262CHAPTER7.KERR-LENSANDADDITIVEPULSEMODELOCKING7.1.2Two-Mirror ResonatorsWe consider the two mirror resonator shown in Figure 7.4.Optical ElementABCD-MatrixLFree Space Distance L01Thin Lens with-01/ ffocal lengthfMirror under Angle0 to Axis and Radius R2cosSagittal PlaneRMirror under Angle10 to Axis and Radius R-21RcosTangential PlaneBrewsterPlateunderAngle to Axis and Thicknessn0d, Sagittal PlaneBrewster Plate under(11Angle to Axis and Thickness0d, Tangential PlaneTable7.1:ABCDmatrices for commonlyused optical elements.trR2R1ZM1M2LFigure7.4:Two-Mirror Resonator with curved mirrors with radi of curvatureRi and R2The resonator can be unfolded for an ABCD-matrix analysis, see Figure7.5
262CHAPTER 7. KERR-LENS AND ADDITIVE PULSE MODE LOCKING Optical Element ABCD-Matrix Free Space Distance L µ 1 L 0 1 ¶ Thin Lens with focal length f µ 1 0 −1/f 1 ¶ Mirror under Angle θ to Axis and Radius R Sagittal Plane µ 1 0 −2 cos θ R 1 ¶ Mirror under Angle θ to Axis and Radius R Tangential Plane µ 1 0 −2 R cos θ 1 ¶ Brewster Plate under Angle θ to Axis and Thickness d, Sagittal Plane µ 1 d n 0 1 ¶ Brewster Plate under Angle θ to Axis and Thickness d, Tangential Plane µ 1 d n3 0 1 ¶ Table 7.1: ABCD matrices for commonly used optical elements. Figure 7.4: Two-Mirror Resonator with curved mirrors with radii of curvature R1 and R2. The resonator can be unfolded for an ABCD-matrix analysis, see Figure 7.5. 7.1.2 Two-Mirror Resonators We consider the two mirror resonator shown in Figure 7.4
2637.1. KERR-LENS MODE LOCKING (KLM)1122f2fifCzLLFigure 7.5: Two-mirror resonator unfolded. Note, only one half of the fo-cusing strength of mirror 1 belongs to a fundamental period describing oneresonator roundtrip.Theproduct of ABCD matrices describing one roundtrip according toFigure 7.5 are then given by上0L()()()(1M(7.13)111wherefi=R/2,and f2=R2/2.To carry outthisproduct and toformulatethe cavity stability criteria, it is convenient to use the cavity parametersgi=1-L/Ri,i=1,2.The resulting cavity roundtripABCD-matrix can bewritten in theform(2g192 - 1)2g2LABM :(7.14)T2g1 (9192-1) /L(2g192-1)ResonatorStabilityThe ABCD matrices describe the dynamics of rays propagating inside theresonator. An optical ray is characterized by the vector r=whereris the distance from the optical axis and r' the slope of the ray to the opticalaxis. The resonator is stable if no ray escapes after many round-trips, whichis the case when the eigenvalues of the matrix M are less than one. Sincewe have a lossless resonator, i.e. detjM = 1, the product of the eigenvalueshastobe1and,therefore,thestableresonatorcorrespondstothecaseofacomplex conjugate pair of eigenvalues with a magnitude of 1. The eigenvalue
7.1. KERR-LENS MODE LOCKING (KLM) 263 Figure 7.5: Two-mirror resonator unfolded. Note, only one half of the focusing strength of mirror 1 belongs to a fundamental period describing one resonator roundtrip. The product of ABCD matrices describing one roundtrip according to Figure 7.5 are then given by M = µ 1 0 −1 2f1 1 ¶ µ 1 L 0 1 ¶ µ 1 0 −1 f2 1 ¶ µ 1 L 0 1 ¶ µ 1 0 −1 2f1 1 ¶ (7.13) where f1 = R1/2, and f2 = R2/2. To carry out this product and to formulate the cavity stability criteria, it