Zhang K., Li D. Electromagnetic Theory for Microwaves and Optoelectronics

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7.8 Coupling of Modes 461

b(z) = a(0)

coshTLsinhT(z−L)−j

∆β

sinhTL sinhT(z−L)

1 +

|κ

sinh

T L

−jβz

. (7.250)

The power-ﬂows of the two modes are P

and P

, which ar proportional to

|a(z)|

and −|b(z)|

, respectively, and are given by

|a(z)|

= a(z)a

∗

(z) = |a(0)|

1 +

|κ

sinh

T (z − L)

1 +

|κ

sinh

T L

, (7.251)

|b(z)|

= b(z)b

∗

(z) = |a(0)|

|κ

sinh

T (z − L)

1 +

|κ

sinh

T L

, (7.252)

and

|a(z)|

− |b(z)|

= |a(0)|

1 +

|κ

sinh

T L

= |a(L)|

The power fed in mode 1 at z = 0 will monotonously transfer to mode 2

due to the coupling. It leads to the decreasing of power of mode 1 in the

direction of power ﬂow, +z. At the same time, the power transferred to

mode 2 propagates along −z, an monotonously increasing in the direction of

power ﬂow, −z, refer to Fig. 7.33(a). The total power |a(z)|

−|b(z)|

remain

unchange along z and is equal to the power at the output end of the coupler,

|a(L)|

For synchronous state, β

= β

, ∆β = 0, T

= |κ

. Suppose

a(0) 6= 0, b(L) = 0, (7.249) and (7.250) become

a(z) = a(0)

coshTz−sinhTLsinhT(z−L)

1 + sinh

T L

−jβz

=a(0)

coshT(z−L)

cosh T L

−jβz

, (7.253)

b(z) = a(0)

cosh T L sinh T (z − L)

1 + sinh

T L

−jβz

= a(0)

sinh T (z − L)

cosh T L

−jβz

. (7.254)

The power-ﬂow distributions along the two modes are

|a(z)|

= |a(0)|

cosh

T (z − L)

cosh

T L

, (7.255)

|b(z)|

= |a(0)|

sinh

T (z − L)

cosh

T L

, (7.256)

refer to Fig. 7.33(b).

An example of contra-directional coupling of two modes is the distributed

feedback (DFB) structure.

462 7. Periodic Structures and the Coupling of Modes

Figure 7.33: Power ﬂow distributions of the contra-directional coupled modes

for asynchronous state (a) and synchronous state (b).

Figure 7.34: Uniform dielectric waveguide (a) and periodic dielectric waveg-

uide (b), (c) and (d).

7.9 Distributed Feedback (DFB) Structures

A distributed feedback (DFB) grating structure is a periodic dielectric waveg-

uide in which the periodicity is due to corrugation of one of the boundaries

shown in Fig. 7.34(b)–(d). Such periodic waveguides are used for a variety of

purposes in optoelectronics, including optical ﬁlters, grating couplers, and,

most important, DFB lasers.

Recall from the preceding sections that forward-wave space harmonics and

backward-wave space harmonics are supported in a periodic structure. Hence,

it is possible to realize phase matching between two modes of opposite power

ﬂows if the phase velocity of a forward-wave harmonic in one mode is close

to the phase velocity of a backward-wave harmonic in the other mode with

opposite group velocities. In this case, appreciable mode coupling is realized

through two speciﬁc space harmonics and is a contra-directional coupling

which leads to coupled exponentially growing and decaying waves in the

structure.

7.9 Distributed Feedback (DFB) Structures 463

7.9.1 Principle of DFB Structures [23, 38, 52, 116]

In the uniform dielectric waveguide shown in Fig. 7.34(a), there is no ap-

preciable coupling between two guided modes of opposite group velocities,

because both of the two modes are forward waves and their phase velocities

are opposite.

