Optical fibre excitation by polychromatic partially coherent sources

Volume 20, number 2

OPTICS COMMUNICATIONS

February 1977

OPTICAL FIBRE EXCITATION BY POLYCHROMATIC PARTIALLY COHERENT SOURCES D.J. CARPENTER and C. PASK Department o f Applied Mathematics, Institute of Advanced Studies, Australian National University, Canberra, A.C.T., 2600 Australia Received 29 October 1976

The general formalism describing modal fields excited in optical fibres by partially coherent sources is given. The limiting cases of spectrally pure, quasi-monochromatic, coherent and totally incoherent sources are discussed.

1. Introduction In this communication we establish the general formalism describing optical fibre excitation by polychromatic partially coherent sources, and show how it reduces to the special cases considered previously. The fibre is characterised by its modal fields and the source field by its mutual coherence function F. We show that the theory for fibre excitation links these two via an illumination matrix of the type introduced by O'Neill [ 1].

2. Derivation of excitation formalism We consider an optical fibre with refractive index n 1 aligned with the positive z-axis and surrounded by an infinite, uniform medium with index n 2 (< n 1)The source field is then taken to be the incident field Ein c in the z = 0 plane. The field of the fibre may be written as [2] Ef = E B + E R ,

(1)

where E B is the bound mode field which for frequency co is given by N

EB = q~=l aq(W)eq(r, co) exp[i(wt - ~qZ)] .

(2)

In eq. (2), aq, eq and/3q are the amplitude, field and propagation constant, respectively, of mode q and r 262

is the position vector in the xy-plane. N = N(¢o) is the number of propagating bound modes. The component E R is the radiation field, which may be written as an integral over continuous modal components

E R =fa R(~, q) e R (r, co, q) exp {i [cot -/3R(q)z ] } dq.

(3)

The bound modes propagate unchanged along an ideal lossless fibre, but E R is the field associated with power which progressively leaves the fibre with increasing distance from the source. We develop the formalism below in terms of E B, returning to E R later. The frequency component Einc(r, ~ ) of the incident field will excite modes with an amplitude given by [2,3] 1/2

aq(6o)=(-~)ffeq(r,~).Einc(r,~)dr,

(4)

z=0

where/a is the permeability (assumed constant) and e 1 (= n 2) is the dielectric constant of the fibre material. Throughout the analysis we are assuming weakly guiding fibres [2, 3] where the difference in refractive indices is small. In this approximation the modes may be taken as linearly polarized and we now consider only one component of polarization (we can then drop the vector notation). The other component is treated in a similar manner and may be added in independently to construct the unpolarized light situation. Einc(r, w) is a frequency component of a partially

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OPTICS COMMUNICATIONS

coherent field. We assume that we have a statistically stationary system and define the spectral component of the coherence function I" by [ 1,4] ,

~

f

t

- r(r,r ,co)8(60 - 6o') _

(5)

where () denotes an ensemble average and * the complex conjugate. Now for each frequency component, the power in mode q is given by ~ aq (co)a~ (60) [2, 3]. Thus the total power P excited by the incident field is

P={f ~1 (aq(co)a~(co))d60,

(6a)

N

= ½f ¢~=1"4'7'7(60) d°""

(6b)

3. Special cases In many situations we can assume the condition of spectral purity [5] so that P(r, r', 60) = r(r, r') ~b(co),

e1

7 ffff

(9)

es(r'60) (Einc(r, co)Ein* e (r,' 60))

eq, (r,t 60) dr dr'

g ffff

with

es(r,60)r'(r,r')e~(r',60)drdr'.

(11)

A further simplification results in the quasi-monochromatic limit which we use here to write the coherence function in the form FQM (r, r', 60) = F (r, r', 6o0) ~ (60,600)

es(r, eo)r(r,r',60 ) e q* ( r ' , 60) dr dr'

z=O

(7)

where Bsq is just O'Neill's illumination matrix [ 1] with the bound modes as basis functions. Our final general formula is

(1 2)

where ~ is a function highly peaked around the major frequency w 0. Defining ~ (~°0) = f ~(60, ~0) d60,

=- ( q /u)B~q (60),

N P = {(el//a) f ~ 1 B q q ( 6 0 ) d ~

(10)

Z=0

Z=0

= - el -

p = ~ (e 1/p)fcb(60) trace (B) d60

Bsq = f f f f

Using eqs. (4) and (5) we find

X

Since this power is not bound to the fibre it becomes of less importance as z increases.

