Lecture 12- ECE 240a Threshold Mirror Loss Ver Chap. 8-9 - - PowerPoint PPT Presentation

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Lecture 12- ECE 240a

Ver Chap. 8-9

ECE 240a Lasers - Fall 2019 Lecture 12 1

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Review Rate Equations

Rate equations dN1 dt = R1(t) + N2 τ21 + σ(ν)Iν hν (N2 − N1) − N1 τ1 dN2 dt = R2(t) − N2 τ21 − σ(ν)Iν hν (N2 − N1) (Note these assume g1 = g2) These are equations (8.3.2a) and (8.3.2b) in Verdeyen (3 rd edition)

ECE 240a Lasers - Fall 2019 Lecture 12 2

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Review Rate Equations

Rate equations dN1 dt = R1(t) + N2 τ21 + σ(ν)Iν hν (N2 − N1) − N1 τ1 dN2 dt = R2(t) − N2 τ21 − σ(ν)Iν hν (N2 − N1) (Note these assume g1 = g2) These are equations (8.3.2a) and (8.3.2b) in Verdeyen (3 rd edition)

ECE 240a Lasers - Fall 2019 Lecture 12 2

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Small Signal Gain

Define intensity gain (or loss) as γ(ν) = σ(ν)(N2 − N1) = A21 λ2 8πn2

• cross-section

g(ν)(N2 − N1) Differential equation for the intensity in an optical amplifier is then dIv dz = γ0(ν)Iν with a solution Iν(z) = Iν(0)eγ0(ν)z Define small signal gain at a distance L as G0 = eγ0(ν)L

ECE 240a Lasers - Fall 2019 Lecture 12 3

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Small Signal Gain

Define intensity gain (or loss) as γ(ν) = σ(ν)(N2 − N1) = A21 λ2 8πn2

• cross-section

g(ν)(N2 − N1) Differential equation for the intensity in an optical amplifier is then dIv dz = γ0(ν)Iν with a solution Iν(z) = Iν(0)eγ0(ν)z Define small signal gain at a distance L as G0 = eγ0(ν)L

ECE 240a Lasers - Fall 2019 Lecture 12 3

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Small Signal Gain

Define intensity gain (or loss) as γ(ν) = σ(ν)(N2 − N1) = A21 λ2 8πn2

• cross-section

g(ν)(N2 − N1) Differential equation for the intensity in an optical amplifier is then dIv dz = γ0(ν)Iν with a solution Iν(z) = Iν(0)eγ0(ν)z Define small signal gain at a distance L as G0 = eγ0(ν)L

ECE 240a Lasers - Fall 2019 Lecture 12 3

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Resonant Frequencies of General Resonator

If one of the mirror is not flat then νm,p,q = c 2nd

• q + (1 + m + p)

π cos−1 (g1g2)1/2

• where

g1,2 = 1 − d R1,2 Example: d/R2 = 1/2, then νm,p,q = c 2nd

• q + (1 + m + p)

4

• ECE 240a Lasers - Fall 2019 Lecture 12

4

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Resonant Frequencies of General Resonator

If one of the mirror is not flat then νm,p,q = c 2nd

• q + (1 + m + p)

π cos−1 (g1g2)1/2

• where

g1,2 = 1 − d R1,2 Example: d/R2 = 1/2, then νm,p,q = c 2nd

• q + (1 + m + p)

4

• ECE 240a Lasers - Fall 2019 Lecture 12

4

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Resonant Frequencies of General Resonator

If one of the mirror is not flat then νm,p,q = c 2nd

• q + (1 + m + p)

π cos−1 (g1g2)1/2

• where

g1,2 = 1 − d R1,2 Example: d/R2 = 1/2, then νm,p,q = c 2nd

• q + (1 + m + p)

4

• ECE 240a Lasers - Fall 2019 Lecture 12

4

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Threshold Condition

Unsaturated small signal gain γ0(ν) per unit length must be larger than loss per unit length Loss has two terms:

Mirror loss (lumped - not distributed) Other scattering losses αs(absorption loss included in γ0(ν))

Total loss A is then given by A = R1R2e−αsℓ where ℓ is the length of the resonator If resonator is a ring, then ℓ is circumference of ring. If resonator is standing wave (He-Ne) then round-trip length or ℓ = 2d where d is distance between mirrors.

ECE 240a Lasers - Fall 2019 Lecture 12 5

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Threshold Condition

Unsaturated small signal gain γ0(ν) per unit length must be larger than loss per unit length Loss has two terms:

Mirror loss (lumped - not distributed) Other scattering losses αs(absorption loss included in γ0(ν))

Total loss A is then given by A = R1R2e−αsℓ where ℓ is the length of the resonator If resonator is a ring, then ℓ is circumference of ring. If resonator is standing wave (He-Ne) then round-trip length or ℓ = 2d where d is distance between mirrors.