DFB structure is just a section of corrugated p eriodic dielectric waveguide

as shown in Fig. 7.34(b)–(d). In the structure, there exists a set of space

harmonics of which each guided mode is composed, and the phase constants

of space harmonics are given by (7.54) as:

= β +

2πn

, (7.257)

where β = β

denotes the phase constant of the fundamental harmonic and

n = 0, ±1, ±2, ±3, ··· denotes the order of the space harmonic. The sum

of the ﬁelds of all space harmonics satisﬁes the boundary conditions of the

periodic structure and constitutes a normal mode.

If the fundamental (n = 0) harmonic of the mo de is a forward wave then

the space harmonics of positive orders must be forward waves too, whereas

the space harmonics of negative orders must be backward waves.

In a periodic waveguide, appreciable coupling between two guided modes

of opposite group velocities is possible if the phase velocity of a forward-

wave harmonic in one mode is approximately equal to that of a backward-

wave harmonic in another mode. We note that the phase velocity of the

n space harmonic with positive group velocity can be close to that of the

−(n + 1) space harmonic with negative group velocity, which is the condition

of synchronous or phase matching,

≈ −β

−(n+1)

, i.e., β +

2πn

≈ −

β −

2π(n + 1)

which gives

βp ≈ π, β

p ≈ −β

−(n+1)

p ≈ (2n + 1)π. (7.258)

Generally speaking, the n = 0 space harmonic and n = −1 space harmonic

are the two strongest harmonics in the structure and the other harmonics can

be neglected. We consider mode coupling via the 0 and −1 space harmonics

only. The phase constants of the 0 and −1 space harmonics are related via

−1

= β −

2π

. (7.259)

Suppose that there are two traveling-wave modes with opposite group

velocities. The fundamental harmonics of these two modes are a and b,

respectively:

a = a

−jβz

, b = b

jβz

. (7.260)

464 7. Periodic Structures and the Coupling of Modes

The −1 space harmonics of these two modes are a

−1

and b

−1

, respectively:

−1

= a

−1m

exp

−j

β −

2π

, b

−1

= b

−1m

exp

β −

2π

. (7.261)

Suppose that the ratios of the −1 harmonic to the fundamental harmonic in

the two modes are s

and s

, respectively, i.e.,

−1m

= s

, b

−1m

= s

, (7.262)

then we have

−1

= s

a exp

2π

, b

−1

= s

b exp

−j

2π

. (7.263)

The coupled-mode equations associated with these two modes with opposite

group velocities via the 0 and −1 space harmonics are written as

= −jβa + χ

−1

, (7.264)

= jβb + χ

−1

, (7.265)

where χ

and χ

are the coupling coeﬃcients connecting the −1 space

harmonic and the fundamental space harmonic of the two guided modes with

opposite group velocities. Substituting (7.263) into the above equations, we

deduce that

= −jβa + χ

b exp

−j

2π

= −jβa + κ

b exp

−j

2π

, (7.266)

= jβb + χ

a exp

2π

= jβb + κ

a exp

2π

, (7.267)

where κ

= χ

and κ

= χ

are the coupling coeﬃcients of the two

modes with opposite group velocities.

With the following variable substitution,

a = A exp

−j

, b = B exp

, (7.268)

where A(z) and B(z) are space envelops of the two modes a and b, the

preceding coupled-mode equations become

= −j

β −

A + κ

B = −jβ

A + κ

B, (7.269)

= j

β −

B + κ

A = −jβ

B + κ

A. (7.270)

7.9 Distributed Feedback (DFB) Structures 465

These two equations are identical with the standard coupled-mode equations

(7.206) and (7.207), in which

= (β − π/p), β

= −(β − π/p).

