To be published in Journal of the Optical Society of America B: Theory of a Far-Off Resonance Mode-Locked Raman Laser in H2 with High Finesse Cavity Enhancement Authors: Yihan Xiong, Sytil Murphy, J. Carlsten, and Kevin Repasky Accepted: 18 April 2007 11 May 2007 Posted: Doc. ID: 79532 Title:

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OSA

Theory of a Far-Off Resonance Mode-Locked Raman Laser in H2 with High Finesse Cavity Enhancement Yihan Xiong, Sytil Murphy and J. L. Carlsten Physics Department, Montana State University, Bozeman, MT 59717

，USA

Published by， Kevin Repasky

ECE Department, Montana State University, Bozeman, MT 59717

USA

OSA Abstract: In this paper we present a theory for far-off resonance mode-locked Raman lasers in H2 with high finesse cavity enhancement. The theoretical derivation for the mode-locked

Raman laser is based on a time-dependent continuous wave (CW) Raman theory.

Numerically calculated results, including the Stokes threshold and intra-cavity fields’

amplitude and phase evolution are discussed in three different regimes depending on the relations between the coherence dephasing rate γ 31 and the repetition rate Ω of the mode-

locked pump laser. The threshold results from the mode-locked pump cases are compared with the CW, single-mode pump field case.

Copyright OCIS codes: 140.3550, 140.4050, 140.3480, 140.4780, 190.4380, 190.5650, 290.5860.

1. Introduction

1

In the last several years, continuous-wave (CW) Raman lasers in high finesse cavities (HFC) filled with H2 have been demonstrated.1-3 With the build-up of pump laser power provided by the HFCs, the lasing threshold can be lower than 1 mW.2,3 Because these lasers operate far off resonance with the “intermediate” states involved in the two-photon Raman process, their gain depends only weakly upon the pump wavelength. Thus using external cavity diode lasers that can be tuned over tens of nm, widely-tunable CW Raman lasers are achievable.3 With the variety of commercially available low-cost diode lasers and with common gases such as

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CH4 and H2 for the Raman medium, CW Raman lasers can now cover the spectrum from the visible to the near IR( ~4 μm ).

OSA Here we discuss the possibility of making a mode-locked Raman laser source, starting

with a mode-locked diode laser as the pump laser. This paper will investigate whether such a

device can operate with approximately the same average pumping power as a CW Raman laser

even with the pumping modes all individually below threshold. We were encouraged by the

earlier work of J. Rifkin, who showed that the correlated Raman gain driven by a multilongtudinal mode laser can be larger than the gain for a monochromatic laser4 because of collaboration between the Raman gain and other four-wave-mixing processes. Thus, we theoretically analyze far-off resonance mode-locked Raman lasers in HFC’s filled with H2.

2. Far-off resonance mode-locked Raman theory The output of a mode-locked laser consists of many in-phase CW fields. Our physical model of the general detuned (i.e. off resonance with the intermediate electronic state) case in a three-level system pumped with a mode-locked laser is shown in Fig. 1, where Ω is the repetition rate of mode-locked laser or spacing between adjacent fields (in our case Ω ∼ GHz ), n labels either the particular pump or Stokes mode, ω pn is the optical frequency of pump mode n 2

and ωsn is the optical frequency of Stokes mode n , ω pn = ω p1 + (n − 1)Ω and ωsn = ωs1 + (n − 1)Ω . Because this system is far off electronic resonance, each longitudinal laser mode from the modelocked laser only populates the upper virtual level which is determined by the individual mode frequency. Then all the population in the different virtual levels undergo a radiative transition to the lower lasing level 3, releasing Stokes photons at the various transition frequencies. For simplification and clarification, an example with two in-phase pump modes is used in the following mathematical derivation. This two-mode theory can then be extended to multi-

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mode mode-locked Raman theory. Since the system is far-off resonance, the rotating wave approximation is not valid here. The complex intra-cavity pump and Stokes field amplitudes can

OSA be written as:

E p = 12 [ E p1 (t )e

− iω p1t

+ E p 2 (t )e

− iω p 2t

] + c.c

(1)

Es = 12 [ Es1 (t )e − iωs1t + Es 2 (t )e− iωs 2t ] + c.c

(2)

where E pn and Esn are the amplitudes of the individual pump and Stokes fields.

