values of V and d in the range 1.5 ≤ V ≤ 2.5 and 2 ≤ d ≤ 4.5.

If only one beam is injected at the input end, the output powers are then obtained by setting A2(0) = 0 and are given by

P1(L) = P0cos2(κL) P2(L) = P0sin2(κL)

where P0 ≡ A02 is the incident power at the first input port. The coupler acts as a beam splitter

and the splitting ratio depends on the parameter κL. If the coupler length is chosen such that κL = π/4 the power is equally divided between the two output ports. Such couplers are called 50:50 or -3dB couplers. It is very important to underline that a coupler introduces a relative phase shift of π/2 between the two output ports as indicated by the factor j in the off-diagonal term in the transfer matrix. This phenomenon plays an important role in the realization of interferometric structures based on fiber couplers.

3.2 All-Fiber Sagnac Interferometers

An all-fiber Sagnac interferometer can be easily realized by connecting a fiber to the output ports of a fiber coupler:

Fig.3.3 Schematic illustration of an all-fiber Sagnac interferometer.

The entering optical field (with power Pin) is split by coupler into two halves which travel

along the same optical path following opposite directions. Accordingly with the fiber coupler transfer matrix, the clockwise signal’s power is ρPin (where ρ is the coupler splitting ratio)

while for the counterclockwise signal power is (1-ρ)Pin and a phase shift of π/2 is introduced

by the coupler. After a round trip the two signals reach the coupler again, the clockwise one experiences a further attenuation ρ towards the output port while the counterclockwise one experiences an attenuation (1-ρ) and a second phase shift of π/2 towards the output port. As a result the two signals combine at the output port with a relative phase displacement of π and they interfere destructively. In particular when ρ = 0.5 the two counterpropagating signals experience the same attenuations, so when they interfere power at the output port is null. In such conditions the interferometer acts as a mirror and all the power is reflected back.

The interferometer maintains almost the same behaviour for any input signal; however, by inserting a nonlinear element into the loop we can introduce a power dependent phase shift and the interferometer can be customized for several applications. The nonlinear element can be a semiconductor or a fiber.

3.2.1 Nonlinear Optical Loop Mirror (NOLM)

In this case the nonlinear element is an optical fiber and the exploited nonlinear phenomenon can be self-phase modulation or cross-phase modulation. Since the Kerr effect has a femtosecond response speed, this scheme is used for ultrahigh-speed time-domain switching.

3.2.1.1 Self-Phase Modulation based NOLM

Fig.3.4 Schematic illustration of a SPM-based NOLM.

Let us consider a CW or a quasi-CW input beam incident at one port of the fiber coupler. The transmittivity of a Sagnac interferometer depends on the power splitting ratio of the coupler. If a fraction ρ of the input power P0 travels along the forwarding path, the loop transmittivity

can be obtained by calculating the total phase shifts acquired by the two waves after a round trip. If A0 is the amplitude of the input field, the amplitudes of the forward (clockwise) and

backward (counterclockwise) propagating fields are given by:

0

A_f = ρA A_b = j 1− ρA₀

Notice the π/2 phase shift for Ab introduced by the coupler. After a round trip both fields

acquire a linear phase shift and a nonlinear SPM- and XPM-based one. The two fields reaching the coupler are:

2 2 2 2 A A exp[ j j | | 2 | | )L] A A exp[ j j | | 2 | | )L] f f f b b b b f ' ' 0 0 = φ + γ( Α + Α = φ + γ( Α + Α

where φ0 = βL is the linear phase shift for a loop with length L and propagation constant β.

