Output Power and Linewidth Ampliﬁers ECE 455 Optical Electronics Threshold

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ECE 455 Lecture 4

Gain Saturation

Optical Output Power and Linewidth Ampliﬁers ECE 455 Optical Electronics Threshold

Output Power

Optimization Eﬃciency Tom Galvin Mode Gary Eden Selection

Linewidth, Stabilization, ECE Illinois and Tuning

Summary Introduction

ECE 455 Lecture 4

Gain Saturation In this section, we will learn how to do the following things: Optical Ampliﬁers Determine the gain of a laser ampliﬁer Threshold Find the threshold gain of a cavity Output Power

Optimization Predict the output power of a laser Eﬃciency

Mode Determine the output mode of the laser Selection Unless otherwise stated, steady state ( d = 0) behavior may be Linewidth, dt Stabilization, assumed. and Tuning

Summary Conventions for this Section

ECE 455 Lecture 4

Gain Saturation

Optical N2 Ampliﬁers Assume the laser can be Threshold described by the diagram Output Power to the right. Laser τ2 Optimization R2 Eﬃciency Some processes have been Mode omitted to simplify the Selection N1 equations τ1 Linewidth, N0 Stabilization, and Tuning

Summary Gain Saturation Derivation I

ECE 455 Lecture 4 Begin by writing the laser rate equations: Gain dN2 N2 I σse g2 Saturation = − − N2 − N1 + R2 = 0 (1) dt τ2 hν g1 Optical Amplifiers Threshold dN1 N1 I σse g2 N2 = − + N2 − N1 + = 0 (2) Output Power dt τ1 hν g1 τ21 Optimization Efficiency dNp N2 I σse g2 = + N2 − N1 = 0 (3) Mode dt τ hν g Selection 21 1 Linewidth, In the absence of an optical field, the stimulated emission term Stabilization, and Tuning may be ignored. In steady state, this yields: Summary N2 Rτ1τ2 N2 = R2τ2; N1 = τ1 = (4) τ21 τ21 Gain Saturation Derivation II

ECE 455 Lecture 4

For simplicity, let τ1 τ2. This is a reasonable model for Gain Saturation systems with a favorable lifetime ratio. With this assumption Optical and Equation 4, we ﬁnd that N1 N2. Thus: Ampliﬁers

Threshold ∆N0 = N2 − N1 ≈ N2 = R2τ2 (5) Output Power Optimization Now consider the situation when the medium is in the presence Eﬃciency

Mode of an optical ﬁeld I . Stimulated emission can no longer be Selection ignored. Linewidth, Stabilization, and Tuning dN2 N2 I σse g2 = − − N2 − N1 + R2 = 0 (6) Summary dt τ2 hν g1 Gain Saturation Derivation III

ECE 455 Lecture 4 Combining this equation with Equation 5, we can write: Gain Saturation dN2 N2 I σse Optical = − − N2 + R2 = 0 (7) Ampliﬁers dt τ2 hν Threshold

Output Power Finally, solving for N2, we obtain:

Optimization Eﬃciency R τ ∆N ∆N = 2 2 = 0 (8) Mode I σse τ2 1 + I /I Selection 1 + hν sat Linewidth, Stabilization, where and Tuning hν Summary Isat = (9) σse τ2 Gain Saturation Derivation IV

ECE 455 Lecture 4 Finally, multiplying both size of Equation 8 by the stimulated Gain Saturation emission cross section, σse (ν), we have Optical Ampliﬁers γ0(ν) Threshold γ(ν) = (10) 1 + I /Isat Output Power

Optimization Efficiency Returning to the original definition of the gain coefficient, we

Mode ﬁnd: Selection 1 dI γ (ν) γ ≡ = 0 (11) Linewidth, I dz 1 + I /I Stabilization, sat and Tuning

Summary What happens in the limit that I → ∞? Homogeneous Saturation

ECE 455 Lecture 4

Gain Saturation Optical Laser Ampliﬁers Threshold γ(ν) γ0(ν) γ0(ν) Output Power

Optimization Eﬃciency Laser γ(ν) Mode Selection Linewidth, ν ν Stabilization, 0 ν 0 ν and Tuning

Summary Inhomogeneous Saturation

ECE 455 Lecture 4

Gain Saturation

Optical Ampliﬁers Laser Threshold γ(ν) Output Power γ(ν) Optimization Eﬃciency Laser Mode Selection Linewidth, ν ν Stabilization, 0 ν 0 ν and Tuning

Summary Gain Saturation Summary

ECE 455 Lecture 4

Gain Saturation

Optical Intense light reduces the population inversion Ampliﬁers Medium with reduced population inversion has reduced Threshold gain Output Power

Optimization Homogeneous saturation reduces the entire gain spectrum Eﬃciency uniformly Mode Selection Inhomogeneous saturation only reduces part of the gain Linewidth, Stabilization, spectrum and Tuning

Summary Application of Gain Saturation: STED Microscopy

ECE 455 Stimulated Emission Depletion (STED) Microscopy is a Lecture 4 method of overcoming the diﬀraction limit

Gain Saturation λ dmin ≈ (12) Optical 2NA Ampliﬁers Threshold of confocal microscopy. Here NA = n · sin(θ) is the numerical Output Power aperture. Optimization Eﬃciency

Mode The basic idea of STED is as follows: Selection 1 Stain a sample with a gain medium (commonly a dye) Linewidth, Stabilization, 2 Excite a volume of gain material with a pump laser and Tuning

Summary 3 De-excite all but a small fraction of the pumped volume using stimulated emission 4 Scan beams around while collecting spontaneous emission signal to generate image STED Microscopy: Excitation and De-excitation

ECE 455 Lecture 4

Gain Saturation

Optical Ampliﬁers

Threshold

Output Power

Optimization Eﬃciency

Mode Selection

Linewidth, Stabilization, and Tuning Dye is excited by a laser at wavelength λp Summary Spontaneous emission is suppressed in regions of the sample illuminated with a de-excitation laser (λd ) Fluorescence only comes from regions not illuminated by the de-excitation laser. STED Microscopy: Excitation and De-excitation

ECE 455 Lecture 4

Gain Saturation

Optical Ampliﬁers

Threshold

Output Power

Optimization Eﬃciency

Mode Selection

Linewidth, Stabilization, and Tuning Left: Pump beam at a shorter wavelength (λp) used to excite Summary ﬂuorescence in sample. Right: De-excitation beam at ﬂuorescence wavelength (λd ), used to de-populate upper state STED Microscopy: Spot Size

