Effect of vibration on the statistical correlation of an optical field produced by multimode optical fibers

Volume 62, number 4

OPTICS COMMUNICATIONS

15 May 1987

EFFECT OF VIBRATION ON T H E STATISTICAL CORRELATION OF AN OPTICAL FIELD P R O D U C E D BY M U L T I M O D E OPTICAL FIBERS QIN Ke-Qi Shanghai Institute of Opticsand Fine Mechanics, AcademiaSinica, P.O. Box 8216, ShanghaL P.R. China Received 3 November 1986

The effect of vibration on the statistical correlation of the optical field produced by a multimode optical fiber is investigated both theoreticallyand experimentally.The statistical distribution of the optical field at the exit face of the multimode optical fiber under vibration is treated as a quasi-homogeneoussource with a time-varyingspatiallystationary phase. The field distribution at the Fresnel-diffractionplane is investigated by analyzing theoretically the propagation of the correlation function and also by experiments. The results show that the correlation coefficientis dependent of the vibration frequencyand the time interval used for recording.

1. Introduction As result of interference and coupling among a large number of different modes of laser light propagating within the core of a multimode fiber, a speckle field is formed at the exit face of the fiber. This phenomenon may be regarded to be a spatially quasi-homogeneous random process [ 1-3]. While part of the fiber vibrates the interference and coupling among the modes within the core must depend on time. The relative phases between the modes at the exit face of the fiber are found to be related to the vibration frequency and time. One can expect that the correlation function of the diffracted field of the fiber under vibration is different from that produced in absence of vibration. We believe that it is the relative phase between the modes that is affected by vibration since the modes distribution which is sensitive to the coupling of the incident beam is known to be independent of vibration [ 4]. Hence the phase change with time may be described by a time varying spatially stationary phase factor. On the basis of the previous consideration the purpose of this paper is to explore the effect of vibration on the statistical correlation of the optical field produced by a multimode fiber. Both theory and experiment show consistent results. The relationship between the correlation coefficient and the vibration frequency and time is given. 0 030-4018/87/$03.50 © Elsevier Science Publishers B.V. (North-HoUand Physics Publishing Division)

In addition effects of vibration for quasi-single mode fiber and for multimode fiber under misalignment coupling at launching end are also studied, mainly experimentally, for comparison. The results of this paper may be useful for applications where the correlation of a vibrating optical fiber is needed.

2. Theory The complex amplitude of light emerging along the direction S from the exit face of the multimode fiber will be denoted by Eo(u, S±; ogs, t) as shown in fig. 1. Here u= (ux, Uy) is a position vector, S± = S - S z denotes the normal component of the unit vector S along a light ray, o9~=21rv~, vs is the vibration frequency and t is the time associated with the vibration. The correlation function of the complex amplitude distribution Eo(u, S±; o9~, t) of the light is defined by

J(ul, u2, S±l, S±2; cos, t) =(E~o(ul,S±~;Ogs, t) Eo(uz,S12;ogs, t ) ) ,

(1)

where the asterisk * denotes the complex conjugation and ( ) stands for the ensemble average. The vibration affects mainly the phases between the modes of emerging light, so at the time t the statis225

Volume 62, number 4

OPTICS COMMUNICATIONS

15 May 1987

Fig. 1. Coordinate system relating to the optical arrangement used for analysis.

tical correlation function is the same as that in the static case except for an additional phase factor which results from the vibration. In general, description propagating characteristics of light in a fiber core by the light ray equation is equivalent to that given by wave equation, provided that the core diameter is much larger than the light wavelength [ 5 ]. Typically low-order modes radiate from the end of the fiber at a small angle relative to the fiber axis. On the other hand, the higher-order modes radiate at large angle. Therefore the different modes can simply be identified by different vector S±. We assume that the relative phases between the adjacent modes are not influenced so seriously by the vibration as those between the non-adjacent modes. Hence eq. (1) may be expressed as

J(u,, u2, S±1, S_L2; cos, t) = ( E*oo(U, ) Eo(tt2 ) ) × exp[i0(Sl~, S±2; cos, t)],

(2)

where ~(S± ~, Sl2; cos, t) denotes the time-varying phase difference between mode S l ~ and S± 2 which results from the vibration with frequency Us= ¢os/2zc. The function (E~o(U~) Eo(u2)) is the correlation function for the static case (absence of vibrations) and can be expressed according to refs. [ 1-3] as

(E*o(U,) Eo(u2 ) )

=Io[½ (u~ +u2)] u(u2 -u~ ), 226

(3)

with Io(u) and /z(u) having the form of gauss±an functions:

I o ( u ) = l e x p ( - 2 l u l 2/qo),

=0,

lul <~a/2,

(4)

lul >a/2.