The equations describing Raman laser action are a simple generalization of the semi-classical

equations of Sargent, Scully and Lamb to a three-level system.5, 6 The result for the general detuned case is the following set of coupled ordinary differential equations: 5-8 E p1 = − Lp1 E p1 + i

E p 2 = − Lp 2 E p 2 + i

ω p1 ε0

ωp2 ε0

μ12 ρ 21e

μ12 ρ 21e

3

iω p1t

+ K ( E pin (1) , t )

iω p1t iΩt

e

+ K ( E pin ( 2 ) , t )

(3)

(4)

Es1 = − Ls1 Es1 + i

Es 2 = − Ls 2 Es 2 + i

ωs 1 ε0

ωs 2 ε0

μ 23 ρ 23eiω

s 1t

μ23 ρ 23eiω t eiΩt s1

(5)

(6)

ρ 21 = −(iω21 + T 1 ) ρ 21 − i μ21 E p ( ρ 22 − ρ11 ) − i μ23 Es ρ31

(7)

* ρ 23 = −(iω23 + T 1 ) ρ 23 − i μ 23 Es ( ρ 22 − ρ33 ) − i μ21 E p ρ31

(8)

2 ( 21)

Published by 2( 23)

OSA * ρ31 = −(iω31 + T 1 ) ρ31 − i μ 23 Es* ρ 21 − i μ 21 E p ρ 23 2 (31)

(9)

* * D21 = − T1(121) ( D21 + 1) − 4 Im( μ21 E p ρ 21 ) − 2 Im( μ23 E s ρ 23 )

(10)

* * D23 = − T1(121) D23 − 2 Im( μ21 E p ρ 21 ) − 4 Im( μ23 E s ρ 23 )

(11)

where Lpn and Lsn are the cavity loss constants for pump and Stokes fields; μij is the dipole

moment matrix element between level i and level j; E pin ( n ) is the input pump field and k (t , E pin ( n ) ) accounts for the pump beam’s “leakage” into the cavity; the ρii are diagonal density-

matrix elements representing the population of level i ; ρij is the off-diagonal density-matrix element representing molecular coherences; ωij is the frequency difference between level i and j; the Dij refer to the population differences and

1 T2( ij )

is the dephasing rate between level i and level

j. We choose to define σ 21 = ρ 21 exp ( iω p1t ) and σ 23 = ρ 23 exp ( iωs1t ) . 4

There are several approximations we can use to simplify these differential equations. First, the virtual level is far below the first electronic level 2 as shown in Fig.1, so the Raman laser is far off the electronic resonance. As such, the coherences associated with level 2 will not be affected by the laser interaction, ρ 21 = ρ 23 = 0 . ρ 22 = 0 because the population of level 2 is not affected either. Another approximation we can make is that the ground-state is not depleted, as typically the case for CW Raman lasers in H2.8,9 Thus, D21 = −1 and D23 = 0 . We also choose to set ρ31 = σ 31e

− i (ω p1 −ωs 1 ) t

, where σ 31 varies much slower than the frequency difference ω p1 − ωs1 .

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With all these approximations, we can write ρ 21 and ρ 23 in terms of ρ31 by setting the left side of

OSA Eq. (7) and (8) to be zero and then plug them into Eq. (9) to get a equation for σ 31 :

σ 31 = [i (ω p1 − ωs1 − ω31 ) − T 1 ]σ 31 + 2 (31)

+(

iμ12 μ21 

ω23

2

Ep E − * p

i μ32 μ23 

ω21

2

i μ21μ23 

ω21

2

[( Es* E p e

i (ω p1 −ωs 1 ) t

)]

(12)

Es E )σ 31 * s

The third term on the right hand side of Eq. (12) represents the AC stark shift due to the pump

and Stokes field and will be neglected because this term will be much smaller than all the frequency width for the Raman laser systems we are studying.9 The general solution to Eq. (12)

shows that the significant terms are as those with variations slower than or comparable to 1 T2( 31)

(Appendix A). Therefore, in using Eq. (1) and (2), we will safely drop all the terms from

Es* E p e

i (ω p1 −ωs 1 ) t

that oscillate at ω pn , ωsn or ω pn − ωsn because even at high pressures,

1 T2( 31)

will

never be comparable to an optical frequency. We can then simplify Eq. (12) to:

σ 31 = (iΔ − γ 31 )σ 31 + ig31 ( Es*1 E p1 +Es*2 E p 2 +Es*1 E p 2 e− iΩt +Es*2 E p1eiΩt )