ECE 455 Lecture 4

Gain Saturation

Optical Ampliﬁers

Threshold

Output Power

Optimization Efficiency Left: Fluorescence pattern with de-excitation beam off. This Mode Selection spot size is the normal diffraction limit for fluorescence

Linewidth, microscopy. Center: Fluorescence pattern when the peak Stabilization, and Tuning intensity of the de-excitation beam is 10 · Isat . Right:

Summary Fluorescence pattern when the peak intensity of the de-excitation beam is 100 · Isat . Note the decrease in size from regular diﬀraction limit. Optical Ampliﬁcation

ECE 455 Lecture 4 Equation 11 is a separable diﬀerential equation and can be Gain integrated as follows: Saturation

Optical Z Iout Z L Ampliﬁers 1 1 + dI = γ0dz (13) Threshold Iin I Isat 0 Output Power

Optimization When integrated, this gives: Eﬃciency Mode Iout Iout − Iin Selection ln + = γ0L (14) Linewidth, Iin Isat Stabilization, and Tuning This is an exact, but transcendental equation, which can be Summary diﬃcult to use. In the next slides, two limits will be considered in which this expression becomes simpler. Small Signal Limit

ECE 455 Lecture 4

Gain Saturation If Iout Isat , then, to an approximation, the upper and lower Optical Ampliﬁers state populations are unaﬀected by the presence of the optical

Threshold ﬁeld. dI γ0I Output Power = ≈ γ0I (15) Optimization dz 1 + I /Isat Eﬃciency Thus: Mode γ0L Selection Iout = Iine (16) Linewidth, Stabilization, A small signal grows exponentially. and Tuning

Summary Saturated Gain Limit

ECE 455 Lecture 4 If Iin Isat , then the intense optical ﬁeld will drive the system to transparency. A ﬁxed amount of power will be extracted per Gain Saturation unit length. This process is commonly known as ’bleaching.’

Optical Ampliﬁers dI γ0I = ≈ γ0Isat (17) Threshold dz 1 + I /Isat Output Power Optimization Iout = Iin + γ0Isat L (18) Eﬃciency Mode A large signal grows linearly. Why? Selection

Linewidth, Stabilization, and Tuning

Summary The power available from a laser ampliﬁer is:

Pav = Aγ0Isat L (19) Gain Saturation Picture

ECE 455 Lecture 4

Gain Saturation

Optical Ampliﬁers

Threshold

Output Power

Optimization Eﬃciency

Mode Selection

Linewidth, Stabilization, and Tuning

Summary The Saturation Intensity

ECE 455 Lecture 4

Gain I = hν only valid with simplifying assumptions of this Saturation sat σse τ2 Optical section Ampliﬁers Rigorous Deﬁnition: Threshold

Output Power Isat ≡ {The intensity which causes the small Optimization Eﬃciency signal gain to fall by half} (20) Mode Selection Characteristic of the gain medium Linewidth, Stabilization, and Tuning Scale of what intensities can be achieved with a given

Summary medium Example: Saturation Intensities

ECE 455 Lecture 4 Problem: Find the saturation intensity of the Helium-Neon (HeNe) laser. The relevant parameters are: λ = 632.8 nm, Gain −8 −17 2 Saturation τ2 = 3×10 s, and σse = 3×10 m

Optical Solution: Simply use Equation 9 Ampliﬁers Threshold hc Isat = Output Power λσse τ2 Optimization −34 Eﬃciency (6.626 × 10 J-s) · (299792458 m/s) = −17 2 −8 Mode (632.8 nm) · (3 × 10 m ) · (3 × 10 s) Selection 2 Linewidth, = 34.88 W/cm (21) Stabilization, and Tuning A typical bore radius for a HeNe laser is w = 0.5 mm. This Summary 0 2 implies an output of P = πIsat r = 273 mW. Commercial HeNe lasers can be purchased with outputs ranging from 10 - 200 mW. Example: Saturation Intensities

ECE 455 Lecture 4 Problem: Find the saturation intensity of the titanium doped sapphire (Ti:Sapph) laser. The relevant parameters are: λ = Gain −6 −23 2 Saturation 800 nm, τ2 = 3.8×10 s, and σse = 3.4×10 m

Optical Solution: Simply use Equation 9 Ampliﬁers Threshold hc Isat = Output Power λσse τ2 Optimization −34 Eﬃciency (6.626 × 10 J-s) · (299792458 m/s) = −23 2 −6 Mode (800 nm) · (3.4 × 10 m ) · (3.8 × 10 s) Selection 2 Linewidth, = 192 kW/cm (22) Stabilization, and Tuning A typical beam waist for a HeNe laser is w = 0.3 mm. This Summary 0 2 implies an output of P = πIsat r = 543 W. Commercial Ti:Sapph systems can be purchased with average output power from 100 mW - 10 W. Threshold Gain Picture

ECE 455 Lecture 4

Gain R R Saturation HR OC Optical Ampliﬁers Gain Medium Threshold

Output Power

Optimization Eﬃciency

Mode Selection

Linewidth, Stabilization, and Tuning

Summary

L Threshold Gain

ECE 455 Lecture 4 If a laser is pumped above threshold, the round trip gain must Gain Saturation be 1! Optical If the round trip gain were less than one, the ﬁeld would Ampliﬁers

Threshold decay continuously, but then output power would decay to

Output Power zero. Optimization If the round trip gain were greater than one, the field Efficiency would grow continuously. Because the gain medium is Mode Selection being pumped with a finite amount of energy, this is Linewidth, impossible. Stabilization, and Tuning The light intensity will saturate the gain. Summary Inside a laser γ = γth < γ0 Threshold of a Two-Mirror Cavity

ECE 455 Lecture 4

Gain 1 2 Saturation

Optical Ampliﬁers

Threshold

Output Power

Optimization Eﬃciency

Mode Selection

Linewidth, Stabilization, and Tuning γthL γthL R1e R2e = 1 (23) Summary 1 γ = − ln (R R ) (24) th 2L 1 2 Threshold of a More Complicated Cavity

ECE 455 Lecture 4

Gain Saturation

Optical If the gain medium on the previous slide experiences a Beer’s Ampliﬁers Law absorption loss, the threshold gain expressions become: Threshold

Output Power (γth−α)Lg (γth−α)Lg R1e R2e = 1 (25) Optimization Eﬃciency 1 Mode γth = α + ln (R1R2) (26) Selection 2Lg Linewidth, Stabilization, and Tuning

Summary Example: Finding the Threshold

ECE 455 Lecture 4 Problem: Find the threshold gain of the following cavity.