/t(u) = e x p ( - 1ul2/2 ~o).

(5)

Here a denotes the fiber core diameter, a>>2 (the wavelength of light), a>>qo, qo>>{o, qo represents the radius of the source (which is defined by the extent to which the average intensity distribution of the source becomes e 2 times the maximum value), and {o represents the spatial correlation length. We further assume that the vibration is an harmonic function of the time which we fake for simplicity to be sin(cost). The phase factor is spatially stationary because of the random process associated with the interference of the modes within the core. On substituting from eqs. ( 3 ) - ( 5 ) into eq. ( 2 ) w e find that J ( u l , / / 2 , S ± I, S ± 2 ; O)s,

t)

-=Io e x p ( - ]ul +u212/2q~) exp( --fu2--U112/2~ 2) × exp[i2ze IS±2 - S ~

Ice sin(COst)],

(6)

where ce is a parameter which may have a little bearing on the vibration intensity. In the following we investigate the statistical correlation in the Fresneldiffraction plane by analyzing the propagating correlation function.

Volume 62, number 4

OPTICS COMMUNICATIONS

The amplitude distribution El (~) at the ~ plane is given by

El(¢;to~,t)=f Eo(u,S±;to~,t)K(u,¢)du,

(7)

where K(u, ¢)=(1/i2/) exp((in/2l)lu-¢l 2) is a function which represents the propagation of light from the exit face of the optical fiber to the Fresneldiffraction plane, l is the distance from the source to the plane. ~ and S± are related by ¢= (l/x/1 - IS~ Iz ) S±. Then the correlation function in the diffracted plane becomes

J(¢l, ¢2; to,, t)

=f f J(ul,u2,S±l,S~2;tos,l) xK'*(U,, ¢, ) K(U2, ¢2) du, du 2

15 May 1987

as (o has in the source plane, q~ represents the extent of the average intensity distribution at the ¢ plane, which is defined as the point at which the average intensity becomes e-2 times the maximum value at origin. By taking the two points in eq. (9) as I~,1= 1¢21=1~1, the average optical intensity is found to be given by =J(¢, ¢; COs,t) = exp(-2[¢12/2q2),

(10)

where = 2 ( rt qo~o)210 / (2l) 2 is the. a~'erage intensity at ¢ = 0. It is seen from eq. (10) that the average intensity is independent of time. Physically this is obvious, because the vibration of the fiber affects the phases between the modes only, and the numerical distribution of the various order-modes within the core is not influenced by vibration. From eq. (9) the correlation coefficient of the light in the ¢ plane is given by ~'l(~l, ¢2; tos, t) =exp[ -i(rc/;tl) (1¢112 -1¢212)]

=(10/(2tl) 2) exp[-- (izt/21) (I¢112- 1¢212)1

xexp( - 1¢1 -~212/2~ 2)

×exp[i2n IS±2 - S ± i I a sin to~t]

X exp(i2n IS±2 -SL~ I a sin tost).

x f f exp(-lu~ +u212/2q2o)

(11)

The integrals contained in eq. (8) can be performed in a similar way as in ref. [2]. Eq. (8) then gives

The first factor in eq. (11) is an imaginary exponential factor which is independent of time and denotes the intrinsic phase of the light at the two points. The second exponential factor describes the correlation coefficient in the static case. The third imaginary exponential factor describes the phase change with time between the mode Sj_l and S±2. Under static conditions, i.e. when tos=0 the correlation coefficient is given by

J(~J, ~2; to~, t) = [2(~qo~0)210/(~l) 2 ]

#static =exp[ --i(•/2l) (l~l I 2 -- 1~212)]

× exp( - l u 2 -u112/2(2) ×exp[i(rc/2l) ( It/2 [ 2 _ Ill112)] xexp[i(2n/2/) (Ul "¢-u2"¢)] dill du2.

(8)

× e x p [ - i ( n / 2 l ) (IG 12- 1¢212)1

Xexp( - I¢1 -¢2 J2/2(2).