is convenient to use the cavity parameters gi = 1 − L/Ri, i = 1, 2. The resulting cavity roundtrip ABCD-matrix can be written in the form M = µ (2g1g2 − 1) 2g2L 2g1 (g1g2 − 1) /L (2g1g2 − 1) ¶ = µ A B C D ¶ . (7.14) Resonator Stability The ABCD matrices describe the dynamics of rays propagating inside the resonator. An optical ray is characterized by the vector r=µ r r0 ¶ , where r is the distance from the optical axis and r0 the slope of the ray to the optical axis. The resonator is stable if no ray escapes after many round-trips, which is the case when the eigenvalues of the matrix M are less than one. Since we have a lossless resonator, i.e. det|M| = 1, the product of the eigenvalues has to be 1 and, therefore, the stable resonator corresponds to the case of a complex conjugate pair of eigenvalues with a magnitude of 1. The eigenvalue
264CHAPTER7.KERR-LENSANDADDITIVEPULSEMODELOCKINGequation to M is given by2g2L(2g192 - 1) - 入(7.15)det [M - >-1|= det2g1 (9192 - 1) /L (2g192 - 1) 2 - 2(2g192 1) 入+ 1 = 0.(7.16)The eigenvalues are入1/2=(2g1921) ±V(2g192 1)2 1,(7.17)exp(±0),cosh = 2g192-1, for [2g192-1|>1(x(+0),=29, 21- (718)The case of a complex conjugate pair with a unit magnitude corresponds toa stable resontor.Therfore. the stability criterion for a stable two mirrorresontoris[29192 - 1]| ≤ 1.(7.19)The stable and unstable parameter ranges are given by(7.20)stable:0≤91·92=S≤1(7.21)unstable：9192≤ 0; or 9192 ≥1.where S = gi -g2, is the stability parameter of the cavity.The stabil-ity criterion can be easily interpreted geometrically.Of importance arethe distances between the mirror mid-points M, and cavity end points, i.e.gi=(R,-L)/R,=-S:/Ri,as shown in Figure7.6.trRGS2?ZM2M1LFigure 7.6: The stability criterion involves distances between the mirror mid-points M, and cavity end points. i.e. gi = (R, - L)/R, = -Si/R
264CHAPTER 7. KERR-LENS AND ADDITIVE PULSE MODE LOCKING equation to M is given by det|M − λ · 1| = det ¯ ¯ ¯ ¯ µ (2g1g2 − 1) − λ 2g2L 2g1 (g1g2 − 1) /L (2g1g2 − 1) − λ ¶¯ ¯ ¯ ¯ = 0, (7.15) λ2 − 2 (2g1g2 − 1) λ +1=0. (7.16) The eigenvalues are λ1/2 = (2g1g2 − 1) ± q (2g1g2 − 1)2 − 1, (7.17) = ½ exp (±θ), cosh θ = 2g1g2 − 1, for |2g1g2 − 1| > 1 exp (±jψ), cos ψ = 2g1g2 − 1, for |2g1g2 − 1| ≤ 1 .(7.18) The case of a complex conjugate pair with a unit magnitude corresponds to a stable resontor. Therfore, the stability criterion for a stable two mirror resontor is |2g1g2 − 1| ≤ 1. (7.19) The stable and unstable parameter ranges are given by stable : 0 ≤ g1 · g2 = S ≤ 1 (7.20) unstable : g1g2 ≤ 0; or g1g2 ≥ 1. (7.21) where S = g1 · g2, is the stability parameter of the cavity. The stability criterion can be easily interpreted geometrically. Of importance are the distances between the mirror mid-points Mi and cavity end points, i.e. gi = (Ri − L)/Ri = −Si/Ri, as shown in Figure 7.6. Figure 7.6: The stability criterion involves distances between the mirror midpoints Mi and cavity end points. i.e. gi = (Ri − L)/Ri = −Si/Ri