5

(13)

where Δ = ω31 − (ω pn − ωsn ) , γ 31 =

1 and g31 = T31

μ21μ 23 4 2ω21

. For simplicity, in the remaining discussion

and simulations we set Δ ≈ 0 or ω31 ≈ (ω pn − ωsn ) , assuming that the two-photon Raman detuning is zero. At long times ( t

1

γ 31

), the pump and Stokes fields will approach steady-state, while the

coherence σ 31 continues to oscillate. The solution for the coherence equation at t

1

γ 31

(or

e −γ 31t → 0 ) is simple to obtain from the general solution to a first linear differential equation:

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σ 31 = ig31[( Es*1 E p1 + Es*2 E p 2 )(

1

) + ( Es*1 E p 2 )(

e − iΩt eiΩt ) + ( Es*2 E p1 )( )] γ 31 − iωRF γ 31 + iω RF

OSA γ 31

(14)

Recall that σ 31 is the slowly varying portion of the total coherence which oscillate near ω31 :

ρ31 = σ 31e

− i (ω p1 −ω s 1 ) t

= σ 31e −iω31t

ρ31 = ig31[( E E p1 + E E p 2 )( * s1

* s2

e− iω31t

γ 31

e −i (ω31 +Ω ) t e −i (ω31 −Ω ) t * ) + ( E E p 2 )( ) + ( Es 2 E p1 )( )] γ 31 − iωRF γ 31 + iωRF * s1

(15)

The first term on the right-hand side of Eq. (15) oscillates at ω31 and is completely resonant with the two photon transition. The second and third terms on the right-hand side oscillate at

ω31 ± Ω and are off-resonance with the two photon transition by an amount equal to ±Ω . These terms are less strong than the on-resonance term because they fall in the wings of the Lorentzian profile of the Raman gain

γ 31 2 γ 31 +Ω2

where γ 31 is the half width half maximum (HWHM) of the

Raman gain, and also have a dispersion term proportional to associated with them.

6

± iΩ 2 γ 31 +Ω2

that affects the phase

As we mentioned earlier, due to far-off resonance and ground-state non-depletion,

ρ 21 = ρ 23 = 0 , ρ 22 = 0 , D21 = −1 and D23 = 0 . With these conditions, set the left side of Eq. (7) and (8) to be zero and ρ 21 and ρ 23 can be expressed in terms of ρ31 , then plug these expressions into Eq. (3)-(6). Basically using the same type of approximations and simplifications used for

σ 31 , Eqs. (3)-(6) can be simplified to: E p1 = − Lp1 E p1 − ig p1 ρ31eiω31t ( E p1 + E p1e− iΩt ) + k (t , E pin (1) )

(16)

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E p 2 = − Lp 2 E p 2 − ig p 2 ρ31eiω31t ( E p1eiΩt + E p 2 ) + k (t , E pin (2) )

(17)

OSA (

Es1 = − Ls1 Es1 − ig s1 ρ31eiω31t

) (E *

(

Es 2 = − Ls 2 Es 2 − ig s 2 ρ31eiω31t

p1

) (E *

+E p 2e −iΩt )

(18)

eiΩt +E p 2 )

(19)

p1

where all the fast optical frequency oscillation terms are dropped in fields equations because optical frequency is much larger than Lpn ( sn ) , which is about 105 with both reflectivities of two mirrors about 0.99988 ( Lpn ( sn ) = −(c / 2l ) ln( R pn ( sn ) ) 1). And g pn =

ω pn μ21μ23 2 ε 0ω21

and g sn =

ω sn μ21μ 23 2 ε 0ω23

are

constants related to the Raman gain. Plugging the solution of ρ31 Eq. (15) into Eqs. (16)-(19), gives after dropping all the fast oscillation terms compared with Lpn ( sn ) including e ± iΩt and e ± i 2 Ωt :

E p1 = − L p1 E p1 − g p1 g31[

Es 1 ( E p1Es*1 + E p 2 Es*2 )

γ 31

7

+

Es 2 ( E p1Es*2 ) γ 31 + iΩ

] + K ( E pin1 , t )

(20)

E p 2 = − L p 2 E p 2 − g p 2 g31[

Es 2 ( E p1Es*1 + E p 2 Es*2 )

γ 31

Es1 = − Ls1 Es1 + g s1 g 31[

+

Es 1 ( E p 2 Es*1 ) γ 31 −iΩ

E p1 ( E *p1 Es 1 + E *p 2 Es 2 )