(12)

Using the relations

Xexp[i2rr IS±2 --S± I I 19lsin to~t]

J_~(g) = ( - 1 )nJn(g), ×exp( - I ¢ , +¢212/2q 2) × exp( - 1¢, - ¢ 2 1 2 / 2 ~ ) ,

(9)

where the two parameters ~ and q~ are defined by

G =2llzcqo,

qt =21/n~o,

and ~ l has the same meaning in the diffraction plane

cos(g sin tot) =Jo(g) +2J2(g) cos 2tot+2J4(g) cos 4tot+ .... sin(g sin tot) = 2J1 (g) sin tot +2J3(g) sin 3tot+ ....

(13) 227

Volume 62, number 4

OPTICS COMMUNICATIONS

where the J+ n ( g ) ( n = 0, 1,2...) is the _+n-order Bessel function, eq. (11 ) becomes / z ( ~ , ~2; co~, t) = e x p [ - i ( ~ / 2 / ) (1¢, 12 -1¢2 J-~)]

15 May 1987

where .U

d (o&,M)=~

sinco~tdt, 0

×exp(-I¢,-¢212/2(~)

O<~a,(co~, At) < 1.

+c*:,

X

J,,(2~ro~lS±~-S±2l)exp(incod).

~. tt =

(14)

~x

The time-averages of nth-order (n>t2) harmonic factor have been neglected in the first approximation. Then eqs. (12), (15) and (17) are the correlation coefficients of the field in the ¢ plane for the static cases ( G = 0 ) , fast vibration (At G>> 1), and slow vibration (At vs ~ 1 ), respectively. For simplicity, when the two points of interest are chosen to be located symmetrically around the origin, we have

In practice a physical value of a system can only be measured within a finite time interval ~It. When the system changes fast or the time interval required for recording is long enough so that dt G>> 1, the result of measurement is equivalent to the time-average of the measurand quantities. Hence the time-average of the correlation coefficient, derived from eq. (14) is given by

and

(#(¢,, ¢:; co~, t) ) j ..... ,

IS~ t - S ~ _~1

×exp( - I ¢ , -¢2 I2/2~) (15)

where ( . . . ) j ..... ~ denotes the nine-average for the fast varying system and can simply be expressed as

I

(19)

(20)

The visibility of the interference pattern behind the ¢ plane which results from light from the two points is [6]

v = I/~(¢,, ¢2; co,, t)l.

(21)

Hence we have from eqs. (12), (15), (17), ( 2 0 ) a n d

(21)

2rt/~os

(...)j, .... ,=(co~/2~)

(I, (¢,) 5 = (I, (¢2) 5

= I¢, -¢~ I/(l 2 + ¼I¢, -¢212) "-~.

=exp[ -i(~r/2l) (IG 12 -1¢2 2)]

x Jo(2Zco~lS~, - 8 ± 2 1 ),

(18)

(''')dt

(16)

I~t~,,c = e x p ( - 1 ¢ , - ~ 2 I 2/2~),

(22)

0

for the periodic harmonic functions. But in cases when dt ~,~ 1 i.e. when the periodic variation of the system is slow or the time interval required to measure the system is short, the time-average may be somewhat different from eq. (16). Approximately we have

{,u(¢,,¢2;co~,t)>, .... ,

I/(l 2 +

¼I¢, -¢21 ~)"2) I,

(23) I'i,..... , =exp( - t¢, -¢21 : / 2 ~ )

+ 461(co~, At)

×exp( - I ¢ , -¢212/2~'~)

xJf(2=o~ I¢, - ¢ 2 I/(l 2 + ~ I¢, -¢212) ,,2)] .2

× [Jo(2~a I & , - s ± 2 I)

228

x }Jo(2no~J¢, -¢2

x [Yg(2~za I¢, -¢2 I / ( l 2 + ~ I,~, -¢e I -~) '"-~)

= e x p [ - i ( z U 2 l ) (l¢l [2 --1¢212)]

+i26,(co~,At) J,(27ralS±,-S±21)],

V,, .... , = e x p ( - i ¢ , -¢212/2¢~)

(24) (17)

Eqs. (22), (23), (24) are the visibiilities of the patterns produced by the interference of the lights from the two points in the ¢ plane for the static case and

Volume 62, number 4

OPTICS COMMUNICATIONS

15 May 1987

for the cases o f fast and slow vibration, respectively. In the next section a comparison will be made between the calculated and measured dependences of the interferogram visibilities. In addition for the case of misalignment coupling o f the launching end of the multimode fiber under vibration and for the case o f quasi-singlemode fiber in vibration will be discussed on the basis of experiments.

