# Tunable and Switchable Erbium-Doped Fiber Laser Using a Multimode-Fiber Based Filter
## 2 Experimental Setup and Principle  
Figure 1 shows the schematic diagram of our tunable and switchable fiber laser. A 980-nm laser diode is used as a pump source to pump a 12-m single-mode erbium-doped fiber (EDF) via a 980/1550 nm wavelength-division multiplexing (WDM). An optical isolator is included to ensure unidirectional operation in the ring cavity. A coupler with a 30% coupling ratio is used for the laser output, where 5% of the output power is directed to an optical spectrum analyzer (OSA) for monitoring the laser spectrum and 95% of the power is connected to a power meter. A piece of MMF (between two SMFs) is coiled into a three-paddle PC. The MMF is a graded-index (GRIN) fiber with a numerical aperture of 0.29. The core diameter and cladding diameter of the MMF are 100 \(\upmu\)m and 140 \(\upmu\)m, respectively. Such a simple SMF/MMF/SMF structure can actually serve as a functional optical bandpass filter, based on optical interference between different modes. The fundamental mode  
propagating in the first section of the SMF will couple into the MMF, where many modes with different effective refractive indices could be excited. The different modes interfere with each other when they recouple back to the second section of the SMF. The measured intensity of the SMF/MMF/SMF structure based filter can be given by [15]. \[I=\sum_{i=1}^{n}I_{i}+\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}2\sqrt{I_{i}I_{j}}\cos \left(\frac{2\pi(n_{i}-n_{j})L}{\lambda}\right), \tag{1}\]  
where \(I_{i}\) and \(I_{j}\) are the power distributed in the \(i\)th-order and \(j\)th-order modes, respectively, \(n_{i}\) and \(n_{j}\) are the refractive indices of the two modes, \(L\) is the length of the MMF, and \(\lambda\) is the wavelength of the light in vacuum. The resonance condition for a constructive interference between any two modes can be written as:  
\[(n_{i}-n_{j})L=m\lambda,\qquad(m=1,2,3\ldots). \tag{2}\]  
The resultant interference pattern of the all-fiber filter is the superposition of all the interference terms from different modes. When only considering two excited dominant modes, Equation (1) becomes:  
\[I=I_{1}+I_{2}+2\sqrt{I_{1}I_{2}}\cos(2\pi(n_{1}-n_{2})L/\lambda). \tag{3}\]  
Equation (3) is the typical sinusoidal expression of two-beam interference where the two beams have the same physical length, but with a different modal index. However, when the MMF is coiled into a three-paddle PC, the two dominant modes experience different stress-induced birefringence intensity while propagating along the MMF, which results in an additional phase shift. The transmission of the MMF in a PC based filter could be expressed as [22]:  
\[T=1+2\sqrt{\kappa(1-\kappa)}\cos(2\pi(n_{1}-n_{2})L/\lambda+\varphi), \tag{4}\]  
with:  
\[\varphi=\arctan(\tan(\delta_{1}-\delta_{2})\cos 2\Omega), \tag{5}\]  
where \(\kappa\) is the coupling coefficient of the fundamental mode, \(\varphi\) stands for the additional phase shift due to the different stress-induced birefringence intensity of the two modes, \(\delta_{1}\) and \(\delta_{2}\) are the stress-induced birefringence intensity of the two modes, and \(\Omega\) stands for the angle of the fast axis orientation of the PC with respect to the x polarization component assuming the same orientation angle for the two modes. Basically, Equation (4) is equivalent to Equation (3), except the additional phase shift term, \(\varphi\). From Equation (5), one can clearly see the phase angle, \(\varphi\), could be changed by \(\Omega\) i.e., adjusting the  
Figure 1: Experimental setup of the tunable and switchable erbium-doped fiber laser (EDFL). EDF: Erbium-doped fiber; WDM: Wavelength division multiplexer; ISO: Isolator; PC: Polarization controller; SMF: Single mode fiber; MMF: Multimode fiber. paddles of the PC. This unique feature could be taken advantage of to turn a conventional MMF mode beating filter into a tunable bandpass filter.