Gain Polarizer R3 = 99% Saturation T = 90% Optical Ampliﬁers R2 = 99% Threshold

Output Power

Optimization Absorber Eﬃciency α = 10-2 cm-1 Mode L = 5 cm Selection a

Linewidth, Stabilization, and Tuning Gain Summary R1 = 90% Lg = 10 cm T1 = 10% Output Example: Finding the Threshold

ECE 455 Lecture 4

Gain Saturation Solution: Begin by setting the round-trip gain to one. Optical Ampliﬁers

Threshold R1exp(γthLg )R2TR3exp(−αLa) = 1 (27)

Output Power

Optimization Eﬃciency La 1 γth = α − ln(R1R2TR3) (28) Mode Lg Lg Selection −1 Linewidth, = 0.0281 cm (29) Stabilization, and Tuning

Summary Example: Forcing TEM00 Operation I

ECE 455 Problem: To stop higher-order modes from lasing, a circular Lecture 4 aperture with radius ra = 1.5w(z) is placed in a laser cavity as

Gain shown below. Find the threshold gain for both the TEM00 and Saturation TEM11 modes with and without the aperture present. Optical Ampliﬁers

Threshold Output Power R = 99 % 1 Aperture R2= 94 % Optimization Eﬃciency Gain Medium Mode Selection

Linewidth, Stabilization, and Tuning

Summary

Lg= 5 cm

L = 18 cm Example: Forcing TEM00 Operation II

ECE 455 Lecture 4 The plot below shows the transmitted intensity as a function of aperture size. Gain Saturation

Optical Ampliﬁers

Threshold

Output Power

Optimization Eﬃciency

Mode Selection

Linewidth, Stabilization, and Tuning

Summary Example: Forcing TEM00 Operation III

ECE 455 Solution: Without the aperture, the threshold gains of each Lecture 4 mode are identical:

Gain 1 −1 Saturation γth = − ln(R1R2) = 0.0072 cm (30) Optical 2Lg Ampliﬁers

Threshold Reading the plot on the previous page, we ﬁnd T00 = 0.9888 Output Power for the TEM00 mode. The threshold is then: Optimization Eﬃciency 1 −1 γth,00 = − ln(R1T00R2T00) = 0.0094 cm (31) Mode 2Lg Selection Linewidth, Similarly for the TEM mode, T = 0.8260. Therefore Stabilization, 11 11 and Tuning 1 Summary −1 γth,11 = − ln(R1T11R2T11) = 0.0454 cm (32) 2Lg

If you pump the laser such that γth,00 ≤ γ0 ≤ γth,11, only the TEM00 mode will lase. Saturation of Gain Media Inside a Cavity

ECE 455 Lecture 4

Gain Saturation

Optical Ampliﬁers

Threshold

Output Power γ0(ν) Optimization Eﬃciency Cavity Modes Mode Selection

Linewidth, Stabilization, γth and Tuning

Summary

νq-2 νq-1 ν0 νq νq+1 ν Saturation of Homogeneous Medium in a Cavity

ECE 455 Lecture 4

Gain Saturation Optical Laser Ampliﬁers

Threshold

Output Power

Optimization Eﬃciency Cavity Modes Mode Selection γ(ν) Linewidth, Stabilization, γth and Tuning

Summary

νq-2 νq-1 ν0 νq νq+1 ν Gain Clamping

ECE 455 Lecture 4 Gain would Gain γ Saturation keep increasing Optical Ampliﬁers outside cavity Threshold

Output Power Gain “clamped”

Optimization Eﬃciency at threshold

Mode Selection th

Linewidth, γ Stabilization, and Tuning

Summary

γth γ0 Spontaneous Emission Power

ECE 455 Lecture 4

Gain Saturation spont P Spontaneous Optical Ampliﬁers emission power Threshold

Output Power in absence of cavity

Optimization Eﬃciency

Mode Selection Excess power is Linewidth, Stabilization, laser output and Tuning

Summary

Pth Ppump Laser Emission Power

ECE 455 Lecture 4

Gain laser Saturation P Power rises Optical Ampliﬁers linearly above Threshold

Output Power threshold

Optimization Eﬃciency

Mode Selection

Linewidth, Power zero Stabilization, and Tuning below threshold Summary

Pth Ppump A Tale of Three Gains

ECE 455 Lecture 4 γ0 - Small Signal Gain

Gain γ0 = σse ∆N0 Saturation Set by material parameters and pumping rate Optical The gain in the absence of a strong optical ﬁeld Ampliﬁers

Threshold γth - Threshold Gain 1 Output Power γth = α − 2L ln (R1R2) = σse ∆Nth Optimization Determined only by the properties of the cavity Eﬃciency The gain which just balances cavity losses Mode Selection γ - Gain 1 dI Linewidth, γ ≡ I dz = σse ∆N Stabilization, and Tuning Set by both the cavity and the pump rate Summary Inside a cavity

γ = γ0 if γ0 < γth γ = γth if γ0 > γth Conventions for this Section

ECE 455 Lecture 4

RHR ROC Gain Saturation Ta Tb

Optical Ampliﬁers

Threshold

Output Power Lg Optimization Eﬃciency L Mode Selection Linewidth, In this section we will consider the output power from a laser Stabilization, and Tuning system as shown above: Summary Window transmissions Ta and Tb Mirror relﬂectivities Rhr and Roc Power taken from Rhr and Toc + Roc = 1 Internal gain medium loss of α The Low Loss Approximation I

ECE 455 Lecture 4 In the laser resonator, there is both a forward and backward propogating wave. Each wave works to saturate the medium. Gain Saturation We may write:

Optical Ampliﬁers dI+ γ0I+ = − γthI+ (33) Threshold dz 1 + I++I− Isat Output Power

Optimization dI− −γ0I− Eﬃciency = − γthI− (34) dz I++I− Mode 1 + I Selection sat Linewidth, In these equations, γth removes the portion of the gain which is Stabilization, and Tuning used to overcome the cavity losses.