amplitude of vibration the factor a is independent o f 1, 1~,-~21 and v~ provided that the relation /it v~/> 1 is satisfied. Consequently the value o f a can be determined from fig. 3 and from the equation c~= 2.4(12+ ~ I ~ --~212) ~/2/2rt [~, -~21- It was found to be ce= 3.5

3. Theoretical and experimental results

Results calculated from eqs. ( 2 2 ) - ( 2 4 ) are shown in fig. 4, where the amplitude of the vibrations is seen to be the same for all the three cases. Hence the value of o~ was chosen to be 3.5. Furthermore without loss of generality we choose 6t (cos, At) =0.3, (~ = 1.6 cm and l = 3 1 cm. An inspection o f fig. 4 shows the following characteristics: (a) In the static case the visibility of the interference pattern is higher than when there are vibrations and for finite values o f 1~l-~2l the visibility V~a~ic~ 0. But with increasing values of I ~ - ~21 the contrast Vslatic will tend to 0. The experimental results are shown in figs. 5 ( a ) - (c). (b) When in fast vibration or when the time interval/it for recording is long enough i.e. when/It Vs>> 1 the visibility will be zero first for I ~ - ~ 2 1 ~ 3.4 cm. It is o f interest to note that when I ~ - ~ 2 1 > 3.4 cm the visibility was not zero but became a little larger than zero, which is consistent with the result of the experiment, as seen in fig. 6. In fig. 6(a) I~ - ~21 --- 3.0 cm, in fig. 6 (b) L~ l - ~21 = 3.4 cm and V~, . . . . ~=0. However when I~x-~21=3.7 cm as shown in fig. 6(c) the visibility Va,~.~ ~ 0 . (c) When the recording time interval/it is comparable to the vibration period l/v~ of the system i.e. /it v~~ 1, the visibility Va, ~s~ ~ of the interference

3.1. Determination of the vibration factor c~ Since J o ( x ) = 0 when x = 2 . 4 , one finds from eq. (23) that the visibility Vat,,~>~=0, when 2rta I ¢ , - ~ 2 1 / ( 1 2 + ¼I¢1-¢212) t/2-~ 2.4. The value of a can be found by measuring 1¢~-¢21 and the corresponding value of l. The experimental set-up is shown in fig. 2. The laser light emitted from a HeNe laser o f wavelength 632.8 nm, operating on its fundamental mode, is focused by a × 10 microscope objective onto the entrance face of a section of a fiber. A graded-index multimode fiber of about 50 m length with a core diameter of 40 # m was used. A part of the fiber nearer to the entrance face of about 40 cm is mechanically connected to a vibrator driven by an oscillator. Two pinhole apertures were places symmetrically around the z-axis with separation I~ - ~2 I in the ~ plane, which is at a distance of l from the exit face of the fiber. A lens with focal-distance o f 40 cm behind the ~ plane focused the light rays to the camera where the interference pattern was formed. The relation between l and J~l-~21 corresponding to V~, ~,~ ~= 0 is shown in fig. 3. The linear relationship of l and 1~1-~21 indicates that for a fixed

qe-Ne

L~',er

"

3.2. Correlation coefficient and vibration characteristics

I

Fig. 2. Experimental setup used to measure the contrast of fringes formed by interference between two light beams from two pinholes. 229

V o l u m e 62, n u m b e r 4

OPTICS COMMUNICATIONS

15 May 1987

3.0" !, 2.0-

/.0

Z Ic,,~ 3o

Fig. 3. Relationship between the separation of the two pinholes and the distance l when the visibility V3,~ ~. ] = 0.

Fig. 5. Photographs of interference fringes, when the fiber is not vibrating, v~=0,3t= 1/15 s. (a) I~,-421 = 3.0 cm, (b) 14t-~21 =3.4 cm, (c) 14]-421 =3.7 cm.

pattern is lower than that in static case but higher than when the v i b r a t i o n s are fast. W i t h finite 1 ~ - ~ 2 1 there are no zero values for the contrast V ~ , ~ , as seen in figs. 7 ( a ) - ( c ) . The previous results show that the correlation coefficient o f the diffracted field o f m u l t i m o d e optical fiber u n d e r v i b r a t i o n is related to the vibrating frequency and the recording time, and m a y be classified as fast (3t u~>>l) a n d slow (3t u ~ 1) vibration. In the case when 3t u~< 1 the correlation

coefficient m a y be expressed by eq. (17) in which the time-average o f the 2 n d - o r d e r or 3rd-order harm o n i c are not neglected. F o r the case when At u~>l 1 it can be described by a m e n d i n g the value offij (09~, At). Hence the effects o f vibrations on the statistical correlation o f the field p r o d u c e d by m u l t i m o d e optical fiber can well be described by the t i m e - d e p e n d e n t spatial stationary phase.