The phase-matching condition is β

≈ β

, that is βp ≈ π. Since the group

velocities of the two modes are opposite, the relation between κ

and κ

= κ

∗

= κ. (7.271)

Introducing the detuning parameter δ, which is a function of frequency,

δ = β −

, (7.272)

and supposing β(ω

) = π/p, i.e., δ(ω

) = 0, we can then expand β(ω) around

ω = ω

as follows:

β(ω) ≈ β(ω

) +

dβ

dω

(ω − ω

) =

ω − ω

. (7.273)

Hence the detuning parameter δ becomes

δ =

ω − ω

. (7.274)

Finally we have the coupled-mode equations for the periodic structure

= −jδA + κ B, (7.275)

= jδB + κ

∗

A. (7.276)

Comparing these equations with the standard coupled-mode equations

(7.206) and (7.207), we have

= δ, β

= −δ, κ

= κ, κ

= κ

∗

Substituting such conditions into the solution of the coupled-mode equations

(7.213), we have the propagation coeﬃcient of the coupled modes in the DFB

structure.

1,2

+ β

− β

− κ

= ±

− |κ|

. (7.277)

The ω–β diagram of the DFB structure is shown in Fig. 7.35, in which

the frequency ω is represented by the detuning parameter δ. We note from

the ﬁgure that when |δ| > |κ|, β is real and the coupled waves are persistent

waves, and when |δ| < |κ|, β becomes imaginary and the coupled waves

degenerate into exponentially growing or decaying waves.

For |δ| < |κ|, β

1,2

are imaginary. Let γ

1,2

= jβ

1,2

, then (7.277) reduces to

1,2

= ∓γ = ∓

|κ|

− δ

. (7.278)

466 7. Periodic Structures and the Coupling of Modes

Figure 7.35: The ω–β diagram of a DFB structure.

Figure 7.36: A segment of a DFB structure.

7.9.2 DFB Transmission Resonator [31, 52]

Consider a segment of DFB structure of length l, as shown in Fig. 7.36. The

load is connected at z = 0 and the input of the DFB is at z = l. The incident

wave with envelope B(z), propagates along −z, whereas the reﬂected wave

with envelope A(z), propagates along +z.

For |δ| < |κ|, the solutions of the coupled-mode equations must be in the

form of exponentially growing or decaying waves. Try the following form of

superposition of growing and decaying exponential functions

A(z) = A

γz

+ A

−

−γz

, (7.279)

B(z) = B

γz

+ B

−

−γz

, (7.280)

as solutions to (7.275) and (7.276). We obtain

jδ ± γ

, (7.281)

7.9 Distributed Feedback (DFB) Structures 467

Figure 7.37: Frequency response of a single-segment DFB structure.

which is similar to (7.219).

Suppose that the structure is matched at the terminal z = 0 in such a

way that A(0) = 0, i.e., A

−

= −A

. Then, from (7.279), (7.280) and (7.281),

we have the space envelopes of the two mods

A(z) = 2A

sinh γz, (7.282)

B(z) = 2A

cosh γz +

jδ

sinh γz

. (7.283)

The reﬂection coeﬃcient at the input port z = l is given by

Γ (l) =

A(l)

B(l)

κ sinh γl

γ cosh γl + jδ sinh γl

. (7.284)

The frequency response of the power reﬂection coeﬃcient |Γ (l )|

of a single

segment DFB structure is shown in Fig. 7.37.

At the center frequency, ω = ω

, β = π/p, δ = 0, γ = |κ|, the space

envelopes (7.282) and (7.282) become

A(z) = 2A

sinh γz = 2A

sinh |κ|z, (7.285)

B(z) = 2A

cosh γz = 2A

cosh |κ|z. (7.286)

The distributions of space envelops A(z), B(z) and power ﬂows A|(z)|

|B(z)|

at the center frequency are shown in Fig 7.38.

The reﬂection coeﬃcient at the center frequency become

Γ (l)|

= tanh |κ|l, (7.287)

468 7. Periodic Structures and the Coupling of Modes

Figure 7.38: The distribution of space envelops A(z), B(z) and A|(z)|

|B(z)|

of coupled waves in a single-segment DFB structure.

and the input impedance of the DFB segment is given by

1 + Γ

1 − Γ

= e

2|κ|l

, (7.288)

where Z

is the characteristic impedance of the uniform waveguide without

the grating. When l is suﬃciently large,

Γ (l)|

= tanh |κ|l → 1,

refer to Fig. 7.37.

