Lecture 12- ECE 240a Review Small Signal Gain Resonant Frequencies Lasing Conditions Lasing Lecture 12- ECE 240a Threshold Mirror Loss Ver Chap. 8-9 Threshold Conditions Homogeneous Gain Media Gain Saturation 1 ECE 240a Lasers - Fall 2019 Lecture 12
Review Rate Equations Lecture 12- ECE 240a Rate equations Review Small Signal Gain dN 1 = R 1 ( t ) + N 2 + σ ( ν ) I ν ( N 2 − N 1 ) − N 1 Resonant Frequencies dt τ 21 hν τ 1 Lasing Conditions Lasing − σ ( ν ) I ν dN 2 = R 2 ( t ) − N 2 Threshold ( N 2 − N 1 ) dt τ 21 hν Mirror Loss Threshold Conditions (Note these assume g 1 = g 2 ) Homogeneous Gain Media These are equations (8.3.2a) and (8.3.2b) in Verdeyen (3 rd edition) Gain Saturation 2 ECE 240a Lasers - Fall 2019 Lecture 12
Review Rate Equations Lecture 12- ECE 240a Rate equations Review Small Signal Gain dN 1 = R 1 ( t ) + N 2 + σ ( ν ) I ν ( N 2 − N 1 ) − N 1 Resonant Frequencies dt τ 21 hν τ 1 Lasing Conditions Lasing − σ ( ν ) I ν dN 2 = R 2 ( t ) − N 2 Threshold ( N 2 − N 1 ) dt τ 21 hν Mirror Loss Threshold Conditions (Note these assume g 1 = g 2 ) Homogeneous Gain Media These are equations (8.3.2a) and (8.3.2b) in Verdeyen (3 rd edition) Gain Saturation 2 ECE 240a Lasers - Fall 2019 Lecture 12
Small Signal Gain Lecture 12- ECE 240a Review Small Signal Define intensity gain (or loss) as Gain Resonant Frequencies λ 2 Lasing γ ( ν ) = σ ( ν )( N 2 − N 1 ) = g ( ν )( N 2 − N 1 ) A 21 Conditions 8 πn 2 Lasing � �� � Threshold cross-section Mirror Loss Threshold Conditions Differential equation for the intensity in an optical amplifier is then Homogeneous Gain Media dI v dz = γ 0 ( ν ) I ν Gain Saturation with a solution I ν ( z ) = I ν ( 0 ) e γ 0 ( ν ) z Define small signal gain at a distance L as G 0 = e γ 0 ( ν ) L 3 ECE 240a Lasers - Fall 2019 Lecture 12
Small Signal Gain Lecture 12- ECE 240a Review Small Signal Define intensity gain (or loss) as Gain Resonant Frequencies λ 2 Lasing γ ( ν ) = σ ( ν )( N 2 − N 1 ) = g ( ν )( N 2 − N 1 ) A 21 Conditions 8 πn 2 Lasing � �� � Threshold cross-section Mirror Loss Threshold Conditions Differential equation for the intensity in an optical amplifier is then Homogeneous Gain Media dI v dz = γ 0 ( ν ) I ν Gain Saturation with a solution I ν ( z ) = I ν ( 0 ) e γ 0 ( ν ) z Define small signal gain at a distance L as G 0 = e γ 0 ( ν ) L 3 ECE 240a Lasers - Fall 2019 Lecture 12
Small Signal Gain Lecture 12- ECE 240a Review Small Signal Define intensity gain (or loss) as Gain Resonant Frequencies λ 2 Lasing γ ( ν ) = σ ( ν )( N 2 − N 1 ) = g ( ν )( N 2 − N 1 ) A 21 Conditions 8 πn 2 Lasing � �� � Threshold cross-section Mirror Loss Threshold Conditions Differential equation for the intensity in an optical amplifier is then Homogeneous Gain Media dI v dz = γ 0 ( ν ) I ν Gain Saturation with a solution I ν ( z ) = I ν ( 0 ) e γ 0 ( ν ) z Define small signal gain at a distance L as G 0 = e γ 0 ( ν ) L 3 ECE 240a Lasers - Fall 2019 Lecture 12
Resonant Frequencies of General Resonator Lecture 12- ECE 240a If one of the mirror is not flat then Review � � q + ( 1 + m + p ) c Small Signal cos − 1 ( g 1 g 2 ) 1 / 2 ν m , p , q = Gain 2 nd π Resonant Frequencies Lasing where Conditions d Lasing g 1,2 = 1 − Threshold R 1,2 Mirror Loss Threshold Example: d / R 2 = 1 / 2 , then Conditions � � Homogeneous q + ( 1 + m + p ) c Gain Media ν m , p , q = Gain 2 nd 4 Saturation 4 ECE 240a Lasers - Fall 2019 Lecture 12
Resonant Frequencies of General Resonator Lecture 12- ECE 240a If one of the mirror is not flat then Review � � q + ( 1 + m + p ) c Small Signal cos − 1 ( g 1 g 2 ) 1 / 2 ν m , p , q = Gain 2 nd π Resonant Frequencies Lasing where Conditions d Lasing g 1,2 = 1 − Threshold R 1,2 Mirror Loss Threshold Example: d / R 2 = 1 / 2 , then Conditions � � Homogeneous q + ( 1 + m + p ) c Gain Media ν m , p , q = Gain 2 nd 4 Saturation 4 ECE 240a Lasers - Fall 2019 Lecture 12