2657.1. KERR-LENS MODE LOCKING (KLM)Thefollowing rules for a stable resonator can bederived from Figure7.6using the stability criterion expressed in terms of the distances Si. Note, thatthe distances and radii can be positive and negativeSiS2(7.22)stable:0<<1.RR2Therules are:.A resonator is stable,if themirrorradii,laid out alongtheoptical axis.overlapA resonator isunstable,if theradii do not overlap or one lies withinthe other.Figure 7.7 shows stable and unstable resonator configurationsSTABLEUNSTABLERR2Figure 7.7: Illustration of stable and unstable resonator configurationsFigurebyMITOCW.For a two-mirror resonator with concave mirrors and Ri ≤ R2, we obtainthe general stability diagram as shown in Figure 7.8. There are two rangesforthemirrordistanceL,withinwhichthecavityisstable,O<L<Ri and
7.1. KERR-LENS MODE LOCKING (KLM) 265 The following rules for a stable resonator can be derived from Figure 7.6 using the stability criterion expressed in terms of the distances Si. Note, that the distances and radii can be positive and negative stable : 0 ≤ S1S2 R1R2 ≤ 1. (7.22) The rules are: • A resonator is stable, if the mirror radii, laid out along the optical axis, overlap. • A resonator is unstable, if the radii do not overlap or one lies within the other. Figure 7.7 shows stable and unstable resonator configurations. Figure 7.7: Illustration of stable and unstable resonator configurations. For a two-mirror resonator with concave mirrors and R1 ≤ R2, we obtain the general stability diagram as shown in Figure 7.8. There are two ranges for the mirror distance L, within which the cavity is stable, 0 ≤ L ≤ R1 and STABLE UNSTABLE R1 R1 R1 R1 R2 R2 R2 R2 R2 R2 Figure by MIT OCW
266CHAPTER7.KERR-LENSANDADDITIVEPULSEMODELOCKINGLo'R'R2R,+R2Figure7.8:Stabileregions (black)for thetwo-mirror resonatorR2 ≤ L ≤ Ri + R2. It is interesting to investigate the spot size at the mirrorsand the minimum spot size inthe cavity as a function of the mirror distanceL.Resonator Mode CharacteristicsThe stablemodes of theresonator reproduce themselves after one round-trip,i.e. from Eq.(7.10) we findAqi + B(7.23)91Cqi + DThe inverse q-parameter, which is directly related to the phase front curva-ture and the spot size of the beam, is determined by1- ADA-D(1=0.(7.24)BB219qThe solution isA-D/(A + D)_1(7.25)2|B2Bq)1/2If we apply this formula to (7.15), we find the spot size on mirror 1入12V(A+ D)"-1=-(7.26)元W?9)1/2or2入L192w4(7.27)T911-9192LR2- L(ΛRi(7.28)R1 -LRi+R2T
266CHAPTER 7. KERR-LENS AND ADDITIVE PULSE MODE LOCKING Figure 7.8: Stabile regions (black) for the two-mirror resonator. R2 ≤ L ≤ R1 +R2. It is interesting to investigate the spot size at the mirrors and the minimum spot size in the cavity as a function of the mirror distance L. Resonator Mode Characteristics The stable modes of the resonator reproduce themselves after one round-trip, i.e. from Eq.(7.10) we find q1 = Aq1 + B Cq1 + D (7.23) The inverse q-parameter, which is directly related to the phase front curvature and the spot size of the beam, is determined by µ1 q ¶2 + A − D B µ1 q ¶ + 1 − AD B2 = 0. (7.24) The solution is µ1 q ¶ 1/2 = −A − D 2B ± j 2 |B| q (A + D) 2 − 1 (7.25) If we apply this formula to (7.15), we find the spot size on mirror 1 µ1 q ¶ 1/2 = − j 2 |B| q (A + D) 2 − 1 = −j λ πw2 1 . (7.26) or w4 1 = µ2λL π ¶2 g2 g1 1 1 − g1g2 (7.27) = µλR1 π ¶2 R2 − L R1 − L µ L R1 + R2 − L ¶ . (7.28)