Study of the behaviour of reflectance and phase change on reflection at thin and very thick films

Optics & Laser Technology 30 (1998) 125±132

Study of the behaviour of re¯ectance and phase change on re¯ection at thin and very thick ®lms L. Ward Coventry University, Coventry, UK Received 6 April 1998; accepted 16 April 1998

Abstract It is shown that equations for the four functions, Rs, Rp, Fs and Fp for solid materials can be expressed as either quartics in e4 (e4=e1ÿe3) and e2 or in polar co-ordinates; in each case, when e2 is plotted against e4, heart-shaped ®gures are produced. Over the intermediate thickness range between the limit of validity of AbeleÁs' equations and solids, the re¯ectances and phase changes show a series of maxima and minima due to interference between rays re¯ected from the top and bottom of the layer in agreement with the exact theory. The plots of these functions in the e4, e2 plane change from circles (in the AbeleÁs regime) to heart-shaped ®gures as the thickness increases. # 1998 Elsevier Science Ltd. All rights reserved.

1. Introduction

dielectric constant and e3 is equal to (no
sin yo)2. Thus:

The exact equations for the re¯ectance (Rs) and phase change on re¯ection (Fs) at a thin absorbing ®lm on a non-absorbing substrate are [1]:

u2 ‡ v2 ˆ …e24 ‡ e22 †1=2

Rs ˆ‰r21
exp…2vx† ‡ r22
exp…ÿ2vx† ‡ 2r1 r2
cos…f1 ÿ f2 ‡ 2ux†Š ‰exp…2vx† ‡ r21 r22
exp…ÿ2vx† ‡ 2r1 r2
cos…f1 ‡ f2 ‡ 2ux†Š

…1†

where e4 ˆ e1 ÿ e3

For very thin ®lms, AbeleÁs [2] expanded the exponential and trigonometrical terms in Eqs. (1) and (2) in powers of x and so obtained his equations for R and F in terms of e1 and e2. Ward [3, 4] showed that these expressions could be rearranged into the equation of a circle viz.: …e4 ÿ a†2 ‡ …e2 ‡ b†2 ˆ c2

tan Fs ˆ‰r1
sin f1 …exp…2vx† ÿ ‡r2 …1 ÿ

r21 †

r22


exp…ÿ2vx††


sin…f2 ÿ 2ux†Š

For Rs,

‰r1
cos f1 …exp…2vx† ‡ r22
exp…ÿ2vx†† ‡ r2 …1 ÿ

r21 †


cos…f2 ‡ 2ux†Š

…2†

where r1, f1 are the re¯ection coecient and phase change at the air±®lm interface, r2, f2 are the re¯ection coecient and phase change at the ®lm±substrate interface, x = 2pd/l, where d is the ®lm thickness and l the wavelength, u and v are de®ned by the equations below 2u2 ˆe1 ÿ e3 ‡ ‰…e1 ÿ e3 †2 ‡ e22 Š1=2 2v2 ˆe3 ÿ e1 ‡ ‰…e1 ÿ e3 †2 ‡ e22 Š1=2

…4†

…3†

e1 and e2 are real and imaginary parts of the complex 0030-3992/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 0 - 3 9 9 2 ( 9 8 ) 0 0 0 2 6 - 7

a ˆ …y2o ‡ y2s †=2; b ˆ ‰ys ÿ yo …1 ‡ Rs =…1 ÿ Rs †Š=x; c ˆ 2
yo
R1=2 s =‰x…1 ÿ Rs †Š: Reasonable values of Rp may be obtained if yo=no
cos yo is replaced by zo=no/cos yo and ys=ns
cos ys by zs=ns/cos ys. In these expressions, yo is the angle of incidence in medium of refractive index, no, and ys is the refracted angle in the substrate of refractive index ns.