Resonant Frequencies of General Resonator Lecture 12- ECE 240a If one of the mirror is not flat then Review � � q + ( 1 + m + p ) c Small Signal cos − 1 ( g 1 g 2 ) 1 / 2 ν m , p , q = Gain 2 nd π Resonant Frequencies Lasing where Conditions d Lasing g 1,2 = 1 − Threshold R 1,2 Mirror Loss Threshold Example: d / R 2 = 1 / 2 , then Conditions � � Homogeneous q + ( 1 + m + p ) c Gain Media ν m , p , q = Gain 2 nd 4 Saturation 4 ECE 240a Lasers - Fall 2019 Lecture 12
Threshold Condition Lecture 12- ECE 240a Review Small Signal Gain Unsaturated small signal gain γ 0 ( ν ) per unit length must be larger than Resonant Frequencies loss per unit length Lasing Conditions Loss has two terms: Lasing Threshold Mirror loss (lumped - not distributed) Mirror Loss Other scattering losses α s (absorption loss included in γ 0 ( ν ) ) Threshold Conditions Total loss A is then given by Homogeneous Gain Media A = R 1 R 2 e − α s ℓ Gain Saturation 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 = 2 d where d is distance between mirrors. 5 ECE 240a Lasers - Fall 2019 Lecture 12
Threshold Condition Lecture 12- ECE 240a Review Small Signal Gain Unsaturated small signal gain γ 0 ( ν ) per unit length must be larger than Resonant Frequencies loss per unit length Lasing Conditions Loss has two terms: Lasing Threshold Mirror loss (lumped - not distributed) Mirror Loss Other scattering losses α s (absorption loss included in γ 0 ( ν ) ) Threshold Conditions Total loss A is then given by Homogeneous Gain Media A = R 1 R 2 e − α s ℓ Gain Saturation 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 = 2 d where d is distance between mirrors. 5 ECE 240a Lasers - Fall 2019 Lecture 12
Threshold Condition Lecture 12- ECE 240a Review Small Signal Gain Unsaturated small signal gain γ 0 ( ν ) per unit length must be larger than Resonant Frequencies loss per unit length Lasing Conditions Loss has two terms: Lasing Threshold Mirror loss (lumped - not distributed) Mirror Loss Other scattering losses α s (absorption loss included in γ 0 ( ν ) ) Threshold Conditions Total loss A is then given by Homogeneous Gain Media A = R 1 R 2 e − α s ℓ Gain Saturation 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 = 2 d where d is distance between mirrors. 5 ECE 240a Lasers - Fall 2019 Lecture 12
Threshold Condition Lecture 12- ECE 240a Review Small Signal Gain Unsaturated small signal gain γ 0 ( ν ) per unit length must be larger than Resonant Frequencies loss per unit length Lasing Conditions Loss has two terms: Lasing Threshold Mirror loss (lumped - not distributed) Mirror Loss Other scattering losses α s (absorption loss included in γ 0 ( ν ) ) Threshold Conditions Total loss A is then given by Homogeneous Gain Media A = R 1 R 2 e − α s ℓ Gain Saturation 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 = 2 d where d is distance between mirrors. 5 ECE 240a Lasers - Fall 2019 Lecture 12
Threshold Condition Lecture 12- ECE 240a Review Small Signal Gain Unsaturated small signal gain γ 0 ( ν ) per unit length must be larger than Resonant Frequencies loss per unit length Lasing Conditions Loss has two terms: Lasing Threshold Mirror loss (lumped - not distributed) Mirror Loss Other scattering losses α s (absorption loss included in γ 0 ( ν ) ) Threshold Conditions Total loss A is then given by Homogeneous Gain Media A = R 1 R 2 e − α s ℓ Gain Saturation 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 = 2 d where d is distance between mirrors. 5 ECE 240a Lasers - Fall 2019 Lecture 12
Distributed Mirror Loss Lecture 12- ECE 240a Express total loss A in terms of a loss per unit length Review Small Signal Gain A = e − 2 α t d Resonant Frequencies Lasing Conditions Taking log e of each side we obtain Lasing Threshold α t = α + α m Mirror Loss Threshold Conditions Term α m is the distributed mirror loss Homogeneous Gain Media 1 1 Gain α m = 2 ℓ g log e Saturation R 1 R 2 Photon lifetime τ p is loss × sec = loss/length x length/time or 1 τ p = α t c where c is speed of light in resonator. Photon lifetime is single best metric to characterize quality of resonator. 6 ECE 240a Lasers - Fall 2019 Lecture 12
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