Summary 1 2 2 γth = αint − ln Ta Tb Rhr Roc (35) 2Lg The Low Loss Approximation II

ECE 455 If losses are low enough in the cavity, it may be assumed that Lecture 4 dI dz = 0. This implies I+ = I−. This is the Low Loss

Gain Approximation. If Equation 33 is set equal to zero and solved Saturation for I+, we ﬁnd: Optical Ampliﬁers Isat γ0 I+ = − 1 (36) Threshold 2 γth

Output Power I+ is the intensity which must be present in the gain medium. Optimization Eﬃciency To ﬁnd how much is coupled out, one must multiply be Tb and T . Mode 2 Selection Toc TbIsat γ0 Linewidth, Iout = − 1 (37) Stabilization, 2 γth and Tuning

Summary To ﬁnd the total power, multiply by the area Toc TbIsat γ0 Pout = A − 1 (38) 2 γth Alternative Derivation of Output Power I

ECE 455 Lecture 4 Once again assume the loss of cavity is small. Then the intracavity intensity is Gain Saturation γ0 Optical I = Isat − 1 (39) Ampliﬁers γth Threshold

Output Power The stimulated emission power from all atoms in the cavity is

Optimization Eﬃciency Pe = σse ∆Nth · I · V Mode Selection = γth · I · V (40) Linewidth, Stabilization, and Tuning But only a fraction of this power will be useful. This fraction is:

Summary 1 1 − ln (R Roc ) − ln (R Roc ) T = 2 hr = 2 hr (41) 1 γ L αLg − 2 ln (Rhr Roc ) th g Alternative Derivation of Output Power II

ECE 455 Combining the results of Equations 39-41 Lecture 4

Pout = Pe T (42) Gain Saturation " 1 # γ0 − 2 ln (Rhr Roc ) Optical = γth Isat − 1 · V · (43) Ampliﬁers γth γthLg Threshold γ0 V 1 Output Power = I − 1 · − ln (R R ) (44) sat γ L 2 hr oc Optimization th g Eﬃciency γ0 1 Mode = AIsat − 1 − ln (Rhr Roc ) (45) Selection γth 2 Linewidth, (46) Stabilization, and Tuning

Summary Therefore the output intensity is 1 γ0 Iout = − ln (Rhr Roc ) Isat − 1 (47) 2 γth Yet Another Derivation of Output Power I

ECE 455 For the next derivation, we’ll look at the cavity as a whole. Lecture 4 This derivation will require a few new variables

Gain φ - total number of photons in cavity Saturation V - mode volume (inside and outside the gain nedium) Optical Ampliﬁers τc - photon lifetime in cold (unpumped) cavity Threshold τoc - photon lifetime if output coupler were the only Output Power source of photon loss Optimization Eﬃciency There are also three equations to keep in mind: Mode Selection 1 1 1 Linewidth, = + (48) Stabilization, τc τoc τnc and Tuning c φhν Summary I = (49) n V ∆N ∆N = 0 (8) 1 + I /Isat Yet Another Derivation of Output Power II

ECE 455 The rate of change of the number of photons can be written as: Lecture 4

dφ Lg I σse φ Gain = V ∆N − + ηseed A21N2 (50) Saturation dt L hν τc Optical Lg c φhν 1 φ Ampliﬁers = V σ ∆N − + η A N (51) L n V hν se τ seed 21 2 Threshold c L c φ Output Power g = φσse ∆N − + ηseed A21N2 (52) Optimization L n τc Eﬃciency Lg c 1 Mode = φ σse ∆N − + ηseed A21N2 (53) Selection L n τc Linewidth, Stabilization, In steady state, the spontaneous emission into the cavity mode and Tuning

Summary is negligible. Therefore: Lg c 1 φ σse ∆N − = 0 (54) L n τc Yet Another Derivation of Output Power III

ECE 455 Lecture 4

Obviously φ = 0 is a mathematical, but not physical solution, Gain Saturation but according to Equations 8 and 49, ∆N is a function of φ. Optical Subsituting them into Equation 54 and solving for φ yields: Ampliﬁers Threshold Isat V Lg n Output Power φ = τ σ ∆N − (55) hν L c se 0 c Optimization Eﬃciency φhν Mode The total output power is Pout = Selection τoc Linewidth, Stabilization, Lg τc 1 n and Tuning P = I V σ ∆N − (56) out sat L τ se 0 τ c Summary oc oc Rigrod Analysis

ECE 455 Lecture 4

Gain Saturation With α = 0, the exact solution for output intensity was derived Optical Ampliﬁers by Rigrod to be: Threshold 1 2 2 Output Power γ0Lg + 2 ln Rhr Roc Ta Tb I = Isat TbToc (57) Optimization r ! q 2 Eﬃciency 2 2 Roc Tb 1 − Rhr Roc Ta Tb 1 + 2 Mode Rhr Ta Selection

Linewidth, Stabilization, Note that this equation is valid for ANY level of coupling. and Tuning

Summary Comparing Models

ECE 455 We’ve seen one and derived three expressions for the output Lecture 4 power of the laser, but how are they related?

Gain First three models assume gain medium is uniformly Saturation saturated, which implies low loss is necessary Optical Ampliﬁers Rigrod analysis assumes α = 0, but otherwise any amount Threshold of loss is accounted for Output Power Models should generally agree with each other, especially Optimization Eﬃciency in the limit of low loss

Mode These are only order of magnitude estimates anyway, they Selection do not include eﬀects such as: Linewidth, Stabilization, Gaussian beam shape and Tuning Non-uniform pumping Summary Nonlinear loss mechanisms Many others The ﬁt to data can be surprisingly good even with these simple models Example: Comparing Output Power Models I

ECE 455 Lecture 4 −2 Problem: A Ti:Sapphire (Isat = 192 kW-cm and α = 0) 2 with cavity parameters Lg = 5 mm, L = 1 m, A = 0.07 mm , Gain Saturation Rhr = 0.99, Roc = 0.94, and γ0 = 1.1γth. Find the output Optical power of this laser using all four models. Ampliﬁers Solution: According to the low-loss approximation: Threshold Output Power −2 2 (0.06) · (1.0) · (192 kW-cm ) Optimization Pout = (0.07 mm ) [1.1 − 1] Eﬃciency 2 Mode = 0.403 W (58) Selection

Linewidth, Stabilization, and Tuning 1 P = (0.07 mm2) − ln(0.99 · 0.94) (192 kW-cm−2) [1.1 − 1] Summary out 2 = 0.4158 W (59) Example: Comparing Output Power Models II

ECE 455 Lecture 4 Next we’ll estimate the power using Rigrod’s analysis. First,

Gain we’ll need to find γ0 Saturation Optical 1 −1 Amplifiers γ = 1.1γ = 1.1 ln(0.99·0.94) = 0.0791 cm (60) 0 th 2(5 mm) Threshold Output Power Next, we insert this into the Rigrod formula to obtain: Optimization Efficiency 2 −2 Mode Pout = (0.07 mm )(192 kW-cm )(0.06)(1.0) Selection −1 1 (0.0791 cm )(5 mm) + 2 ln(0.99 · 0.94) Linewidth, × Stabilization, √ q and Tuning 0.94 1 − 0.99 · 0.94 1 + 0.99 Summary = 0.4158 W (61) Example: Comparing Output Power Models III

ECE 455 Lecture 4 Finally, we’ll estimate the output power using the rate equation approach.