The reflected and transmitted fields can be obtained by using the coupler transmission matrix:

A A j A _j A f t r b ' ' ⎛ _{ρ 1− ρ ⎛}⎞ _⎞ ⎛ ⎞ = ⎜ _{⎟⎜ ⎟} ⎜ _{⎟ ⎜ ⎟} 1− ρ ρ ⎝ ⎠ _{⎝ ⎠}⎝ ⎠

Then the transmittivity TS = |At|2/|A0|2 of the Sagnac interferometer is given by:

{

}

S 0

T = − ρ(1− ρ) +1 2 1 cos[(1 2 ) P L]− ρ γ

where P0 = |A0|2 is the input power. For ρ = 0.5 TS = 0 and the loop reflects any input signal;

in fact when power is equally divided, the two counterpropagating waves experiment the same nonlinear phase shift and their interference is completely destructive.

Fig.3.5 Nonlinear response of an all-fiber Sagnac interferometer.

Fig.3.5 shows the nonlinear response of an all-fiber Sagnac interferometer for ρ = 0.4 (solid line) and ρ = 0.45 (dashed line) [3]. Similar characteristics can be obtained by using ρ = 0.5 as well, but introducing a lumped loss into the fiber loop, as will be clear in the next paragraph.

3.2.1.1.1 SPM-based NOLM with ultrashort pulses

So far we have considered CW input beams but our primary interest is to investigate the behaviour of such a device when using ultrashort pulses [4]. In this case the XPM-induced phase shift can be ignored for optical pulses short enough that they overlap for a relative short time compared with the round-trip time. This is due to the fact that a pulse train signal experiences SPM based on its instantaneous peak power and XPM based on the counterpropagating signal’s mean power; thus, for low duty-cycle peak power is much higher than mean power and XPM contribution can be ignored.

Fig.3.6 SPM-based NOLM scheme (left) and input signal (right).

Fig.3.6 shows a generic SPM-based NOLM structure involving Not-PM devices; γ [W-1_km-1_]

is the fiber nonlinear coefficient, L [km] its length and a is the loss used to unbalance the loop to make counterpropagating beams experience different phase shifts. To make the two fields interfere it is necessary they are polarized on the same axis, so a polarization controller is required to be inserted into the loop.

Let Ein[W

JG _1/2

] be the NOLM input field with a generic splitting ratio ρ at the coupler. For clockwise and counterclockwise paths, the counter-propagating signals and can be expressed as: 1 E JG 2 E JG 1 i E = ρ ⋅E JG JG n E2 = 1− ρ ⋅Ein JG JG

Due to the presence of the lumped loss and a generic splitting-ratio the field intensities are unbalanced and experience different phase shifts. The clockwise travelling signal, whose initial power is ρ·Pin, experiences SPM and the induced phase shift is given by:

c P Lin eff

φ = γρ

where Leff (the effective fiber length) is computed as:

L 1 e L_eff −α − = α

α being the fiber attenuation per unit of length. On the other side also the counterclockwise travelling signal, whose power after the lumped loss is a·(1- ρ)·Pin (a < 1 is the attenuation

introduced by the concentrated loss), experiences SPM and the induced phase shift is:

cc a(1 )P Lin eff

φ = γ − ρ

After having covered the entire loop (each one proceeding along its own way) the two signals travel again through the coupler. The output signal is then the result of two fields:

(1) (2) out out out

EJG =EJG +EJG

where E(1)out originates from the clockwise signal: JG

(1)

out in in

EG = a E exp[ jρG φ + γρ₀ j P L ]_eff

and E(2)out originates from the counterclockwise signal: JG

(2)

out in _in

E = a (1− ρ)E exp[ jφ + γ0 j a(1− ρ)P Leff + ]

JG JG

π

φ0 = βLeff is the linear phase shift( being β the propagation constant). The presence of π is due

to the two crossed-passes through the coupler, each one responsible of a phase shift of π/2. The output signal’s power is 2.

out out

If the coupler splitting-ratio is 0.5 then E1

JG

and E2

JG

experience the same attenuations but different phase shifts during the round-trip and Pout = 0 for a relative phase displacement of

2nπ.

Fig.3.7 Example of a 50/50 coupler based NOLM.