Es 2 = −Ls 2 Es 2 + gs 2 g31[

γ 31

+

E p 2 ( E*p1Es 1 + E*p 2 Es 2 )

γ 31

] + K ( E pin 2 , t )

E p 2 ( E *p 2 Es 1 ) γ 31 + iΩ

+

]

E p1 ( E*p1Es 2 )

γ 31 −iΩ

]

(21)

(22)

(23)

From these equations, it can be seen there are two ways to generate the desired Stokes mode. For

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generating Stokes mode 1, the first term in the square brackets on the right-hand side of Eq. (22)

OSA represents the interaction between pump mode 1 and the on-resonance piece of the coherence, in

which E p1 ( E *p1 Es1 ) is the regular Raman process and E p1 ( E *p 2 Es 2 ) is a four-wave-mixing process (phase mismatch can be ignored [Appendix B]). The second term in the square brackets results from the interaction between pump mode 2 and the off-resonance piece of the coherence. Now we can easily extend the two-mode case to the general mode-locked case, ei βΩt ∑ Es*α E p (α − β ) M

σ 31 = ig31

M −1

∑

α =1

γ 31 + i βΩ

β =− M +1

∑

(24)

M

M Es β

E pn = − Lpn E pn − g pn g31 ∑

α =1

Es*α E p [α −( β −n )]

γ 31 + i ( β − n ) Ω

β =1

∑

+ K ( E pin ( n ) , t )

(25)

M

M E pβ

Esn = − Lsn Esn + g sn g31 ∑ β =1

8

α =1

( Es*[ α −( β −n )] E pα )*

γ 31 + i ( β − n ) Ω

(26)

where M is the number of pump modes and n labels the particular mode concerned with. For this case, the coherence is composed of an on-resonance piece and many off-resonance pieces separated the resonant piece by ±Ω , ±2Ω , ±3Ω

. As we mentioned before, both on-resonance

and off-resonance coherences can interact with different pump modes to generate desired Stokes mode.

3. Numerical analysis with different conditions

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In this section, we present calculations relevant to our current experimental work. In our experimental set-up, the pump laser wavelength is approximately 800 nm and the Stokes light is

OSA at about 1200 nm. The HFC mirrors have a reflectivity (transmissivity) of R pn ( sn ) =0.99988 ppm

( Tpn ( sn ) =40 ppm) at both the pump and Stokes wavelengths, giving a cavity finesse of about

26,000 for both wavelengths. The cavity length is l=17.78 cm, yielding a free spectral range

(FSR) of 844 MHz. In order to simultaneously couple all the pump laser’s oscillating modes into the HFC, these modes must also be spaced by 844 MHz (or some integral multiple thereof). Therefore the repetition rate of the mode-locked laser is also Ω =844 MHz.

Due to the difference in wavelength of the various modes, dispersion in the H2 also needs

to be considered. With 100 longitudinal modes spanning 84 GHz (or 0.18 nm at the ~800 nm pump wavelength in vacuum), the additional refractive shift in the cavity resonant frequencies between mode #1 and #100 is only about 0.6 kHz at 10 atm.10-12 For comparison, the linewidth of the HFC is about 28 kHz. Thus the dispersion due to H2 can be neglected in this case. The cavity loss coefficient is defined as Lpn ( sn ) = −(c / 2l ) ln( R pn ( sn ) ) .1 Over the ~90 GHz range of optical frequencies, Lpn and Lsn (~105 Hz) can be considered the same for all the

longitudinal modes. The optical-pumping constant K (t , E pin ( n ) ) = ( c / l ) Tpn E pin ( n ) and the

9

Raman gain coefficient G = (1/ 8 ) γ 31α c ( ε 0 / μ0 )

1/ 2

(

λp λ p + λs

),

1,9,14

where α =1.5×10-11 m/W at 800

nm is the plane-wave gain coefficient,15 E pin ( n ) is the input pump field amplitude and c is the speed light in vacuum. Also we have g pn g31 =

ω pn ωs 1

ω

G , g sn g31 = ωsns1 G .

In this section, using all the numbers and coefficients mentioned earlier along with τ , the mode-locked pulse duration (

1

τ

≈ M Ω ), we compare Ω with the dephasing rate γ 31 in three

different regimes: low pressure ( γ 31

Ω ), high pressure ( γ 31

M Ω ) and medium pressure

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( γ 31 does not satisfy either γ 31

Ω or γ 31

M Ω ) to determine the Stokes threshold.