Gain Saturation 2 · (0.995 m + 1.76 · 0.005 m) τ = = 99.8 ns (62) Optical c 8 Ampliﬁers (3 × 10 m/s)(1 − 0.99 · 0.94) Threshold 2 · (0.995 m + 1.76 · 0.005 m) Output Power τ = = 115.4 ns (63) oc (3 × 108 m/s)(1 − 0.94) Optimization Eﬃciency These can now be placed into Equation 56 Mode Selection −2 2 Linewidth, Pout = (192 kW-cm )(0.07 mm )(5 mm) Stabilization, and Tuning 5 mm 99.8 ns 1 1.76 × (0.0755 cm−1) − Summary 1 m 115.4 ns 111.2 ns 3 × 108 m/s = 0.580 W (64)

We see that all of the models predict similar output powers. Example: Dust on the Optic

ECE 455 Problem: A piece of dust drifts near one face of the crystal Lecture 4 and is burnt by the high-intesity light. The smoke coats one

Gain side of the crystal, causing a 0.1% scattering loss. Assume the Saturation laser is pumped with the same power as before. Find the new Optical Ampliﬁers output power of the laser.

Threshold Solution: We’ll use Rigrod’s analysis

Output Power 2 −2 Pout = (0.07 mm )(192 kW-cm )(0.06)(1.0)(0.95) Optimization −1 1 2 Efficiency (0.0791 cm )(5 mm) + ln(0.99 · 0.94 · 0.95 ) × 2 Mode √ q Selection 2 0.94·0.952 1 − 0.99 · 0.94 · 0.95 1 + 0.99 Linewidth, Stabilization, and Tuning = 0.292 W (65) Summary The Lesson: Outside the cavity, a 0.1% mirror loss would only have resulted in a 0.1% power loss. Inside the cavity, the effect of the loss is magnified. What is the output power with a 1% scattering loss? Example: Dust on the Optic Redesign

ECE 455 Lecture 4 Problem: Suppose the laser is redesigned so that 2 A = 0.0167 mm and γ0 = 1.38γth, with all other parameters Gain remaining the same. Find the power before and after the dust Saturation

Optical burns on the optic Ampliﬁers Solution: We’ll use Rigrod’s analysis, we ﬁnd the output Threshold powers to be are: Output Power

Optimization Eﬃciency Pout = 0.4150 W (66)

Mode Selection After the dust burns on, the output power is reduced to: Linewidth, Stabilization, and Tuning Pout = 0.3770 W (67) Summary The Lesson: Lasers operating well above threshold are less sensitive to perturbations. Small mode areas with high intensities are favored. Output Coupler Optimization I

ECE 455 Lecture 4

Gain Before we start with equations, a few quick observations: Saturation All power scattered or coupled out at R is wasted Optical hr Ampliﬁers because the output is taken at Roc . Hence Rhr should be Threshold as high as possible Output Power In the limit that Roc is the only loss in the cavity, Roc Optimization Eﬃciency should be chosen as large as possible. Mode Selection If other loss mechanisms are present, than an optimal

Linewidth, value exists for the output coupler. The power needs to be Stabilization, and Tuning taken out of the cavity before it gets lost to scattering or

Summary reabsorption. Output Coupler Optimization II

ECE 455 Recall that in the Low-Loss approximation, the output power of Lecture 4 a laser is Toc Isat γ0 Gain Iout = − 1 (37) Saturation 2 γth Optical Ampliﬁers The threshold gain may be written

Threshold 2Lg γth = 2αLg − ln(Rhr ) − ln(Roc ) (68) Output Power = 2αLg − ln(Rhr ) − ln(1 − Toc ) (69) Optimization Eﬃciency 0 = α + Toc (70) Mode Selection where the approximation ln(1 − Toc ) ≈ Toc has been used and Linewidth, 0 Stabilization, α ≡ 2αLg − ln(Rhr ) for convenience. and Tuning Summary The equation for the output power is now: Toc Isat 2Lg γ0 Iout = 0 − 1 (71) 2 α + Toc Output Coupler Optimization III

ECE 455 Lecture 4 We can take the derivative of Equation 71! Gain dIout Isat 2Lg γ0 2Lg γ0Toc Saturation = 0 − 1 − 0 2 = 0 (72) dToc 2 α + Toc (α + Toc ) Optical Ampliﬁers

Threshold The optimum value for the transmission is

Output Power 01/2 0 Optimization Toc,opt = 2Lg γ0α − α (73) Eﬃciency 1/2 Mode ≈ 2Lg (γ0α) − 2αLg (74) Selection Linewidth, Equation 74 is valid in the limit that R → 1. The Stabilization, hr and Tuning corresponding output intensity for this coupling is: Summary 2 1/2 1/2 Iout,opt = Isat Lg γ0 + α (75) Output Coupler Optimization III

ECE 455 Lecture 4

Gain Saturation

Optical Ampliﬁers There is no one Top,opt for a give laser type. It changes Threshold depending on Output Power

Optimization Desired output power Eﬃciency Unavoidable cavity losses (Rhr and α) Mode Selection

Linewidth, Stabilization, and Tuning

Summary A Tale of Many Eﬃciencies

ECE 455 Lecture 4

Gain Saturation

Optical Ampliﬁers

Threshold There are many eﬃciencies in common use in laser physics Output Power

Optimization Energy is lost to heat, not collected in the output beam Eﬃciency

Mode Selection

Linewidth, Stabilization, and Tuning

Summary Wall Plug Eﬃciency

ECE 455 Lecture 4 Output power of laser divided by the amount of power the power company Gain charges you for. Saturation

Optical Includes all losses, including electrical Amplifiers circuits, diffraction, spontaneous Threshold emission, mode overlap, auxiliary Output Power equipment and more. Optimization Pout Efficiency η = The CO2 laser can have a wall plug PPowerBill Mode Selection efficiency around 30% and diode lasers (76) Linewidth, can have wall plug efficiencies around Stabilization, and Tuning 60%. Summary Most lasers types have low wall plug efficiencies. Example: The Ar+ laser has a wall plug efficiency of about 0.5%. Optical to Optical Efficiency

ECE 455 Lecture 4 Frequently used in Gain optically pumped lasers Saturation because it can be Optical Amplifiers conveniently measured Threshold Affected by such things as: Output Power Pout Optimization η = (77) Efficiency Surface reflections OO Pin Mode Pump-mode overlap Selection Pumping of states other Linewidth, than laser transition Stabilization, and Tuning Scattering and Summary absorption Spontaneous emission Quantum efficiency Quantum Efficiency

ECE 455 Lecture 4 Sets the upper bound on the efficiency of the laser. Gain Saturation Difference between energy N2 Optical τ21 Amplifiers levels is typically released N1 Threshold as heat.