126

L. Ward / Optics & Laser Technology 30 (1998) 125±132

Squaring Eq. (8) leads to:

For Fs, b ˆ ys …1=x ‡ yo =tan Fs †, …5a† a ˆ yo =x
tan Fs , c2=[(n2oÿe3)(tan2 Fs+yo
x
tan Fs) + y2o]/(x
tan Fs)2 which may be rewritten as c ˆ yo …1 ‡ tan2 Fs ‡ yo
x
tan Fs †1=2 =x
tan Fs

…5b†

and, for Fp, yo may again be replaced by zo and ys by z s. Plotting the radii of the circles vs 1/x (Fig. 1) produced a straight line whose slope was 2yoR1/2 s /(1 ÿ Rs); the limit of validity of AbeleÁs' equations was shown by departure from this straight line; this occurs at x z0.25. Over the range of values of Fs usually encountered (3.0 20.3 rad), the terms in tan Fs and tan2Fs in the numerator of Eq. (5b) may be ignored. Then, for a given value of Fs, a plot of c vs 1/x will also be linear with a slope of yo/tan Fs. The purpose of the present paper is two-fold: (a) to express the equations for R and Fr for solids in terms of e1 and e2 and (b) to investigate the behaviour of R and Fr at intermediate thicknesses between those covered by AbeleÁs' equations and solids.

2. Method 2.1. Solids For solids, when x is very large, the expression for Rs becomes: 2

2

Rs ˆ r21 ˆ ‰… yo ÿ u† ‡ v2 Š=‰… yo ‡ u† ‡ v2 Š

…u2 ‡ v2 †2 ‡ 2
y2o …u2 ‡ v2 † ‡ y4o ˆ 4
u2
y2o
W 2 u2+v2=(e24+e22)1/2

Now, Thus:

2u2=e4+(e24+e22)1/2.

and

e24 ‡e22 ‡ y4o ‡ 2
y2o …e24 ‡ e22 †1=2 ˆ 2
y2o W 2 …e4 ‡ …e24 ‡ e22 ††1=2

…10†

If the square roots are multiplied out, Eq. (10) becomes a quartic in e4 and e2. It is better, however, to express e4 and e2 in terms of polar co-ordinates L and a de®ned by e4=L
cos a and e2=L
sin a. Eq. (10) then becomes: L2 ‡ 2
y2o
L ‡ y4o ˆ 2
y2o
LW 2
‰1 ‡ cos aŠ

…11†

For incidence from air, yo<1 so that y4o is very small and can be ignored. Thus, from Eq. (11), L ˆ 2
y2
W 2
…cos a ‡ 1 ÿ 1=W 2 †

…12†

When W is large, the term in 1/W2 can also be ignored. Expressions for e4 and e2 may then be obtained using e4=L
cos a and e2=L
sin a. Eq. (12) generates a heartshaped ®gure as illustrated in Fig. 2 which shows e2 vs e4 for Rs=0.65 to 0.8 at yo=608; only the positive e2 region (shown as full lines) represents real values of e4 and e2. This ®gure shows the characteristic cusp of the quartic equations; comparison with the curve using the exact equations for x = 20.0 showed that there was a very close similarity between them. 200

…6†

Once again, u and v were found [5] to be related by the equation of a circle for both the optical functions, R and Fr. …7†

For Rs, f ˆyo …1 ‡ Rs †=…1 ÿ Rs †, g ˆ0 and r2 ˆ 4
Rs
y2o =…1 ÿ Rs †2

=

0.

9

Rs

Radius of circle

…u ÿ f †2 ‡ …v ‡ g†2 ˆ r2

…9†

100

Rs

.8 =0

For Fs, f ˆ 0,

Rs=

g ˆ yo =tan Fs and r2 ˆ y2o =sin2 Fs

In the cases of p-polarisation, yo is replaced by zo. One of the objectives of this article is to re-express Eq. (6) in terms of e4 and e2. Eq. (6) can be transformed to give Eq. (8) below: …u2 ‡ v2 † ‡ y2o ˆ 2
u
yo
…1 ‡ Rs †=…1 ÿ Rs † ˆ 2
u
yo
W …8† where W = (1 + Rs)/(1 ÿ Rs).