OSA Dephasing γ 31 is also the half width half maximum (HWHM) of the Lorentzian distribution

describing Raman gain.15 The different possible relationships between γ 31 and Ω are illustrated

in Fig.2.

1. Low pressure ( γ 31

Ω)

Mathematically when γ 31

Ω < MΩ ≈

1

τ

, only the term with β = n needs to be kept in

Eqs.(24)-(26). All the other terms can be considered to be zero due to the comparatively large Ω

in the denominator. So the coherence and field equations can be simplified to:

σ 31 = ig31 ∑ M

Es*α E pα

M

E pn = − Lpn E pn − g pn g31 Esn ∑ α =1

10

(27)

γ 31

α =1

Es*α E pα

γ 31

+ K ( E pin ( n ) , t )

(28)

M

Esn = − Lsn Esn + g sn g31 E pn ∑ α =1

Physically when γ 31

( Es*α E pα )*

(29)

γ 31

Ω , all the off-resonance pieces of the coherence are small

compared with the on-resonance piece, as shown in Fig. 2(a). In other words, the mode-locked temporal pulse width is much shorter than the coherence dephasing time, τ

1

γ 31

, and the

coherence growth is not fast enough to follow the short pulses. Thus only the average power

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determines the growth of the coherence and Stokes modes, and the mode-locked case will have the same gain and same threshold as the CW case 1-3. Figure 3 shows a plot of the average intra-

OSA cavity Stokes intensity versus the average input pump intensity for one, two and three in-phase pump modes. For this plot, the following definition has been used for the average intensity M

I = E = ∑ E( n ) , where I is either the input pump or intra-cavity Stokes intensity ( I pin or I s ) 2

2

n =1

respectively, E is either the input pump or intra-cavity Stokes field ( E pin or Es ) respectively, and

the index n represents the nth mode of either field. In this regime, the three curves completely overlap, indicating that the threshold is independent of the number of modes and that the average output is proportional to the square root of the average input pump power.1

We also give plots of the intra-cavity fields’ amplitude and phase evolution with equal and unequal amplitudes of the three pump modes in Fig. 4 and 5. Once the system reaches steady-state, all the Stokes fields’ phases have evolved to be in-phase, so that mode-locked 2

2

2

Stokes is generated. With equal amplitude pump modes ( E pin1 = E pin 2 = E pin 3 =

2 1 E pin ), 3

final Stokes phase is the vectorial summation of all the Stokes fields’ initial phases as shown in Fig. 4. If the initial pump modes are different in amplitude the Stokes field’s phase evolution will

11

not be a simple vecotrial summation of all the Stokes field’s initial phase, but the steady-state average Stokes intensity will be the same as an equal amplitude case, as shown in Fig. 5.

2. High pressure ( γ 31

MΩ)

Mathematically when γ 31

M Ω , we dropped the i β Ω term in the denominator. Thus,

the coherence and field equations are simplified to: ei βΩt ∑ Es*α E p (α − β ) M

Published by σ 31 = ig31

M −1

∑

α =1

β =− M +1

γ 31

(30)

OSA ∑ M

M Es β

E pn = − Lpn E pn + K ( E pin ( n ) , t ) − g pn g31 ∑ β =1

∑

α =1

Es*α E p [ α −( β −n )]

γ 31

(31)

M

M E pβ

Esn = − Lsn Esn + g sn g31 ∑ β =1

Physically when γ 31

α =1

( Es*[ α −( β −n )] E pα )*

γ 31

(32)

M Ω , from the Lorenztian distribution, all the off-resonance pieces

of the coherence will contribute about the same as the on-resonance piece as shown in Fig. 2.(b). It also means that the mode-locked pulses are like CW light when compared with the even

shorter coherence dephasing time. In other words the coherence growth is fast enough to follow the short pulses. Thus, the high peak power from the mode-locked laser will contribute to the Raman gain. Thus, in the regime of γ 31

M Ω , the mode-locked case will have much smaller

average threshold power than the CW case as shown in Fig.6. While at high input pump power, the CW case has more average Stokes output than mode-locked case. The reason is as following. Consider two possible pumping scenarios that have the same average power. The first is a 12

constant input pump with intensity of 1, which produces constant Stokes of intensity of 1. The second is an input pump intensity that is four times bigger but is only on for forth of the timing, giving an average input power of 1 as well. However, because for the time it is on it is four times bigger and Stokes average output intensity is proportional to the square root of average input pump intensity, it produces only twice the Stokes, which when averaged over the time span is 2/4=0.5, less than CW case. With two equal intensity pump modes, the pulse peak intensity is twice bigger than the