Under the condition |δ/κ| < 1, γ is real, so that the ﬁelds in the DFB

structure are growing or decaying exponential functions, which corresponds

to a stop band. When |δ/κ| > 1, γ is imaginary, β is real, and the ﬁelds in

the DFB structure are persistent waves, which corresponds to a pass band.

In the case of pass band, if the following condition is satisﬁed,

sinh γl = j sin βl = 0, i.e., sin

− |κ|

l = 0,

we must have

− |κ|

l = nπ, n = 0, 1, 2, 3 ···,

and

Γ (l) = 0.

The reﬂection coeﬃcient is zero at a series of frequencies within the pass

band, |δ| > |κ|, see Fig. 7.37. At these frequencies, according to (7.282),

with γ purely imaginary, the reﬂection wave A(z) in the +z direction has a

sinusoidal distribution within the structure and vanishes at the two ends of

the structure. These frequencies are the resonant frequencies of the periodic

DFB structure acting as a distributed Fabry–Perot transmission resonator.

7.9 Distributed Feedback (DFB) Structures 469

Figure 7.39: Quarter-Wave Shifted DFB resonator.

7.9.3 The Quarter-Wave Shifted DFB Resonator

The single-segment periodic DFB structure has a set of transmission reso-

nances in its pass band (|δ| > |κ|), i.e., a pass-band resonance but acts as

a nearly perfect reﬂector in the stop band (|δ| < |κ|). If we place two seg-

ments of periodic DFB structures on a line and separate them by a quarter

wavelength or an odd multiple of wavelengths, it is possible to achieve trans-

mission at δ = 0 and β = π/2 within the stop band, which is known as a

Quarter-Wave Shifted DFB resonator, see Fig. 7.39(a), and the resonance

becomes a stop-band resonance [38, 40, 116]. At the center frequency, δ = 0,

β = 2π/λ = π/p, i.e., λ/4 = p/2we only have to shift one of the periodic

structures by a distance of half a period or to reverse one of the periodic

structures as shown in Fig. 7.39b.

The Q factor of the Quarter-Wave Shifted DFB resonator can be made

higher than that of the single-segment DFB structure, because as a mirror the

reﬂection of the DFB structure can be made nearly perfect if the structure

is made suﬃciently long. The frequency response of the quarter-wave shifted

DFB resonator is shown in Fig. 7.40. The Q factor of the stop-band resonance

is much higher than that of the pass-band resonance, especially when |κ|l is

suﬃciently large.

The DFB laser, i.e., a laser diode using a DFB resonator instead of a

normal F–P resonator is an outstanding dynamic single-mode semiconductor

laser and is an important active device in optoelectronics, especially in optical

ﬁber communications.

470 7. Periodic Structures and the Coupling of Modes

Figure 7.40: Frequency response of the quarter-wave shifted DFB resonator.

7.9.4 A Multiple-Layer Coating as a DFB Transmission

Resonator

The multi-layer HR or AR coating with alternating refraction index intro-

duced in Section 2.6.3 is a typical DFB structure. Following the DFB ap-

proach, the reﬂection coeﬃcient and the input impedance of the structure

are given in (7.287) and (7.288). On the other hand, as a multi-layer coating,

from (2.262) in Section 2.6.3, we have

2(l/p)

where mp = l. Suppose that the indices of the two layers are n

= n + ∆n

and n

= n − ∆n, with ∆n/n being very small. The ab ove equation thus

reduces to

n + ∆n

n − ∆n

2(l/p)

≈

1 +

4∆n

(l/p)

. (7.289)

To relate the two equations, (7.288) and (7.289), we have to evaluate the

coupling coeﬃcient κ for a multi-layer structure. Consider a pair of layers of

indices n

= n + ∆n and n

= n − ∆n forming a segment with a length of

one period of the periodic structure immersed in a medium of index n. The

thickness of each layer is one quarter wavelength at the resonant frequency,

so that p = λ/2n. See Fig. 7.41.

The input impedance of the layer pair is

n + ∆n

n − ∆n

≈ 1 +

4∆n