0.7

R s = 0 .5

0 0

5

10

15

1 – x Fig. 1. Radius of circle vs 1/x using AbeleÁs' approximation for Rs at yo=608.

L. Ward / Optics & Laser Technology 30 (1998) 125±132

127

80

60

R s = 0.80 ε2

40

R s = 0.75

R s = 0.70

20

R s = 0.60

0 –10

0

10

20

30

40

50

60

70

80

ε4

–20

–40

–60

Fig. 2. e2 vs e4 for Rs=0.65, 0.70, 0.75 and 0.80 at yo=608.

A rough approximation to the solution of the equation for Rp may be obtained as before, by substituting zo=no/cos yo for yo in Eq. (12) and this produces similarly shaped curves to Fig. 2. The equation for Fs for a solid material is: tan Fs ˆ tan f1 ˆ 2
yo
v=‰ y2o ÿ …u2 ‡ v2 †Š

…13†

which also transforms to a quartic equation in e4 and e2, as follows: …u2 ‡ v2 † ˆ y2o ÿ 2
yo
v=tan Fs Squaring Eq. (14) then leads to

…14†

…u2 ‡ v2 †2 ÿ 2
y2o …u2 ‡ v2 † ‡ y4o ˆ 4
y2o
v2 =tan2 Fs 2

e4+(e24+e22)1/2

From Eq. (3), 2v = ÿ u2+v2=(e24+e22)1/2. Thus,

…15†

and, as before,

e24 ‡e22 ÿ 2
y2o …e24 ‡ e22 †1=2 ‡ y4o ˆ 2
… yo =tan Fs †2
‰ÿe4 ‡ …e24 ‡ e22 †1=2 Š

…16†

This gives another quartic relation between e4 and e2. Applying the transformation to polar coordinates as before leads to: L2 ÿ 2
y2o
L ‡ y4o ˆ 2
L
… yo =tan Fs †2 …1 ÿ cos a†

…17†

128

L. Ward / Optics & Laser Technology 30 (1998) 125±132

Again neglecting the y4o term, Eq. (17) becomes: L ˆ 2
… yo =tan Fs †2 ‰1 ÿ cos a ‡ tan2 Fs †Š

…18†

and also e4=L
cos a and e2=L
sin a. An approximate solution for Fp may again be obtained by substituting zo for yo in Eq. (18). Fig. 3 shows e2 vs e4 for values of Fp between 6.0 and 6.2 rad at yo=308: once again, a similar set of curves was obtained for Fs; it will be noted that these heartshaped curves are reversed about the e2-axis compared with those for R. Comparison with the curves using

the exact equations for x = 20.0 again shows good agreement between the two methods of obtaining Fs and Fp. The area enclosed within these ®gures may be found „p by performing 0 e2de4 In the case of Rs, the value of this integral is 3
p
y4o
W4; when Rs=0.8 and yo=608, the enclosed area = 3865e4e2 units. The area under the curves will be used later to assess the behaviour of the function with thickness. For Rp, zo is substituted for yo whilst for F, W is replaced by 1/tan F to give the enclosed area as 3
p
y4o/tan4 F; when zo=608 and F = 3.03 rad, the area is 3950e4e2 units.

φ p = 6.2 rad 500

ε2

400

300

φ p = 6.15 rad

200

φ p = 6.10 rad

100

6.05 6.0 0 –600

–500

ε4

–400

–300

–200

–100

0

–100

–200

–300

–400

–500

Fig. 3. e2 vs e4 for Fp=6.0 to 6.2 rad at yo=308.