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average pump intensity for CW case, which should lead to twice smaller in threshold. However the ratio between two thresholds shown in Fig. 6 is about 1.33, which is smaller than 2. That is

OSA because the mode-locked pulse temporal width is much smaller than the cavity build up time (tens of μ s ). Therefore the pulses can not build up as high an intra-cavity power as CW case. In

other words, taking into the consideration of cavity build up time, we expect the threshold for two equal intensity mode case is not quite twice smaller than CW case. If these two modes have

unequal intensity, for example, two modes intensity differs by a lot, then the threshold for this two mode case will almost have the same threshold plot as CW case. So in this region ( γ 31

M Ω ), only when the modes have same intensity, the gain enhancement gets maximized

or threshold gets minimized.

In Fig. 7, the time evolution of the pump and Stokes field amplitude and phases can be seen. The Stokes phases all evolve to the same steady-state value, indicative of mode-locked Stokes being generated.

3. Medium pressure (from γ 31 ∼ Ω to γ 31 ∼ M Ω )

13

If γ 31 does not satisfy either γ 31

Ω or γ 31

 M Ω , then this is a intermediate pressure

case. From Fig. 2.(c.1) and 2.(c.2), some of the off-resonance pieces of coherence will contribute while some of them are too small to contribute to the build-up of the coherence needed in generating Stokes. Thus, the gain is bigger than the low pressure case but smaller than the high pressure case, giving an average threshold that is smaller than the low pressure case but bigger than the high pressure case as shown in Fig. 8. In this regime, the Stokes phases do not evolve to the same steady-state values and the pump phases evolve slightly away from their initial values,

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as shown in Fig. 9. This is due to the extra dispersion phase term

± i βΩ 2 γ 31 + ( β Ω )2

introduced from the

OSA off-resonance piece of the coherence, which degrades the pulse shape, peak power and Raman gain.

4. Conclusion

From these numerical calculations, we can tell the Raman gain or threshold for the far-off

resonance mode-locked Raman laser in H2 with HFC enhancement is dependent on the relations between γ 31 and Ω . In the low-pressure regime ( γ 31

 Ω ), only the on-resonance pieces of

coherence contribute to coherence build-up and mode-locked Stokes generation, and the modelocked pumping will have the same threshold value as CW pumping. When pumping with the

average power above threshold, even if the individual modes from the mode-locked pump source are below threshold; the corresponding Stokes modes can still emit Stokes due to the four-wavemixing-process. In the high-pressure regime ( γ 31

M Ω ), all the on-resonance and off-resonance pieces

of coherence contribute about the same to Raman process, providing large gain enhancement to

14

generate mode-locked Stokes and thus significantly lowering the threshold compared with CW pumping. In the medium-pressure regime, where γ 31 does not satisfy either γ 31 (i.e. γ 31 can vary from

 Ω or γ

31

MΩ

∼ Ω to ∼ M Ω ), besides the on-resonance pieces of the coherence, part

of the off-resonance pieces also help to build up the coherence and generate Stokes light, which leads to enhanced gain compared with the low-pressure mode-locked case or CW pumping. Further, for γ 31 varying from

∼ Ω to ∼ M Ω ,

some extra dispersion terms are introduced from

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these off-resonance pieces of the coherence, degrading the pulse shape, peak power and Raman

OSA gain. Stokes output in this case is “near mode-locked”.

15

Reference: 1. J. K. Brasseur, P.A. Roos, K.S. Repasky and J. L. Carlsten, “Characterization of a continuous-wave Raman laser in H2,” J. Opt. Soc. Am. B Vol. 16, No. 8, 1305-1312 (1999). 2. P. A. Roos, J. K. Brasseur, and J. L. Carlsten, “Diode-pumped, non-resonant, cw Raman laser in H2 using resonant optical feedback stabilization,” Opt. Lett. 24, 1130 (1999). 3. L. S. Meng, K. S. Repasky, P. A. Roos, and J. L. Carlsten, “Widely tunable continuous wave Raman laser in diatomic hydrogen pumped by an external cavity diode laser,” Opt. Lett. 25, 472 (2000).

