Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality Lecture 20- ECE 240a Distributed Feedback Lasers 1 ECE 240a Lasers - Fall 2019 Lecture 20
Distributed Feedback Lasers Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality Noise and linewidth depend on photon lifetime and quality of resonator Cleaved facet lasers are not high “Q” cavities Can create higher Q cavities using distributed feedback Forward Wave Backward Wave Periodic perturbation couples the two waves and provides the feedback mechanism for the laser 2 ECE 240a Lasers - Fall 2019 Lecture 20
Mode Coupling Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality Mode coupling is a common form of energy redistribution that occurs between the modes of a system. For lasers a periodic perturbation ∆ ε ( x , y , z ) can couple the forward and backward waves produces the necessary feedback for lasing. Mode coupling can be analyzed using coupled-mode theory. This approach provides an intuitive way of understanding many energy-redistribution effects, both for lasers and other structures. 3 ECE 240a Lasers - Fall 2019 Lecture 20
Power and Mode Orthogonally Lecture 20- ECE 240a The normalized transverse field components for each mode in the DFB Lasers waveguide for the diode laser can be used as an orthonormal basis to Mode Or- thogonality express the transverse dependence of an arbitrary field. Start with spatial part of the electric field for mode j e j ( x , y ) e − j β j z , E j ( r ) = (1) where e j ( x , y ) is a normalized field that produces a unit-power transfer along the z -axis. The corresponding expression for the magnetic field is h j ( x , y ) e − j β j z . H j ( r ) = (2) The index j is defined so that the following relations hold β − j = − β j e j = e − j h − j = − h j (3) If j is positive, then the mode propagates in the positive z -direction. If j is negative, then the mode propagates in the negative z -direction. 4 ECE 240a Lasers - Fall 2019 Lecture 20
Mode Expansions Lecture 20- ECE 240a The normalized modes form an orthonormal basis . DFB Lasers To derive relationship, define the average cross-power density between Mode Or- two fields as thogonality 2Re � k ( r ) � 1 2Re [ S jk ] = 1 . e j ( r ) × h ∗ S jk ( r ) = , (4) where S jk is the complex cross-power density. If j = k , then we recover the power density in a single mode. Now use this expression for the power density in � P ( z , t ) = S ave ( r , t ) · d a A with d a = � z d A . The orthogonality condition for the normalized fields e j ( x , y ) and h k ( x , y ) can be written as � � k ( x , y ) � 1 e j ( x , y ) × h ∗ · � = ± δ jk , (5) z d A 2 A where the positive sign is for modes propagating in the positive z -direction and the negative sign is for modes propagating in the negative z -direction. 5 ECE 240a Lasers - Fall 2019 Lecture 20
Power Flow Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality Examining (5), only the field components that lie in a plane transverse to � z will produce a cross-product term in the z -direction and a thus a real nonzero contribution to the power density. Therefore rewrite (5) in terms of the transverse field components e tj ( x , y ) and h tk ( x , y ) � � tk ( x , y ) � 1 e tj ( x , y ) × h ∗ = ± δ jk , (6) d A 2 A where the subscript reminds us that field vectors are transverse to the direction of propagation. Equation (6) is the basic equation for the orthogonality of modes. 6 ECE 240a Lasers - Fall 2019 Lecture 20
Example - Power Flow for a TE Mode Lecture 20- ECE 240a Consider a TE mode in a slab waveguide given with an electric field given by DFB Lasers = E y ( x ) e − j βz � Mode Or- E y ( x , z ) (7) y thogonality The magnetic field is determined using time-harmonic form of Maxwell’s equations and (7) j ∇ × Ey ( x , z ) � H ( x , z ) = y ωµ 0 � z � β j ∂Ey ( x ) � � e − j βz . = − Ey ( x ) x + (8) ωµ 0 ωµ 0 ∂x � �� � � �� � Hx ( x ) Hz ( x ) The average power density S ave ( r ) of the field is Re � E ( r ) × H ∗ ( r ) � Re � y × H ∗ ( x , z ) � 1 1 Ey ( x , z ) � S ave ( r ) = = 2 2 � � ∂E ∗ � Ey ( x ) � 2 � y ( x ) 1 j β ∂x � = − Ey ( x ) x + Re z 2 ωµ 0 ωµ 0 � Ey ( x ) � 2 � β = (9) z . 2 ωµ 0 Only the transverse magnetic field component along � x produces a real power density transfer along the z -axis. The axial field component along � z produces a standing wave with a reactive power component - not real power transfer. 7 ECE 240a Lasers - Fall 2019 Lecture 20