ECE 240a Lasers - Fall 2019 Lecture 12 5

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Threshold Condition

Unsaturated small signal gain γ0(ν) per unit length must be larger than loss per unit length Loss has two terms:

Mirror loss (lumped - not distributed) Other scattering losses αs(absorption loss included in γ0(ν))

Total loss A is then given by A = R1R2e−αsℓ where ℓ is the length of the resonator If resonator is a ring, then ℓ is circumference of ring. If resonator is standing wave (He-Ne) then round-trip length or ℓ = 2d where d is distance between mirrors.

ECE 240a Lasers - Fall 2019 Lecture 12 5

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Threshold Condition

Unsaturated small signal gain γ0(ν) per unit length must be larger than loss per unit length Loss has two terms:

Mirror loss (lumped - not distributed) Other scattering losses αs(absorption loss included in γ0(ν))

Total loss A is then given by A = R1R2e−αsℓ where ℓ is the length of the resonator If resonator is a ring, then ℓ is circumference of ring. If resonator is standing wave (He-Ne) then round-trip length or ℓ = 2d where d is distance between mirrors.

ECE 240a Lasers - Fall 2019 Lecture 12 5

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Threshold Condition

Unsaturated small signal gain γ0(ν) per unit length must be larger than loss per unit length Loss has two terms:

Mirror loss (lumped - not distributed) Other scattering losses αs(absorption loss included in γ0(ν))

Total loss A is then given by A = R1R2e−αsℓ where ℓ is the length of the resonator If resonator is a ring, then ℓ is circumference of ring. If resonator is standing wave (He-Ne) then round-trip length or ℓ = 2d where d is distance between mirrors.

ECE 240a Lasers - Fall 2019 Lecture 12 5

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Distributed Mirror Loss

Express total loss A in terms of a loss per unit length A = e−2αtd Taking loge of each side we obtain αt = α + αm Term αm is the distributed mirror loss αm = 1 2ℓg loge 1 R1R2 Photon lifetime τp is loss×sec = loss/length x length/time or τp = 1 αtc where c is speed of light in resonator. Photon lifetime is single best metric to characterize quality of resonator.

ECE 240a Lasers - Fall 2019 Lecture 12 6

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Distributed Mirror Loss

Express total loss A in terms of a loss per unit length A = e−2αtd Taking loge of each side we obtain αt = α + αm Term αm is the distributed mirror loss αm = 1 2ℓg loge 1 R1R2 Photon lifetime τp is loss×sec = loss/length x length/time or τp = 1 αtc where c is speed of light in resonator. Photon lifetime is single best metric to characterize quality of resonator.

ECE 240a Lasers - Fall 2019 Lecture 12 6

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Distributed Mirror Loss

Express total loss A in terms of a loss per unit length A = e−2αtd Taking loge of each side we obtain αt = α + αm Term αm is the distributed mirror loss αm = 1 2ℓg loge 1 R1R2 Photon lifetime τp is loss×sec = loss/length x length/time or τp = 1 αtc where c is speed of light in resonator. Photon lifetime is single best metric to characterize quality of resonator.

ECE 240a Lasers - Fall 2019 Lecture 12 6

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Distributed Mirror Loss

Express total loss A in terms of a loss per unit length A = e−2αtd Taking loge of each side we obtain αt = α + αm Term αm is the distributed mirror loss αm = 1 2ℓg loge 1 R1R2 Photon lifetime τp is loss×sec = loss/length x length/time or τp = 1 αtc where c is speed of light in resonator. Photon lifetime is single best metric to characterize quality of resonator.

ECE 240a Lasers - Fall 2019 Lecture 12 6

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Distributed Mirror Loss

Express total loss A in terms of a loss per unit length A = e−2αtd Taking loge of each side we obtain αt = α + αm Term αm is the distributed mirror loss αm = 1 2ℓg loge 1 R1R2 Photon lifetime τp is loss×sec = loss/length x length/time or τp = 1 αtc where c is speed of light in resonator. Photon lifetime is single best metric to characterize quality of resonator.

ECE 240a Lasers - Fall 2019 Lecture 12 6

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Lasing Condition

Gain per unit length for a mode must be larger than loss per unit length γ0(ν) > αt Recall that γ0 = ∆N0σ(ν) where ∆N0 = N2 − N1 is unsaturated pop. difference and σ(ν) is cross section. We can then write γ0(ν) > αt ∆N0σ(ν) > αt ∆N0 > αt σ(ν) . = Nt where Nt is the threshold population inversion. Now use τp =

1 αrc to write threshold density

Nt = 1 cτpσ(ν)