100

L. Ward / Optics & Laser Technology 30 (1998) 125±132

2.2. Intermediate thicknesses

60

At intermediate thicknesses, the terms in exp(ÿ2vx) in Eqs. (1) and (2) may be ignored; then Eq. (1) may be written;

…20†

Eq. (20) shows that Rs in this region is equal to the re¯ectance of the solid material together with a damped periodic term. Thus, plotting Rs vs x should produce a damped wave asymptotic to the value of Rs (solid). The solution for Fs in this region is also equal to the phase change at the solid material plus a damped periodic term. tan Fs ˆtan f1 ‡ ‰exp…ÿ2vx†=cos y1 Š  r2 =r1
…1 ÿ r21 †
‰sin…f2 ÿ 2ux†Š  ‰cos…f2 ‡ 2ux†Š

…21†

This region was investigated by calculating R and F using the exact equations for these quantities (Eqs. (1) and (2)). As x increased beyond the AbeleÁs limit, the 6000

5000

Area under curve

4000

Solid 3000

2000

1000

0 2

x = 0.7

x = 1.00

x = 1.30

10 0

r21 Š

1

40

20

‡ 2r1 r2
exp…ÿ2vx†‰cos…f2 ÿ f1 ‡ 2ux†

0

50

x = 0.20

…19†

ÿ cos…f2 ‡ f1 ‡ 2ux†


ε2

30

‰r2 ‡ 2r1 r2
exp…ÿ2vx†
cos…f2 ÿ f1 ÿ 2ux†Š Rs ˆ 1 ‰1 ‡ 2r1 r2
exp…ÿ2vx†
cos…f2 ‡ f1 ‡ 2ux†Š ˆr21

129

3

x Fig. 4. Area under curve vs x for Rs=0.8, yo=608.

0

10

20

30

40

50

60

70

80

90

100

ε4 Fig. 5. e2 vs e4 for four values of x at yo=608, Rs=0.8.

plots of e2 vs e4 tended rapidly from the circular form towards the heart-shaped ®gures obtained for solids. Plotting the areas enclosed by these ®gures against x showed an initial constant negative slope followed by a series of maxima and minima converging on the area for the solid (Fig. 4, which is for Rs=0.8 and yo=608). These correspond to the maxima and minima of the interference pattern formed by rays re¯ected from the top and bottom of the ®lm; from Fig. 4, the spacing of successive maxima (or minima) is Dx = 0.38. The cosine term in Eq. (20) will be cyclic with 2ux = 2pm where m is an integer; for m = 1, u = p/Dx = 3.1416/3.8 = 8.3. It may seem strange that a single value of u can represent the whole range of values of e4 and e2 covered by a particular value of Rs, in this case 0.8; however, a closer examination of the heart-shaped ®gures (Fig. 5 which shows a set of curves using four values of x) showed that the areas of (i) the negative e4 region and (ii) most of the positive e4 region did not change signi®cantly with x. Over these regions k, and hence v, is large (>3) and so the exponential term will dominate and changes in the cosine term will be swamped. All the changes in area occurred in the region around large values of e4 where u is large and v small; in this region, the exponential term increases more slowly and the oscillatory term will remain more signi®cant to larger values of x. The contributions to the overall re¯ectance from four points in the e2, e4 plane along the line corresponding to Rs(solid)=0.80 at yo=608 are shown in Fig. 6 (e4=0, e2=40; 20, 50; 60, 40; 70, 20); almost all the oscillatory terms are con®ned to the points 60, 40 and 70, 20 where n>8 and this is in line with the value calculated above. It will be noted that the maxima and minima are reversed between Figs 4 and 6; this is because a minimum in area corresponds to a maximum in re¯ectance. It was found that changing the angle of incidence made little di€erence to the amplitudes of the oscillatory terms. Similar results are obtained for Rp.