Modal Expansions Lecture 20- ECE 240a Express an arbitrary transverse field as expansion of modes � DFB Lasers E t ( x , y ) = a k e tk ( x , y ) (10) Mode Or- thogonality k � H t ( x , y ) = a k h tk ( x , y ) (11) k The z -component of each field follows from Maxwell’s equations and has the same e − j βz dependence. Define e k ( x , y ) as the vector sum of the transverse field component and the axial field component. Full field, which includes both forward and backward propagating waves for the laser field is now � a k e k ( x , y ) e − j β k z E ( r ) = (12) k � a k h k ( x , y ) e − j β k z H ( r ) = (13) k Summation runs over both positive and negative values of k with the sign of the fields and the propagation constant following the sign convention given in (3). 8 ECE 240a Lasers - Fall 2019 Lecture 20
Expansion Coefficients Lecture 20- ECE 240a DFB Lasers Mode Or- To determine the coefficient a k , we cross h ∗ thogonality tj ( x , y ) on the right of each side of (10) and integrate over the cross-sectional area to yield � � � E t ( x , y ) × h ∗ e tk ( x , y ) × h ∗ tj ( x , y ) d A = tj ( x , y ) d A (14) a k A A � �� � k 2 δjk Applying the orthogonality condition given in (6), we have � ± 1 E t ( x , y ) × h ∗ = tj ( x , y ) d A a j 2 A � � j ( x , y ) � ± 1 E ( x , y ) × h ∗ = · � (15) z d A 2 A where the second expression uses the complete field. 9 ECE 240a Lasers - Fall 2019 Lecture 20
Mode Coupling Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality For an ideal waveguide with no perturbations , the coefficients a j do not depend on z . In a DFB laser waveguides, a deliberate periodic perturbation is introduced to the waveguide that produce mode coupling between the forward and backward waves. This coupled provides the feedback. Note that periodic perturbation is for both the index and the gain. 10 ECE 240a Lasers - Fall 2019 Lecture 20
Formalism of Mode Coupling Lecture 20- ECE 240a Start with time-harmonic form of Maxwell’s equations DFB Lasers ∇ × e = − j ωµ 0 h (16) Mode Or- thogonality ∇ × h = (17) j ωε e where e and h are normalized fields , and ε is the (complex) permittivity of an laser waveguide The fields in a DFB waveguide with a periodic perturbation are given by ∇ × E = − j ωµ 0 H (18) j ωε ′ E , = ∇ × H (19) where the permittivity of the perturbed waveguide ε ′ is written as ε ′ ( x , y , z ) = ε ( x , y ) + ∆ ε ( x , y , z ) with ∆ ε ( x , y , z ) being the difference between the permittivity of the perturbed waveguide and the permittivity of the unperturbed waveguide. Now form the dot product of e ∗ with each side of (19), form the dot product of H with the complex conjugate of each side of (16), and subtract j ω � ε ′ e ∗ · E − µ 0 h ∗ · H � ∇· ( e ∗ × H ) = , (20) where we have used the vector identity ∇ · ( A × B ) = A · ( ∇ × B ) − B · ( ∇ × A ) to simplify the left side. 11 ECE 240a Lasers - Fall 2019 Lecture 20
Mode Coupling 2 Lecture 20- ECE 240a Now repeat this process for E field DFB Lasers − j ω � ε e ∗ · E − µ 0 h ∗ · H � Mode Or- thogonality ∇ · ( E × h ∗ ) = (21) . Adding (20) to (21) yields ∇ · ( E × h ∗ + e ∗ × H ) − j ω ∆ ε e ∗ · E , = (22) � �� � S where S is the complex cross-power density of the unperturbed field and the perturbed field. Each mode of the ideal unperturbed waveguide can be written as j ( x , y ) e j β j z , e ∗ e ∗ j ( r ) = (23) where e j ( x , y ) is the transverse dependence of the field for mode j and β is complex in a waveguide used for a laser. In general this field has both a transverse field component and an axial field component. 12 ECE 240a Lasers - Fall 2019 Lecture 20
Mode Coupling 3 Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality For a fixed value of z , the perturbation ∆ ε ( x , y , z ) is only a function of the transverse coordinates. Accounting for the z -dependence of the perturbation we can write � a k ( z ) e k ( x , y ) e − j β k z E ( r ) = (24) k � a k ( z ) h k ( x , y ) e − j β k z H ( r ) = (25) k where a k ( z ) is the z -dependent coefficient of the expansion that is presumed to vary slowly with respect to the spatial variation of e − j β k z . Aside from a scaling constant, the term a k ( z ) is the spatial part of the complex signal envelope that is generated by the mode-coupling process. 13 ECE 240a Lasers - Fall 2019 Lecture 20
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