Mode-locking (partial)
Mode-locking (partial) Mode-locking (partial)
9Mode Locking1. A. Siegman, Lasers [University Science Books, Sausalito, CA, 1986], Chapter272. A. Yariv, Quantum Electronic [ Wiley, New York, 1989 ], Chapter 203. O. Svelto, Principles of Lasers translated by D. C. Hanna [Plenum Press, NewYork, NY, 1998], Chapter 8.9.1 Mode Locked PulsesConsider a laser with a large number N 1 of oscillating axial modes having thesame polarization. If ∆ν osc is the oscillation bandwidth (frequency range over whichgain exceeds loss) and δν ax is the axial mode separation, the number of oscillatingmodes N is given byN = ∆ν osc(9.1)∆ν axFor simplicity asssume that all oscillating modes have nearly the same amplitudeand that N is an odd integer so that (N − 1)/2 is also integer. Recall that axialmode frequencies are separated by ∆ν ax = c/2L =1/τ R . If ν o = Mc/2L (M is anumber ∼ 10 7 ) is central mode frequency (average frequency), then the frequenciesand wave numbers for the modes can be written asν m = ν o + r∆ν ax⎫⎬−(N − 1)2≤ m ≤(N − 1)2(9.2)k m ≡ 2πν m= 2πν o+ 2πm∆ν ax ⎭c c cThe electric field of the laser can then be written as the superposition of standingwave fields of different modesE(z,t) =iE o(N−1)/2m=−(N−1)/2e −i2πνmt+iϕm sin k m zwhere ϕ m is the initial phase of the rth mode. In general ϕ m varies randomly frommode to mode. For phase locked modes 1 the relation ϕ m+1 − ϕ m = ϕ o holds, where1 This is only one possible way mode phases can be locked and is referred to as fundamental modelock equation. many other possibilities exist. For example if mode phases satisfy the relationϕ m+1 − ϕ m = ϕ m − ϕ m−1 + π is second harmonic locking etc.225
9<strong>Mode</strong> Locking1. A. Siegman, Lasers [University Science Books, Sausalito, CA, 1986], Chapter272. A. Yariv, Quantum Electronic [ Wiley, New York, 1989 ], Chapter 203. O. Svelto, Principles of Lasers translated by D. C. Hanna [Plenum Press, NewYork, NY, 1998], Chapter 8.9.1 <strong>Mode</strong> Locked PulsesConsider a laser with a large number N 1 of oscillating axial modes having thesame polarization. If ∆ν osc is the oscillation bandwidth (frequency range over whichgain exceeds loss) and δν ax is the axial mode separation, the number of oscillatingmodes N is given byN = ∆ν osc(9.1)∆ν axFor simplicity asssume that all oscillating modes have nearly the same amplitudeand that N is an odd integer so that (N − 1)/2 is also integer. Recall that axialmode frequencies are separated by ∆ν ax = c/2L =1/τ R . If ν o = Mc/2L (M is anumber ∼ 10 7 ) is central mode frequency (average frequency), then the frequenciesand wave numbers for the modes can be written asν m = ν o + r∆ν ax⎫⎬−(N − 1)2≤ m ≤(N − 1)2(9.2)k m ≡ 2πν m= 2πν o+ 2πm∆ν ax ⎭c c cThe electric field of the laser can then be written as the superposition of standingwave fields of different modesE(z,t) =iE o(N−1)/2m=−(N−1)/2e −i2πνmt+iϕm sin k m zwhere ϕ m is the initial phase of the rth mode. In general ϕ m varies randomly frommode to mode. For phase locked modes 1 the relation ϕ m+1 − ϕ m = ϕ o holds, where1 This is only one possible way mode phases can be locked and is referred to as fundamental modelock equation. many other possibilities exist. For example if mode phases satisfy the relationϕ m+1 − ϕ m = ϕ m − ϕ m−1 + π is second harmonic <strong>locking</strong> etc.225
228 Laser Physics0 t 0 !t 0 +" R " R t 0 +" R !t 0 +2" R 2" R t 0 +2" R ! t 0 +3" R 3" Rt0 z 0 L !z 0 +2L 2L zFIGURE 9.2Pulse sequence observed at a fixed point z o inside the cavity and pulse location ata fixed instant t o .M 1, R 1M 2, R 2z=0 z=L0 t 0t 0+! Rt 0+2! R timeFIGURE 9.3Output pulse sequence from a mode-locked laser.How many pulses are circulating inside the cavity at any time? To answer thisquestion let us look at pulse position a fixed time t o . Then the right and left goingpulse locations are given by (z o = ct o )z →= z o and z ←= −z o +2L. (9.10)It is easily checked that only one pulse (either right or left going) is present in thecavity (0
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