130

L. Ward / Optics & Laser Technology 30 (1998) 125±132

(<0.2) the curves do not cross the e2-axis and represent transitional cases between the AbeleÁs circles and the heart-shaped curves. In these cases changes in area take place at large negative values of e4 where u is small but v, and hence k, is large; these conditions are not conducive to the production of oscillations. When x>0.3 the curves cross the e2-axis and become heartshaped but subsequent changes in area with increasing x are small so only minor oscillations will occur. This is borne out in Fig. 9 which shows the contributions from ®ve points on the curve for Fs(solid)=3.03 rad and yo=608; only at the point e4=10, e2=20 was one full cycle observed with half cycles at 0, 40 and ÿ20,

The phenomenon of oscillatory changes in re¯ectance, and the related quantity transmittance, with d/l has been exploited in the determination of n and k for a ®lm by the envelope method [6, 7] where R is measured as a function of wavelength; peaks and troughs are observed in the values of R, and from the envelopes formed by these, n and k can be calculated provided the ®lm thickness, d, hence d/l, is known. The variation with x of the area under the curves for Fs=3.03 rad at yo=608 is shown in Fig. 7; this ®gure shows only a few oscillatory variations in area compared with the case for Rs (Fig. 4). Fig. 8 shows e2 vs e4 for four di€erent values of x. When x is small 1.0

ε 4 = 70, ε 2 = 20 0.9

60, 40

20, 50 Solid

0.8

ε4 = 0 ε 2 = 20 0.7

Rs

0.6

0.5

0.4

0.3 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

x Fig. 6. Rs vs x at yo=608 for four points in the e4, e2 plane at Rs(solid)=0.8.

L. Ward / Optics & Laser Technology 30 (1998) 125±132

131

6000

Area under curve

5000

Solid

4000

3000

2000 0

1

2

3

x Fig. 7. Area under curve vs x for Fs=3.03 rad at yo=608.

expressed as either quartics in e4 and e2 or in polar co-

50. The spacing of successive peaks and troughs in Fig. 7 is Dx = 1.00 giving an e€ective value of u = 3.14 which is a reasonable average value for the region of e4 between +10 and ÿ20. Similar results are obtained for Fp. When x becomes greater than about 6 (d/l = 1.0) the values of R and Fr cease to change signi®cantly with thickness but approximate to those for the solid material; this is expected from the exact equation as the negative exponential and cosine terms become negligible as x increases.

ordinates; in each case, heart-shaped ®gures are produced when plotting e2 vs e4. Over the intermediate thickness range between the limit of validity of AbeleÁs' equations and solids, the re¯ectances and phase changes show a series of maxima and minima due to interference between rays re¯ected from the top and bottom of the layer; these are rapidly damped by the negative exponential term in x. The persistence of this oscillatory pattern as x increases depends on the relative values of v (expo-

3. Conclusions

nential term) and u (oscillatory term) in the exact

It has been shown that expressions for the four functions, Rs, Rp, Fs and Fp for solid materials can be

equations for R and Fr and only if v is small will the oscillatory term persist above values of x of about 1.0. 60 50

ε2

x = 0.3 40

x = 0.1

30 x = 0.2

20

x = 1.0

10

–120

–100

–80

–60

–40

–20

0

ε4 Fig. 8. e2 vs e4 for four values of x at yo=608, Fs=3.03 rad.

20

132

L. Ward / Optics & Laser Technology 30 (1998) 125±132 3.25

3.20 ε 4 = 10, ε 2 = 20 3.15

0, 40 3.10

–20, 50 3.05 –60, 45

φ s RAD

3.00

ε 4 = –80, ε 2 = 0

2.95

2.90

2.85

2.80

2.75

2.70 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

x Fig. 9. Fs vs x for ®ve points in the e4, e2 plane at yo=608 at Fs(solid)=3.03 rad.

Acknowledgement

The author wishes to acknowledge the invaluable assistance he received from members of the Mathematics Department of Coventry University in connection with this work.

References [1] Ward, L. In: The optical constants of bulk materials and thin ®lms, 2nd ed. Bristol and Philadelphia: IOP Publishing, 1994. pp. 169±170. [2] AbeleÁs F. J Opt Soc Am 1957;47:473. [3] Ward L. Opt Laser Tech 1995;27:125. [4] Ward L. Opt Laser Tech 1996;28:229. [5] Ward L. Opt Laser Tech 1993;25:393. [6] Manu®cier JC, Gasiot J, Fillari J. J Phys E 1976;9:1002. [7] Swanepoel R. J Phys E 1983;16:1214.