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4. J. Rifkin, M. L. Bernt, D. Macpherson and J. L. Carlsten, “Gain enhancement in z XeCl-

OSA pumped Raman amplifier,” J. Opt. Soc. Am. B Vol. 5, No. 8, 1607-1612 (1988).

5. M. Sargent, M. O. Scully and W. E. Lamb, “Semiclasscial Laser Theory,” in Laser Physics, (Addison-Wesley 1974), pp.96-114.

6. R. W. Boyd, “Nonlinear Optics in the Two-Level Approximation,” in Nonlinear Optics, second edition (Academic 2003), pp.261-307.

7. J. V. Moloney, J.S. Uppal and R.G. Harrison, “Origin of chaotic relaxation oscillations in an optically pumped molecular laser,” Phys. Rev. Lett. Vol. 59, 2868-2871 (1997).

8. J. K. Brasseur, Ph.D. thesis, “Construction and noise studies of a continuous wave Raman laser,” Physics Department, Montana State University, pp. 19-31. November (1998).

9. L. Meng, Ph.D. thesis, “Continuous-wave Raman Laser in H2: Semiclassical theory and diode-pumping Experiments,” Physics Department, Montana State University, pp. 17-22. August (2002). 10. Washburn, E. W.,“International critical tables of numerical data, physics, chemistry and technology,” (Edward Wight), 1881-1934. Vol. 7 pp.11.

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11. “American Institute of Physics Handbook,” second edition, (McGraw-Hill), pp.6-95. 12. At 10 atm, λ p (1) =800 nm, n1 =1.00127005738702; 100 modes away λ p (100) =800.18 nm, n100 = 1.00127005036724. 100 × ⎡⎣c / ( 2n1l ) − c / ( 2n100l ) ⎤⎦ =0.59 kHz. 13. K.S. Repasky, J. K. Brasseur, L. Meng and J. L. Carlsten, “Performance and design of an offresonant continuous-wave Raman laser,” J. Opt. Soc. Am. B Vol. 15, No. 6, 1667-1673, (1998). 14. W. K. Bischel and M. J. Dyer, “Wavelength dependence of the absolute Raman gain

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coefficient for the Q (1) transition in H2,” J. Opt. Soc. Am. B, vol. 3, No. 5, May (1986).

OSA 15. W. K. Bischel and M. J. Dyer, “Temperature dependence of the Raman lindewidth and line

shift for the Q (1) and Q (0) transitions in normal and para-H2,” Phys. Rev. A, Vol. 33, No. 5, May (1986).

17

Appendix A: For the linear first order differential equation y ′ + f (t ) y = g (t ) , the general solution t

y (t ) = e

is

− ∫ f ( x ) dx

t

[∫ e∫

z

f ( x ) dx

g ( z )dz + C ]

.

Rewrite

the

Eq.

(13)

as

σ 31 = (−γ 0 )σ 31 + i( A + B1e− iω t +B2 eiω t +C1e −iω t +C2 eiω t ) , where A, B1, B2, C1 and C2 are constants 1

and

1

A ≈ B1 ≈ B2 ≈ C1 ≈ C2 , t

σ 31 = e

− ∫ γ 0 dx

z

t

[∫ e∫

γ 0 dx

2

2

γ 0 ∼ ω1  ω2 .

The

solution

for

this

equation

is

i ( A + B1e − iω1 z +B2 eiω1z +C1e −iω2 z +C 2 eiω2 z )dz + C ]

t

= ie −γ 0t [ ∫ e −γ 0 z ( A + B1e− iω1 z +B2 eiω1z +C1e −iω2 z +C2 eiω2 z )dz + C ] = −ie−γ 0t [

Ae −γ 0t

γ0

+

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B1e − (γ 0 +ω1 ) t B2 e − (γ 0 −ω1 ) t C1e − (γ 0 +ω2 ) t C2 e − (γ 0 −ω2 ) t + ] + + (γ 0 − ω2 ) (γ 0 + ω1 ) (γ 0 − ω1 ) (γ 0 + ω2 )

OSA If

A ≈ B1 ≈ B2 ≈ C1 ≈ C2

C2 e − (γ 0 −ω2 ) t (γ 0 − ω2 )

and

γ 0 ∼ ω1  ω2

,

then

C1e −( γ 0 +ω2 ) t (γ 0 + ω2 )

B1e − (γ 0 +ω1 ) t (γ 0 + ω1 )

and

−γ 0 t B1e − (γ 0 −ω1 ) t B1e − (γ 0 +ω1 ) t B2 e − (γ 0 −ω1 ) t − γ 0t Ae , so σ 31 = −ie [ + + ] . This is the same γ0 (γ 0 − ω1 ) (γ 0 + ω1 ) (γ 0 − ω1 )

solution of equation σ 31 = (−γ 0 )σ 31 + i ( A + B1e −iω1t +B2 eiω1t ) . This shows how we can drop all the fast oscillation terms.