ECE 240a Lasers - Fall 2019 Lecture 12 7

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Lasing Condition

Gain per unit length for a mode must be larger than loss per unit length γ0(ν) > αt Recall that γ0 = ∆N0σ(ν) where ∆N0 = N2 − N1 is unsaturated pop. difference and σ(ν) is cross section. We can then write γ0(ν) > αt ∆N0σ(ν) > αt ∆N0 > αt σ(ν) . = Nt where Nt is the threshold population inversion. Now use τp =

1 αrc to write threshold density

Nt = 1 cτpσ(ν)

ECE 240a Lasers - Fall 2019 Lecture 12 7

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Lasing Condition

Gain per unit length for a mode must be larger than loss per unit length γ0(ν) > αt Recall that γ0 = ∆N0σ(ν) where ∆N0 = N2 − N1 is unsaturated pop. difference and σ(ν) is cross section. We can then write γ0(ν) > αt ∆N0σ(ν) > αt ∆N0 > αt σ(ν) . = Nt where Nt is the threshold population inversion. Now use τp =

1 αrc to write threshold density

Nt = 1 cτpσ(ν)

ECE 240a Lasers - Fall 2019 Lecture 12 7

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Lasing Condition

Gain per unit length for a mode must be larger than loss per unit length γ0(ν) > αt Recall that γ0 = ∆N0σ(ν) where ∆N0 = N2 − N1 is unsaturated pop. difference and σ(ν) is cross section. We can then write γ0(ν) > αt ∆N0σ(ν) > αt ∆N0 > αt σ(ν) . = Nt where Nt is the threshold population inversion. Now use τp =

1 αrc to write threshold density

Nt = 1 cτpσ(ν)

ECE 240a Lasers - Fall 2019 Lecture 12 7

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Figure

Gain per unit length Total loss per unit length 3 Allowed Resonator Modes Possible lasing modes Gain per unit length Total loss per unit length 2 Allowed Resonator Modes Possible lasing modes

Initially spontaneous emission emitted into 4π steradians and follows the gain profile Some of this radiation couples into one of the allowed modes of the resonator

Amount is proportional to the gain profile

ECE 240a Lasers - Fall 2019 Lecture 12 8

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Homogeneous Gain Media

Total loss per unit length Gain reduces (saturates) Gain =Loss@lasing frequency Ideal single lasing mode Intermediate State Saturated State Modes within gain profile begin to be amplified

Intermediate state, gain starts to reduce, but there are still several modes where gain > loss Gain continues to reduce via gain saturation.

Only one mode above threshold Gain= loss at that frequency

One mode consequence of homogeneous broadening. All atoms behave in same manner. Note that cavity mode spectral shape also changes

ECE 240a Lasers - Fall 2019 Lecture 12 9

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Gain Saturation

As signal is amplified, N2 is eventually depleted and γ must decrease relative to small-signal value γ0 in absence of gain saturation. Nonlinear effect is called gain saturation and occurs in all gain media. Gain G must therefore decrease relative to G0

ECE 240a Lasers - Fall 2019 Lecture 12 10

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Gain Saturation

As signal is amplified, N2 is eventually depleted and γ must decrease relative to small-signal value γ0 in absence of gain saturation. Nonlinear effect is called gain saturation and occurs in all gain media. Gain G must therefore decrease relative to G0

ECE 240a Lasers - Fall 2019 Lecture 12 10

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Gain Saturation

As signal is amplified, N2 is eventually depleted and γ must decrease relative to small-signal value γ0 in absence of gain saturation. Nonlinear effect is called gain saturation and occurs in all gain media. Gain G must therefore decrease relative to G0

ECE 240a Lasers - Fall 2019 Lecture 12 10

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Modeling Gain Saturation

Model gain saturation by modifying γ to be a function of the intensity Iν(z) dIν(z) dz = γIν(z) dIν(z) dz 1 Iν(z) = γ0(ν) 1 + [Iν(z)/Is(ν)] where Is(ν) is the saturation intensity. (Will derive expressions from rate equations later.) When Iν(z) = Is(ν), γ = γ0/2, and the gain per unit length is reduced to half the value relative to the input to the amplifier.

ECE 240a Lasers - Fall 2019 Lecture 12 11

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Modeling Gain Saturation

Model gain saturation by modifying γ to be a function of the intensity Iν(z) dIν(z) dz = γIν(z) dIν(z) dz 1 Iν(z) = γ0(ν) 1 + [Iν(z)/Is(ν)] where Is(ν) is the saturation intensity. (Will derive expressions from rate equations later.) When Iν(z) = Is(ν), γ = γ0/2, and the gain per unit length is reduced to half the value relative to the input to the amplifier.