This configuration is widely used for pedestal suppression. In fact if a pulse enters the NOLM then only the central part, if its peak power is large enough, is transmitted out while low power components, like pedestal or noise on space bits, are reflected back. As a result the 50:50 NOLM can reduce the pulsewidth and clean noise on space bits.

Fig.3.8 Example of pedestal suppression.

It should be also clear that, if the peak power of the input signal is strong enough to reach the first maximum of the characteristic, this NOLM can also reduce noise on mark level.

3.2.1.2 Cross-Phase Modulation based NOLM

Fig.3.9 XPM-based NOLM scheme.

An important class of applications is based on XPM effects occurring when a control signal is injected into the loop such that it propagates in only one direction inducing a nonlinear phase shift only on the clockwise propagating wave. The control signal (or pump) is used to unbalance the interferometer; as a result, using a 50:50 coupler a low power signal is reflected in absence of the control but transmitted when the control pulse is applied. The signal and the control must be at different wavelengths so that the second one can be suppressed using a bandpass filter. Such a device acts as a wavelength converter as it creates a copy of the control signal at a different wavelength. Another possible application is channel demultiplexing in OTDM systems [3]:

Fig.3.10 Demultiplexing of an OTDM signal.

the control signal consists of a pulse train at the single-channel bit-rate. It is injected into the loop such that it propagates only in the clockwise direction. The OTDM signal enters from the input and is splitted into counterpropagating directions by a -3dB coupler. The clock signal is

timed such that it overlaps with pulses belonging to the specific OTDM channel we want to demultiplex. As a result, it introduces nonlinear XPM-induced phase shift only for those pulses. The power of the control an the loop length are made large enough to introduce a relative phase shift of π. Thus, pulses belonging to the desired channel are transmitted out of the loop while remaining channels are reflected back.

3.2.1.2.1 Analytic model

The most widely used configuration is composed by two 50:50 couplers. The fiber has a nonlinear coefficient γ and length L. In our approach we can neglect fiber losses, walk-off between signal and pump, and SPM (the signal input power is sufficiently low). Two polarization controllers are required for aligning the counterpropagating halves between themselves and the control pump with the clockwise propagating signal.

Let Ein be the signal input field and

JG

1

EJG and E2

JG

the two counterpropagating halves:

1 2

E =E = 0.5Ein

JG JG JG

while Ep

JG

is the instantaneous pump field and Pp its power. The XPM-based nonlinear phase

shift induced on the clockwise half is

c LPp

φ = γ

while for the counterclockwise half is

cc LPp

φ = γ

As a result, the output field associated to E1

JG and E2 JG are: (1) out in p (2) out in p 1 E E exp[ j j P L] 2 2 1 E E exp[ j j P L ] 2 2 0 0 = φ + γ = φ + γ + π G G G G

and the outgoing signal field can be written as:

(1) (2) out out out

EJG =EJG +EJG

If Pin = |Ein|2 is the instantaneous input power, then the output power is given by:

(

)

{

}

When the mean input power is low enough (for low values of duty-cycle) counterpropagating XPM can be neglected and

(

)

3.2.2 Terabit Optical Asymmetric Demultiplexer (TOAD)

Fig.3.11 Principle of TOAD.

The use of a semiconductor optical amplifier (SOA) as a nonlinear element in place of the fiber reduces the loop length to some centimetres. However the response of such a device is relatively slow because it is based on a nonlinear phenomenon that requires recombination of electron-hole pairs within the active region of the amplifier. The SOA is put in an asymmetric position into the loop and the nonlinear effect is a phase displacement induced by SOA saturation due to the presence of a control pulse. This effect depends only on the SOA asymmetric position and sizes. If the control pulse passes through the SOA between copropagating and counterpropagating pulses, only the last is affected by a phase shift due to the SOA saturation. So, if the phase difference between the pulses is π, then the two components interfere constructively at the coupler. The maximum system bit-rate is limited by the SOA recovery time [4].