Output Power Quantum efficiencies in R2 Laser τ10 Optimization Efficiency excess of 100% are possible. Thermal energy Mode N Selection may be removed by the 0 Linewidth, Stabilization, laser; making the system a and Tuning refrigerator! νL λpump Summary ηQ = = (78) S. R. Bowman IEEE J. νpump λL Quantum Electron. 35 115 (1999). Slope Efficiency

ECE 455 Lecture 4

Gain Describes how much power Saturation

Optical is added to the output for Amplifiers an infinitesimal increase in Threshold the pump power Output Power Units depend on pumping dPout Optimization ηslope = (79) Efficiency scheme dPpump Mode W/W for optical Selection pumping Linewidth, Stabilization, W/mA for electrical and Tuning pumping Summary Other units possible Example: Calculating Efficiency

ECE 455 Problem: An Erbium-doped ﬁber laser is pumped at 980 nm Lecture 4 and lases at 1540 nm. Calculate the quantum, and the optical

Gain to optical, and the slope eﬃciencies at 250 mW and 500 mW Saturation pump power on the diagram (based on data from [1]). Optical Ampliﬁers

Threshold

Output Power

Optimization ) Eﬃciency

Mode 2.5 Selection

Linewidth, Stabilization, and Tuning

Summary 0.86 Laser Power (mW Power Laser

120 250 500 Pump Power (mW) Example: Calculating Eﬃciency

ECE 455 Solution: The quantum eﬃciency is a property of the gain Lecture 4 medium, independent of input or output power.

Gain 980 nm Saturation ηQE = = 0.6364 (80) Optical 1540 nm Ampliﬁers The optical to optical eﬃciencies are: Threshold

Output Power 0.86 mW ηOO (250 mW) = = 0.00344 (81) Optimization 250 mW Eﬃciency 2.5 mW Mode ηOO (500 mW) = = 0.005 (82) Selection 500 mW

Linewidth, (83) Stabilization, and Tuning For this idealized data, the slope does not change between the Summary points. The slope eﬃciency is therefore: 2.5 − 0.86mW η = = 0.00656 (84) slope 500 − 250mW Selecting a Mode

ECE 455 Lecture 4 Our picture of laser oscillation looks like this: 1 The laser cavity deﬁnes a discrete number of optical Gain Saturation modes, each with a its own characteristic frequency and

Optical physical shape. Ampliﬁers 2 Threshold Associated with each of the modes above is a gain. In

Output Power addition to frequency dependent material gain, the gain is

Optimization affected by frequency dependent mirror losses and mode Efficiency dependent diffraction losses. Mode Selection 3 The mode with the highest net gain will depopulate the Linewidth, upper state of the lasing transition. Most other modes will Stabilization, and Tuning no longer be able to lase. Summary 4 If the mode selected is unable to fully depopulate the upper state (inhomogeneous broadening, spatial hole burning), then several modes may oscillate simultaneously. Multiple Mode Lasers

ECE 455 Lecture 4 In an ideal, homogeneously-broadened laser, only one mode will oscillate. However, if various modes are not competing for the Gain Saturation same atoms, multiple modes may lase simultaneously. Optical Ampliﬁers

Threshold Consider the following three scenarios: Output Power 1 Not all emitters are capable of emitting the frequency of Optimization the mode with highest gain Eﬃciency

Mode 2 Some emitters fall in the node of a mode’s standing wave Selection pattern Linewidth, Stabilization, 3 The pumped area in the transverse direction is larger than and Tuning

Summary the mode size Secondary modes may or may not take power from the highest-gain mode 1 Inhomogeneous Broadening

ECE 455 Lecture 4 The gain of an inhomogeneous medium saturates differently Gain Saturation than a homogeneous medium. It can be show that the gain Optical goes as: Amplifiers γ γ = 0 (85) Threshold 1/2 1 + I Output Power Isat Optimization Efficiency For a linear cavity, we use the low loss approximation to find Mode the output power of a given mode: Selection Linewidth, " 2 # Stabilization, Toc AIsat γ0 and Tuning P = − 1 (86) out 2 γ Summary th Inhomogeneous Saturation

ECE 455 Lecture 4

Gain Saturation

Optical Ampliﬁers Laser Threshold γ(ν) Output Power γ(ν) Optimization Eﬃciency Laser Mode Selection Linewidth, ν ν Stabilization, 0 ν 0 ν and Tuning

Summary Saturation of Inhomogeneous Medium in a Cavity

ECE 455 Lecture 4

Gain Saturation Optical Laser Ampliﬁers γ (ν) Threshold 0 Output Power γ(ν) Optimization Eﬃciency Cavity Modes Mode Selection

Linewidth, Stabilization, γth and Tuning

Summary

νq-2 νq-1 ν0 νq νq+1 ν 2 Multiple Longitudinal Modes

ECE 455 Lecture 4 qth mode Gain Saturation

Optical Ampliﬁers … …

Threshold

Output Power

Optimization Eﬃciency th Mode (q+1) mode Selection

Linewidth, Stabilization, and Tuning … …

Summary 2 Multiple Longitudinal Modes

ECE 455 Let λ be the wavelength of the mode with the highest gain Lecture 4 q In linear cavities, the forward and reverse propagating

Gain waves form a standing wave pattern Saturation E0 ı 2π z −ı 2π z 2π Optical λq λq Ampliﬁers E(z) = e + e = E0cos z (87) 2 λq Threshold

Output Power Population inversion is only saturated in regions of high Optimization intensity Eﬃciency ∗ 2 Mode E (z)E(z) |E0| 2 2π Selection I (z) = = cos z (88) 2η 2η λ Linewidth, q Stabilization, and Tuning Standing wave pattern of adjacent longitudinal modes ◦ Summary (λq−1 and λq+1) must at some point be 180 out of phase with each other Adjacent longitudinal modes therefore encounter unsaturated gain medium 2 Multiple Longitudinal Modes

ECE 455 Lecture 4

Gain Saturation Optical Prevents entire gain volume from being used Amplifiers Threshold Output power lowered if laser is forced to oscillate with Output Power only one transverse mode Optimization Efficiency Effect known as spatial hole burning Mode Ring lasers are immune to spatial hole burning because Selection there is no standing wave pattern Linewidth, Stabilization, and Tuning

Summary 3 Multiple Transverse Modes

ECE 455 Lecture 4

Gain Saturation

Optical Ampliﬁers

Threshold

Output Power

Optimization Eﬃciency Mode TEM mode unable to saturate gain medium much Selection 00

