Lecture 20- ECE 240a Distributed Feedback Lasers 1 ECE 240a Lasers - - PowerPoint PPT Presentation

Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality

Lecture 20- ECE 240a

Distributed Feedback Lasers

ECE 240a Lasers - Fall 2019 Lecture 20 1

Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality

Distributed Feedback Lasers

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

ECE 240a Lasers - Fall 2019 Lecture 20 2

Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality

Mode Coupling

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.

ECE 240a Lasers - Fall 2019 Lecture 20 3

Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality

Power and Mode Orthogonally

The normalized transverse field components for each mode in the waveguide for the diode laser can be used as an orthonormal basis to express the transverse dependence of an arbitrary field. Start with spatial part of the electric field for mode j Ej(r) = ej(x, y)e−jβjz, (1) where ej(x, y) is a normalized field that produces a unit-power transfer along the z-axis. The corresponding expression for the magnetic field is Hj(r) = hj(x, y)e−jβjz. (2) The index j is defined so that the following relations hold β−j = −βj ej = e−j h−j = −hj (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.

ECE 240a Lasers - Fall 2019 Lecture 20 4

Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality

Mode Expansions

The normalized modes form an orthonormal basis . To derive relationship, define the average cross-power density between two fields as Sjk(r) . = 1 2Re[Sjk] = 1 2Re ej(r) × h∗

k(r)

, (4) where Sjk 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) =

• A

Save(r, t)·da with da = z dA. The orthogonality condition for the normalized fields ej(x, y) and hk(x, y) can be written as 1 2

• A
• ej(x, y) × h∗

k(x, y)

· z dA = ±δjk, (5) 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.

ECE 240a Lasers - Fall 2019 Lecture 20 5

Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality

Power Flow

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 etj(x, y) and htk(x, y) 1 2

• A
• etj(x, y) × h∗

tk(x, y)

dA = ±δjk, (6) 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.

ECE 240a Lasers - Fall 2019 Lecture 20 6

Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality

Example - Power Flow for a TE Mode

Consider a TE mode in a slab waveguide given with an electric field given by Ey(x, z) = Ey(x)e−jβz y (7) The magnetic field is determined using time-harmonic form of Maxwell’s equations and (7)

H(x, z) = j ωµ0 ∇ × Ey(x, z) y =

• −

β ωµ0 Ey(x)

• Hx(x)
• x +

j ωµ0 ∂Ey(x) ∂x

• Hz(x)
• z

e−jβz. (8)

The average power density Save(r) of the field is

Save(r) = 1 2 Re E(r) × H∗(r) = 1 2 Re Ey(x, z) y × H∗(x, z) = 1 2 Re

• −

j ωµ0 Ey(x) ∂E∗ y (x) ∂x x + β ωµ0

• Ey(x)2

z

• =

β 2ωµ0

• Ey(x)2

z. (9)

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.

ECE 240a Lasers - Fall 2019 Lecture 20 7

Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality

Modal Expansions

Express an arbitrary transverse field as expansion of modes Et(x, y) =

• k

aketk(x, y) (10) Ht(x, y) =

• k

akhtk(x, y) (11) The z-component of each field follows from Maxwell’s equations and has the same e−jβz dependence. Define ek(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 E(r) =

• k

akek(x, y)e−jβkz (12) H(r) =

• k

akhk(x, y)e−jβkz (13) Summation runs over both positive and negative values of k with the sign

• f the fields and the propagation constant following the sign convention

given in (3).

ECE 240a Lasers - Fall 2019 Lecture 20 8

Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality

Expansion Coefficients

To determine the coefficient ak, we cross h∗

tj(x, y) on the right of each

side of (10) and integrate over the cross-sectional area to yield

• A

Et(x, y) × h∗

tj(x, y)dA

=

• k

ak

• A

etk(x, y) × h∗

tj(x, y)dA

• 2δjk

(14) Applying the orthogonality condition given in (6), we have aj = ±1 2

• A

Et(x, y) × h∗

tj(x, y)dA

= ±1 2

• A
• E(x, y) × h∗

j(x, y)

· z dA (15) where the second expression uses the complete field.

ECE 240a Lasers - Fall 2019 Lecture 20 9

Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality

Mode Coupling

For an ideal waveguide with no perturbations , the coefficients aj 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.

ECE 240a Lasers - Fall 2019 Lecture 20 10

Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality

Formalism of Mode Coupling

Start with time-harmonic form of Maxwell’s equations ∇ × e = −jωµ0h (16) ∇ × h = jωεe (17) where e and h are normalized fields , and ε is the (complex) permittivity

• f an laser waveguide

The fields in a DFB waveguide with a periodic perturbation are given by ∇ × E = −jωµ0H (18) ∇ × H = jωε′E, (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

• f 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 ∇·(e∗ × H) = jω ε′e∗ · E − µ0h∗ · H , (20) where we have used the vector identity ∇ · (A × B) = A · (∇ × B) − B · (∇ × A) to simplify the left side.

ECE 240a Lasers - Fall 2019 Lecture 20 11

Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality

Mode Coupling 2

Now repeat this process for E field ∇ · (E × h∗) = −jω εe∗ · E − µ0h∗ · H . (21) Adding (20) to (21) yields ∇ · (E × h∗ + e∗ × H)

• S

= −jω∆ε e∗ · E, (22) 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 e∗

j(r)

= e∗

j(x, y)ejβjz,

(23) where ej(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.