3.3. Misalignment coupling of the incident light beam {,0

The m o d e distribution m a y be sensitive to the )

0.l" 0.7

\~/i

,~-~3lcrrl

CV~ '

~,= ,., o,~

OJI "~ 03"'

U

o.2 O.I

0

i

~

I

'-

E

'

I"

1*~4l--~

Fig. 4. The visibility V o f the interference pattern as a function of I ~ - 4 2 [ , when the d i s t a n c e l a n d the a m p l i t u d e of v i b r a t i o n s are fixed.

230

Fig. 6. Photographs of interference fringes, formed when the fiber is in fast v i b r a t i o n (3tu~>>l). u ~ = 1 2 0 Hz, 3 t = 1 / 8 s. (a) 14~ - 421 = 3.0 cm, (b) I~, - 421 = 3.4 cm, (c) 14, - 421 = 3/7 cm.

Volume 62, number 4

OPTICS COMMUNICATIONS

15 May 1987

Fig. 8. Comparison of the interference patterns formed by quasisingle mode fibers. (a) In the static case: vs=0Hz, At= 1/15 s, [~,-~21 =5.0 cm. (b) In vibration with vs= 100 Hz, At= 1/15 s, I~,-¢_,1 =5.0 cm.

Fig. 7. Interference fringes formed when the fiber is in slow vibration (At vs~, 1), vs= 15 Hz, At= 1/15 s. (a) I~ -¢21 = 3.0 cm, (b) 1¢, -¢21 = 3.4 cm, (c) let -~-~l = 3.7 cm. misalignment of the incident beam [4]. The poorer the alignment is, the more the high-order modes are excited, and at the same the time the fundamental modes decrease. N o w the numerical distribution of modes can be incomplete and may change with time. The phase change between the modes may be very strong and the statistical correlation is quite sensitive to the vibration o f the fiber. Experiments show that no matter whether the fiber vibrates fast or slowly the contrast of the interference pattern is sensitive to the vibration and is much lower than when there is good alignment. The visibility tends to zero even for small values o f I¢~ --~21"

3.4. Quasi-singlernode optical fiber For the sake o f comparison the effect of vibration on statistical correlation o f optical field produced by quasi-singlemode fiber was investigated experimentally. The experimental set-up was almost the same as that shown in fig. 2. The results show the following characteristics: (i) The decrease of the contrast that is due to the vibration of the quasi-singlemode fiber is less serious than in multimode fiber. (ii) In the static case the contrast of the interferogram associated with the correlation coefficient of the optical field is [ 3 ]

Vst~,ic =

2J~(x/2bk/4l) I~, - ~ 2 I ) , x/-2bld4l) I~, - ~ 2 I

(25)

where b is the effective core diameter of the quasi-

singlemode fiber and k = 2n/2. The visibility vanishes the first time when (x/2bk/4l) I ¢l - ~21 ,~ 3.8 3. It is interesting to note that the visibility does not vanish under the same condition when the fiber is in vibration as is seen from fig. 8. In other words the correlation coefficient of optical field decreases as a whole due to the vibration of the fiber, but the maximum transverse coherence length (which is defined as the value of I~, - ~21 when the visibility vanishes first time) is larger than that in the static case. Further theoretical and experimental investigations are in progress.

4. Conclusion Theoretical and experimental results concerning the effects o f vibration on the statistical correlation of an optical field produced by multimode optical fiber has been presented. A time-varying spatial stationary phase factor is introduced to describe the phase time-varying between the modes at the exit face of the fiber. It is found that the correlation coefficient is related to the vibration frequency and the time interval used for recording. There is a clear distinction between effect of fast and slow vibrations.

References [ 1] W.H. Carter and E. Wolf, J. Opt. Soc. Am. 67 (1977) 785. [2] N. Takai and T. Asakura, J. Opt. Soc. Am. A 2 (1985) 1282. [ 3 ] K.Q. Qin, J.W. Chen and R.W. Wang, Chinese Physics- laser (to be published). [4] T. Tsuji, T.Asakura and H. Fojii, Opt. Quantum Electron 16 (1984) 9. [ 5 ] T. Okoshi, Optical fibers (Academic Press 1982) §6. [6] M. Born and E. Wolf, Principle of optics (Pergamon Press, 1975, 5th Ed) §10.4. 231