Linewidth, beyond w(z) Stabilization, and Tuning If pumped region is larger than w(z), other transverse

Summary modes may oscillate Multiple transverse modes are generally undesirable because higher order modes diverge faster and can’t be focused 3 Transverse Mode Intensity Overlap

ECE 455 The following equation represents one possible metric for the Lecture 4 overlap of the TEMmp and TEMm0p0 intensity proﬁles Gain RR p Saturation Imp(x, y)Im0p0 (x, y)dxdy α 0 0 = (89) Optical mp,m p RR RR 0 0 Ampliﬁers Imp(x, y)dxdy Im p (x, y)dxdy

Threshold α 00 10 20 01 11 21 02 12 22 Output Power

Optimization 00 1.00 0.80 0.68 0.80 0.64 0.55 0.68 0.55 0.47 Eﬃciency 10 0.80 1.00 0.80 0.64 0.80 0.64 0.55 0.68 0.55 Mode Selection 20 0.68 0.80 1.00 0.55 0.64 0.80 0.47 0.55 0.68

Linewidth, 01 0.80 0.64 0.55 1.00 0.80 0.68 0.80 0.64 0.55 Stabilization, and Tuning 11 0.64 0.80 0.64 0.80 1.00 0.80 0.64 0.80 0.65 Summary 21 0.55 0.64 0.80 0.68 0.80 1.00 0.55 0.65 0.80 02 0.68 0.55 0.47 0.80 0.64 0.55 1.00 0.80 0.68 12 0.55 0.68 0.55 0.64 0.80 0.65 0.80 1.00 0.80 22 0.47 0.55 0.68 0.55 0.65 0.80 0.68 0.80 1.00 3 Beam Proﬁle of Laser with Multiple Transverse Modes

ECE 455 Lecture 4

Gain Saturation

Optical Ampliﬁers

Threshold

Output Power

Optimization Eﬃciency

Mode Selection

Linewidth, Stabilization, and Tuning

Summary Not easy to determine beam quality by eye Intensities of each mode chosen randomly for ﬁgure Ampliﬁed Spontaneous Emission (ASE)

ECE 455 Lecture 4

Gain Saturation Photons spontaneously emitted along the length of the Optical cavity are ampliﬁed Ampliﬁers

Threshold Although in the same general direction as the output, not

Output Power in the same cavity mode Optimization Important when γL 1 Eﬃciency g

Mode Generally undesirable Selection Adds noise Linewidth, Stabilization, Diﬃcult to focus and Tuning Suppressed by laser or signal Summary Phase Noise

ECE 455 Lecture 4

Gain Saturation

Optical Ampliﬁers Threshold Output Power

Optimization Eﬃciency

Mode Selection

Linewidth, Stabilization, and Tuning

Summary Phase Noise

ECE 455 Lecture 4 While the laser is operating, the spontaneous emission is Gain Saturation clamped at its threshold value Optical Ampliﬁers During operation, a photon may spontaneously emit into

Threshold the lasing mode Output Power This photon’s phase is unrelated to the phase of the lasing Optimization Eﬃciency mode Mode But the gain this photon will experience is the same as the Selection already lasing mode Linewidth, Stabilization, and Tuning The phase of the output undergoes a random walk Summary This is the ultimate limit on the minimum linewidth of a laser Schawlow-Townes Linewidth

ECE 455 Lecture 4

Gain Saturation The minimum linewidth of a laser was derived by Schawlow Optical Ampliﬁers and Townes to be:

Threshold πhν(∆ν )2 Output Power c ∆νlaser = (90) Optimization Pout Eﬃciency Mode Where ∆νc is the resonator bandwidth (FWHM). In practice, Selection lasers have a much larger than this limit due to vibrations, Linewidth, Stabilization, power supply instabilities, and other considerations. and Tuning

Summary Example: Linewidth

ECE 455 Problem: A helium neon laser operates at 632.8 nm, is L = Lecture 4 30 cm long, has end mirror reﬂectivities of Roc = .98 and Rhr

Gain = .999, and a power of 10 mW. Find the linewidth of the laser Saturation as predicted by Schalow and Townes. Optical Amplifiers Solution: The first step is to find the cavity FWHM: Threshold 1/2 ! c 1 − (Rhr Roc ) Output Power ∆ν = (91) c 2nL π(R R )1/4 Optimization hr oc Efficiency = 844 kHz (92) Mode Selection The Schalow-Townes linewidth is then: Linewidth, Stabilization, 2 and Tuning πhc(∆νc ) ∆νlaser = (93) Summary λPout = 7.02 × 10−5 Hz (94) A real HeNe laser has a bandwidth around 5 GHz. Other sources of noise dominate. Mechanical Vibrations

ECE 455 Lecture 4 Mechanical vibrations may slightly change the distance between the end mirrors Gain Saturation A change of λ/(2F ) is all it takes to bring a frequency Optical from resonance to anti-resonance (where F is ﬁnesse) Ampliﬁers

Threshold Small changes in mirror separation will result in output

Output Power freqeuncy changing slightly. Output power may change as

Optimization the dominant mode tunes through the gain profile. See Efficiency slide 80. Mode Selection If the mirrors are pulled far enough apart that the qth Linewidth, Stabilization, mode no longer has the highest gain of all modes, laser and Tuning will hop to another mode. See slide 81. Summary Change in cavity resonance will sweep laser across gain bandwidth Vibrations are more difficult to control in large lasers Mechanical Vibrations: Tuning

ECE 455 Lecture 4

Gain Saturation Laser Optical Ampliﬁers Threshold Resonance Shift Output Power

Optimization Eﬃciency Cavity Modes Mode Selection γ(ν) Linewidth, Stabilization, γth and Tuning

Summary

ν νq-2 νq-1 0 νq νq+1 ν Mechanical Vibrations: Hopping

ECE 455 Lecture 4

Gain Saturation

Optical Ampliﬁers Laser Threshold Output Power Resonance Shift Optimization Eﬃciency Cavity Modes Mode Selection γ(ν) Linewidth, Stabilization, γth and Tuning

Summary

νq-2 νq-1 ν0 νq νq+1 ν Frequency Stabilization

ECE 455 Lecture 4

Gain Saturation

Optical Ampliﬁers

Threshold

Output Power

Optimization Eﬃciency

Mode Selection

Linewidth, Stabilization, and Tuning

Summary Lamb Dip Stabilization

ECE 455 Lecture 4

Gain Saturation

Optical Ampliﬁers

Threshold Doppler-broadened absorption cell inside laser cavity Output Power Loss is lowest when laser Optimization Eﬃciency Iodine-stabilized HeNe lasers are common frequency