ECE 240a Lasers - Fall 2019 Lecture 20 12

Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality

Mode Coupling 3

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 E(r) =

• k

ak(z)ek(x, y)e−jβkz (24) H(r) =

• k

ak(z)hk(x, y)e−jβkz (25) where ak(z) is the z-dependent coefficient of the expansion that is presumed to vary slowly with respect to the spatial variation of e−jβkz. Aside from a scaling constant, the term ak(z) is the spatial part of the complex signal envelope that is generated by the mode-coupling process.

ECE 240a Lasers - Fall 2019 Lecture 20 13

Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality

Mode Coupling Coefficients -1

Substitute ej(x, y) and h∗

j(x, y) into (22) and integrate over the

cross-sectional area A.

• A

∇ · (E × h∗

j + e∗ j×H

• Sj

) dA = −jωejβjz

• A

∆ε e∗

j·E dA. (26)

Apply the following identity to the left side

• A

∇ · Sj dA = ∂ ∂z

• A

Sj · z dA +

• LA

Sj · n dL, (27) where A is the transverse area in the (x, y) plane LA is a line integral around this area, and n is the outward normal unit vector to the contour of the line integral. The integral on A is evaluated over an infinite area and thus the line integral is evaluated at infinity and is zero for any guided mode

ECE 240a Lasers - Fall 2019 Lecture 20 14

Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality

Mode Coupling Coefficients -2

Set line integral to zero Substitute the form of unperturbed modes given in (23) into the right side of (27), Substitute the form of the perturbed field given in (24) and (25) on the right side of (27). Making these substitutions and interchanging the summation and the integration, we can write the left side of (22) as

∂

∂z − j(βj − βk)

k

ak(z)

• A
• e∗

j × hk + ek × h∗ j

• ·

z dA

• ±2δjk
• =

±2 daj(z) dz , (28)

where the orthogonality condition given in (5) has been used. Substituting this expression on the left side of (22) we obtain ±daj(z) dz = −1 2jωejβjz

• A

∆ε(x, y, z)e∗

j(x, y) · E(z, y, z)dA,(29)

where the functional dependence of each argument is written explicitly.

ECE 240a Lasers - Fall 2019 Lecture 20 15

Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality

Mode Coupling Coefficients -3

The right side of (29) depends on the form of the term ∆ε(r)E(r) = ∆P(r) where ∆P(r) is the change in the polarization caused by the perturbation. For a linear change, we can write the perturbation as ∆ε(r)E(r) = ∆ε(r)

• k

ak(z)ek(x, y)e−jβkz. (30) Substituting this expression on the right side of (29) and interchanging the summation and integration we have ±daj(z) dz = −j

• k

κjk(z) ej(βj−βk)zak(z), (31) where κjk(z) . = ω

• A

∆ε(x, y, z)e∗

j(x, y) · ek(x, y)dA

(32) is the coupling coefficient between mode j and mode k caused by the presence of the perturbation ∆ε.

ECE 240a Lasers - Fall 2019 Lecture 20 16

Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality

Periodic Perturbation

Write periodic perturbation as ∆ε(x, z) = ǫℓ(x)e−jKℓz where K = 2π/Λ is the fundamental spatial period and ℓ is a harmonic Suppose waveguide is single mode. Let a1(z) be the forward wave for the mode and let a2(z) be the backward wave

da1(z) dz = −jκ12a2(z) ej∆βz da2(z) dz = −jκ∗

12a1(z) e−j∆βz

where κ12 . = ω

• A

εℓ(x)e∗

1(x, y) · e2(x, y)dA

and ∆β = 2β − ℓK

For efficient coupling ∆β = 0. This is called phase matching

ECE 240a Lasers - Fall 2019 Lecture 20 17

Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality

Application to Lasers

In a laser, both the index and the gain are modified so write ε(x, y, z) = ε0(x, y, z)

• Unperturbed

+ ∆ε(x, y, z)

• Real perturbation

+j γ(x, y, z)

• gain

Assume gain is uniform in z. Define g as the gain coupling between the two waves so that g12(z) . = 2ω

• A

γ(x, y)e∗

j(x, y) · ek(x, y)dA

ECE 240a Lasers - Fall 2019 Lecture 20 18

Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality

Modification of Coupled Equations

Including the gain the in the coupled equations, we can write

da1(z) dz = −jκ12a2(z) ej∆β′z da2(z) dz = −jκ∗

12a1(z) e−j∆β′z

where κ12 . = ω

• A

εℓ(x)e∗

1(x, y) · e2(x, y)dA

and ∆β′ = ∆β + jg and ∆β = 2β − ℓK

ECE 240a Lasers - Fall 2019 Lecture 20 19

Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality

Solution

The form of the solution is a1(z) = a1(0)ej∆βz/2

s cosh [s(L − z)] + j

2 (∆β + jg) sinh [s(L − z)]

s cosh(sL) + j

2 (∆β + jg) sinh(sL)

• a2(z)

= a2(0)e−j∆βz/2

• −jκ∗ sinh [s(L − z)]

s cosh(sL) + j

2 (∆β + jg) sinh(sL)

• where

s =

• |κ|2 − [(∆β + jg) /2]2

∆β = 2β − ℓK

ECE 240a Lasers - Fall 2019 Lecture 20 20

Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality

Plot

ECE 240a Lasers - Fall 2019 Lecture 20 21

Lecture 20- ECE 240a DFB Lasers Mode Or- thogonality

Lasing Condition

The structure with gain will lase with the denominator goes to zero or s cosh(sL) + j 2 (∆β + jg) sinh(sL) = 0 Equation does not have analytical solutions.

ECE 240a Lasers - Fall 2019 Lecture 20 22