Appendix B: To produce Es1 , the four-wave-mixing term E p1 ( E *p 2 Es 2 ) should have a phase L

mismatch term as

1 L

∫e

− i [( k p 1 − k s 1 ) − ( k p 2 − k s 2 )] z

dz . While [(k p1 − k s1 ) − (k p 2 − k s 2 )] is on the order of 10−4

0

and L is about 0.18 m, the whole integral goes to 1. So no phase mismatch terms need to be considered.

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Figure Caption 1. Mode-locked laser interaction with hydrogen. 2. Three different regimes in the relation between γ 31 and Ω . (a)Low pressure ( γ 31 (b) High pressure ( γ 31

Ω ),

M Ω ) and (c) Medium pressure, two cases: γ 31 ∼ Ω

and γ 31 ∼ M Ω . 3. Intra-cavity Stokes Intensity versus outside input pump Intensity. In this simulation

γ 31 =10 MHz and Ω =844 MHz. The three curves are totally overlapped, which means in

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the region where γ 31 << Ω , there is no gain enhancement for the mode-locked case.

OSA 4. Intra-cavity fields’ magnitude and phase evolution with three equal in-phase pump mode.

Since the pump modes are in-phase, we set (θ p1 )initial = (θ p 2 )initial = (θ p 3 )initial = 0 and they

keep the same all the time as seen in the figure. The initial Stokes phase are random, we

set (θ s1 )initial = 0.3757π , (θ s 2 )initial = 0.9813π and (θ s 3 )initial = 0.8186π . Once the system

reaches steady-state, the final phase of Stokes 0.746π is the vectorial summation of all

the Stokes initial phases, mode-locked Stokes formed. At steady-state, with E pin = 3000 , the total average Stokes intensity is I s = Es1 + Es 2 + Es 3 = 5.78 × 109 . 2

2

2

5. Intra-cavity fields’ amplitude and phase evolution with three unequal in-phase pump modes. Since the three pump mode are not equal in magnitude, the final Stokes phase are not the vectorial summation of all the Stokes initial phases, but still reach the same phase. At

steady-state,

with

E pin = 3000

,

the

total

average

( I s = Es1 + Es 2 + Es 3 = 5.78 × 109 ) is same as previous case. 2

2

2

19

Stokes

intensity

6. Intra-cavity Stokes field magnitudes versus outside input pump field. In this simulation

γ 31 =100 GHz and Ω =844 MHz. It is obvious that the more modes, the less threshold. 7. Intra-cavity fields’ amplitude and phase evolution with three in-phase pump modes. Since the pump modes are in-phase, we set (θ p1 )initial = (θ p 2 )initial = (θ p 3 )initial = 0 . The pump phases do not evolve from their initial value. In this example we set the initial random Stokes phases to (θ s1 )initial = 0.8894π , (θ s 2 )initial = −0.7691π and (θ s 3 )initial = −0.4161π . The Stokes’ phases evolve from their initial value to a final value that is the vectorial

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summation of the initial Stokes phases because of the same amplitude in input pump

OSA fields. At steady-state, with E pin = 3000 , the total average Stokes intensity is

( I s = Es1 + Es 2 + Es 3 = 1.561×1010 ). 2

2

2

8. Intra-cavity Stokes field versus outside input pump field. In this simulation γ 31 =1 GHz and Ω =844 MHz. The gain enhancement in this plot is not as big as shown in Fig. 6.

9. Intra-cavity fields’ magnitude and phase evolution with three in-phase pump modes.

Since pump modes are in-phase, we set (θ p1 )initial = (θ p 2 )initial = (θ p 3 )initial = 0 and they

evolve slightly away from their initial values. Here we set the initial random Stokes

phases (θ s1 )initial = 0.03π , (θ s 2 )initial = −0.506π and (θ s 3 )initial = 0.8902π . When the system reaches steady-state, the Stokes fields are not quite in-phase due to the extra phase terms

introduced from off-resonance pieces of coherence, so the peak power and Raman gain are somehow degraded.

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