ECE 240a Lasers - Fall 2019 Lecture 12 11

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Graph of Solution

103 102 101 1 10 102 103 5 10 15 20 0.5 1 1.5 2 1 2 3 4 5 Input Power (P/Psat) Input Power (P/Psat) Gain (dB) Output Power (P/Psat) G0 = 20 dB G0 = 20 dB

ECE 240a Lasers - Fall 2019 Lecture 12 12

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Gain Saturation and Rate Equations

Assume τ1 = 0 Then N1 = 0 and dN2 dt = R2(t) − N2 τ2 − σ(ν)Iν hν N2

ECE 240a Lasers - Fall 2019 Lecture 12 13

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Saturated Gain

Solve dN2 dt = R20(t) − N2 τ2 − σ(ν)Iν hν N2 = R20(t) + N2 τ2

• 1 + Iν σ(ν)τ2

hν

• =

R20(t) + N2 τ2

• 1 + Iν

Is

• where

Is . = hν/σ(ν)τ2 is defined as the saturation intensity. Solution N2(t) = R20τ2 1 + Iν/Is

• 1 − exp
• − t

τ2

• 1 + Iν

Is

• ECE 240a Lasers - Fall 2019 Lecture 12

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Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Plot

1 2 3 4 5 0.0 0.5 1.0 1.5 2.0

Iν/Is=0 Iν/Is=1 Iν/Is=2 Increasing Optical Intensity

Time (t/τ2) N2(t)

When Iν = Is steady state value half of value w/o field present Time constant with field present also shorter When Iν ≫ Is steady-state value greatly reduced as well as time constant

ECE 240a Lasers - Fall 2019 Lecture 12 15

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Below and Above Threshold Conditions

Below threshold, all of the light is spontaneous emission - little or no stimulated emission Spontaneous emission goes into 4π steradians and increases “out the side” of the laser as the pumping rate R2 increases At threshold stimulated emission begins to dominate

Above threshold all of the pump rate goes into stimulated emission - goes

• ut of the cavity through mirrors

Spontaneous emission (∝gain) then clamps - emission out side of cavity stays the same

True for CW conditions - study transient effects next week

ECE 240a Lasers - Fall 2019 Lecture 12 16

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

General CW Conditions

CW Rate Equations 0 = R1(t) + N2 τ21 + σ(ν)Iν hν (N2 − N1) − N1 τ1 0 = R2(t) − N2 τ2 − σ(ν)Iν hν (N2 − N1) Matrix Form

 

• 1

τ2 + σ(ν)Iν hν

• − σ(ν)Iν

hν

−

• 1

τ21 + σ(ν)Iν hν

• 1

τ1 + σ(ν)Iν hν

• 



• N2

N1

• =
• R2

R1

• Solve for N2 − N1

N2 − N1 = R2τ2

• 1 − τ1

τ21

• − R1τ1

1 + τ1 + τ2 − τ1τ2

τ21 σ(ν)Iν hν

• where difference in population now saturates (different than without field

present)

ECE 240a Lasers - Fall 2019 Lecture 12 17

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

CW Small Signal Gain

Assume Iν small enough so that denominator ≈ 1. Then γ0(ν) = σ(ν)∆N = σ(ν)

• R2τ2
• 1 − τ1

τ21

• − R1τ1
• Saturation Intensity then modified to

Is = hν στ2 1 1 + τ1

τ2 (1 − τ2/τ21)

ECE 240a Lasers - Fall 2019 Lecture 12 18

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

High Saturation Regime

Start with saturated gain equation and add frequency dependence dIν(z) dz = γIν(z) dIν(z) dz 1 Iν(z) = γ0(ν) 1 + g(ν) [Iν(z)/Is(ν)] where g(ν) is the lineshape function Take limit as output intensity becomes much greater than saturation

• intensity. Single pass saturated gain is now

I2 = I1 +

• γ0(ν)

g(ν)

• Ig

where input intensity I1 ≫ Is and Ig is intensity that depends on lineshape

ECE 240a Lasers - Fall 2019 Lecture 12 19

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

Saturated Gain Per Unit Length

∆P area × length = γ0(ν)Is g(ν) = hν

 

R2τ2

• 1 − τ1

τ21

• − R1τ1

1 + τ1 + τ2 − τ1τ2

τ21 σ(ν)Iν hν

• 



Expression now only depends on lifetimes and pumping rates All quantum is abstracted into these time constants.

ECE 240a Lasers - Fall 2019 Lecture 12 20

Lecture 12- ECE 240a Review

Small Signal Gain Resonant Frequencies

Lasing Conditions

Lasing Threshold Mirror Loss

Threshold Conditions Homogeneous Gain Media

Gain Saturation

How to Make a Good Laser

Conditions for a “good” laser Small τ1 (lower state does not fill up - produces a larger ∆N) Branching ratio τ2/τ21 → 1 means that all of the upper state goes into the lasing state - none leaks When these conditions are met then Is = hν στ2 and ∆N = R2τ2 Therefore the key parameters for making a good laser are Rate I can pump into the upper state The depletion of the lower state.

ECE 240a Lasers - Fall 2019 Lecture 12 21