Mode standards Selection

Linewidth, Stabilization, and Tuning

Summary Tuning a Laser

ECE 455 Lecture 4

Gain Saturation

Optical Tunable lasers useful for spectroscopy Ampliﬁers Use etalons, gratings, or other ﬁlters to increase loss at all Threshold wavelengths which are not desired Output Power

Optimization Carefully designed so frequency tunes continuously and Eﬃciency doesn’t ’hop’ Mode Selection Commonly used with broadband gain media, including: dye Linewidth, Stabilization, lasers, Ti:Al2O3, and external cavity diode lasers(ECDLs) and Tuning

Summary Grating Feedback Conﬁgurations

ECE 455 Lecture 4 Output Gain Saturation

Optical Ampliﬁers Grating Threshold Laser Diode Lens Output Power Optimization Littman-Metcalf Eﬃciency

Mode Output Selection

Linewidth, Stabilization, and Tuning Grating Summary Laser Diode Lens Litrrow Fabry-Perot Etalon

ECE 455 Lecture 4

Gain Saturation

Optical Ampliﬁers

Threshold Designed to have an FSR much larger than the gain bandwidth

Output Power May be used to both tune laser and narrow linewidth Can also

Optimization be used to stabilize absolute frequency, but are eﬀected by Eﬃciency temperature changes Mode Selection

Linewidth, Stabilization, and Tuning

Summary Lasing Oﬀ Line Center I

ECE 455 Lecture 4 Consider output power predicted by the low loss

Gain approximation Saturation Optical Toc TbIsat γ0 Amplifiers P = A − 1 (38) out 2 γ Threshold th Output Power Lasers can be forced to lase at a specific wavelength with Optimization Efficiency an etalon, diffraction grating, or other filter. The output Mode power would then be: Selection

Linewidth, Stabilization, Toc Tb hc σse (λ)∆N and Tuning Pout (λ) = A − 1 (95) 2 λσse (λ)τ γth Summary T T hc ∆N 1 = A oc b − (96) 2 λτ γth σse (λ) Lasing Oﬀ Line Center II

ECE 455 Lecture 4

Gain In general the variation of σ is much faster than the Saturation se

Optical variation of λ Amplifiers Even if the filter is perfectly lossless, output power will still Threshold drop significantly as the laser is tuned away from the gain Output Power center Optimization Efficiency In macroscopic cavities, the FSR is generally much smaller Mode Selection than the linedwidth, hence there will always be a mode Linewidth, close to line center. Stabilization, and Tuning In Equation 96, as σse → 0, Pout → −∞. What does this Summary mean? Example: Lasing Off Line Center I

ECE 455 Problem: A prism-tuned dye laser is shown below, along an Lecture 4 absorption and relative emission spectrum of the dye. Find the

Gain output power as a function of wavelength. Assume the tuning Saturation mechanism has a ﬂat spectral response. Other necessary Optical Ampliﬁers parameters are Lg = 5 mm, Roc = .9, Rgrating = 0.68, n=1.3, 16 −3 Threshold ∆N0 = 10 cm , rbeam = 100 µm, and τ21 = 4.08 ns. Output Power

Optimization Eﬃciency

Mode Selection

Linewidth, Stabilization, Output Dye Cell Grating and Tuning Beam Coupler Summary Expander Example: Lasing Oﬀ Line Center II

ECE 455 Lecture 4

Gain Saturation

Optical Ampliﬁers

Threshold

Output Power

Optimization Eﬃciency

Mode Selection

Linewidth, Stabilization, and Tuning

Summary Example: Lasing Oﬀ Line Center III

ECE 455 Solution: The ﬁrst step will be to calculate the gain spectrum. Lecture 4 First, the lineshape function is derived from the ﬂuorescence

Gain data: Saturation Iem(λ) gλ(λ) = (97) Optical R I (λ)dλ Ampliﬁers em Threshold which can be evaluated with a computer. The next step is Output Power convert the lineshape into a stimulated emission cross section Optimization Eﬃciency λ2 1 λ4 Mode σse (λ) = A21 2 g(ν) = 3 gλ(λ) (98) Selection 8πn τ21 8πn c Linewidth, Stabilization, where the formula on the right contains only known quantities. and Tuning

Summary The saturation intensity is: hc Isat (λ) = (99) λσse (λ)τ21 Example: Lasing Oﬀ Line Center IV

ECE 455 Lecture 4 The small-signal gain of the system is:

Gain γ0(λ) = σse (λ)∆N (100) Saturation

Optical Ampliﬁers The threshold gain is: Threshold 1 Output Power γth(λ) = α(λ) − ln Roc Rgrating (101) 2Lg Optimization Eﬃciency

Mode Finally, the output power is: Selection Linewidth, Isat (λ) γ0(λ) Stabilization, P(λ) = AToc − 1 (102) and Tuning 2 γth(λ) Summary Note all of the quantities which are dependent on wavelength. A plot of the power vs wavelength is shown on the next slide. Compare with the ﬂuorescence spectra on slide 90. Example: Lasing Oﬀ Line Center

ECE 455 Lecture 4

Gain Saturation

Optical Ampliﬁers

Threshold

Output Power

Optimization Eﬃciency

Mode Selection

Linewidth, Stabilization, and Tuning The Lessons: Summary Compared to the ﬂuorescence, the peak power has been shifted to the red (longer wavelengths) because of the absorption at shorter wavelengths Output power is a nonlinear function of gain. Master Oscillator Power Ampliﬁer

ECE 455 Lecture 4

Gain Saturation

Optical Ampliﬁers High-powered lasers require a larger gain medium and

Threshold pump Output Power More diﬃcult to stabilize large lasers rather than small Optimization Eﬃciency ones Mode Build a small laser which is highly stable, and then amplify Selection it Linewidth, Stabilization, and Tuning

Summary Summary

ECE 455 Lecture 4

Gain A population inversion increases the photon density of the Saturation optical field, but the photon density decreases the Optical Amplifiers population inversion Threshold An ideal homogeneously broadened laser will oscillate at Output Power exactly one frequency Optimization Efficiency Our output power equations are only approximate. We

Mode assumed uniform pumping as well as uniform saturation of Selection the entire gain medium. Neither of these is is a Linewidth, Stabilization, particularly good assumption. and Tuning

Summary The linewidth in unstabilized lasers is usually limited by ’technical noise’ ECE 455 [1] Lecture 4

Gain Saturation

Optical Ampliﬁers

Threshold

Output Power

Optimization Eﬃciency

Mode Selection

Linewidth, Stabilization, and Tuning

Summary