Then eq. (8) becomes

N

=

February 1977

(1 3)

we now have P = ~(el//a) 5(600) trace [B(600) ] ,

(14)

where B(600) is given by (7) with 6o = to0. Eq. (14) has been evaluated for. the optical fibre with circular cross-section [6].

(Sa)

3.1. Coherent source

= {(el/.)f

trace (B) d¢o.

(8b)

Only the diagonal elements of B are required when calculating total power. The off-diagonal elements of B relate to the correlation of the various modal fields. The radiation field can be treated in a similar manner except that now Bqq (6o) will be replaced by a term of the form D(q, 60) and the power associated with the radiation field becomes

In the limit of complete coherence we will have V(r,r', co) = F(r, oo) F*(r',60), and then B becomes

Bsq (60)

= bs

(60) b~ (co),

(16)

e s (r, co) F(r, co) dr.

(1 7)

where

b s (60) = f f

eR = { (~l /U) f f D(q, co) da dco .

(15)

z=0 263

Volume 20, number 2

OPTICS COMMUNICATIONS

3.2. Totally incoherent source

to have r / = 1 so that eqs. (22) and (8) give

The totally incoherent source may be modelled by

F(r,r', co) = $ (r - r') G(r, ¢o)

(18)

(see [4] for a discussion of the physical validity of this formula). Then B in eq. (7) takes on the form

(I 9)

Bsq (¢o) = H e s (r, ¢o) G(r, 6o) e~ (r, ¢o) dr. z=O

For a uniform source with cross-section S in the z = 0 plane

Bsq(~) = G(¢o) f f e s ( r , co)e~l(r, co)dr. S - G(w)(e 1/pill2 Csq (S),

(20a) (20b)

where Csq is studied in detail in [7] for the circular cross-sectional fibre. For a very large source centred on the fibre, S in eq. (20) may be approximated by the whole z = 0 plane since the modal fields decay rapidly away from the fibre, and then the mode orthogonality condition [2, 3] gives

Bsq (w) = G(w)(e 1/U) 112 5pq.

(21)

Thus a large uniform incoherent source excites all modes equally and with zero correlation. For an incoherent source which just covers the fibre entrance face

Bqq (¢o) = G(w)(e 1/U) 112~?q(w),

(22)

where rlq is a well-known function being the fraction of mode q's power which propagates within the fibre [2, 3]. Eq. (22) was derived in [7] in the quasimonochromatic limit and using an envelope function approach as discussed by Wolf [8] [see his eq. (3.12)].

3. 3. Multimode fibre with incoherent source In the multimode limit most modes are expected

264

February 1977

P = ~ (e 1//a)

fN(o )

(23)

Also in this limit we expect [9] N(6o) ~ Ko~2~

(24)

where K is a constant. As an example, suppose that G(¢o) is a non-zero constant only for ¢o0 - A ~< 6o ~< ~0 + A, then eqs. (24) and (23) yield COO+A

P~--½(el/IJ) 1/2 K

f

6°2d6°

to0--A

= (el//a)l/2 KA(w 2 + -~A2).

(25)

Eq. (23) is only strictly true for ~ ~ 0% when N most closely approximates a continuous function. When z ~ 0%p in eq. (25) is the total power, but for smaller z values there is a strong contribution to the total power from the leaky modes, i.e. those modes which become below cutoff as the frequency changes [ 10].

References [I] E.L. O'Neill, Introduction to Statistical Optics (Addison-Wesley,Reading, Mass., 1963). [2] D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974). [3] A.W. Snyder, IEEE, MTT-17 (1969) 1138. [4] M.J. Beran and G.B. Parrent, Jr., Theory of Partial Coherence, (Prentice-Hall, Englewood Cliffs, N.J., 1964). [5] L. Mandel, J. Opt. Soc. Amer. 51 (1961) 1342. [6] D.J. Carpenter and C. Pask, Quant. and Opt. Electron. 8 (1976) 545. [7] A.W. Snyder and C. Pask, J. Opt. Soc. Amer. 63 (1973) 806. [8] E. Wolf, Proc. Roy. Soc. A225 (1954) 96. [9] C. Pask, A.W. Snyder and D.J. Mitchell, J. Opt. Soc. Amer. 65 (1975) 356. [10] C. Pask and A.W. Snyder, Opto-electronics 6 (1974) 297.