Multi-layer Anti-reflection Coating

Abstract

A non-absorbing, substantially colorless, multi-layer anti-reflection coating for use on a substrate having an index of refraction of 1.43 to 2.00 comprises a first layer of a low-index filming material deposited on the substrate and having a thickness less than .lambda./4, a second layer of a high-index filming material deposited on the first layer and having a thickness of less than .lambda./4, a third layer of a low-index filming material deposited on the second layer and having a thickness approximately between 5 .lambda./16 and 7.lambda./16, a fourth layer of a high-index filming material deposited on the third layer and having a thickness of less than .lambda./4, a fifth layer of a low-index filming material deposited on the fourth layer and having a thickness of less than .lambda./4, a sixth layer of a high-index filming material deposited on the fifth layer and having a thickness of more than .lambda./2, and a seventh layer of a low-index filming material deposited on the sixth layer and having a thickness of .lambda./4, wherein .lambda. is a selected wavelength of near ultraviolet range to near infrared range.

Description

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to an anti-reflection coating comprising seven thin layers.

2. Description of the Prior Art

It is known from German Pat. No. 742,463 (1943), for example, that a multi-layer coating comprising in combination two or three different stable materials deposited into thin layers of less than .lambda./4 in thickness, where .lambda. is the wavelength of light, can be equivalently expressed as a single-layer coating having a value of refractive index between the maximum and the minimum refractive index of the deposited materials and within the range over which the refractive indices of the existing filming materials vary with the variation in the wavelength of light, if the wavelength range is represented by the concept of wave number range (1-.sigma., 1+.sigma.), where .sigma. = .lambda..sub.s /.lambda. for the standard wavelength .lambda..sub.s (usually 5500A). Especially, it was taught by L. I. Epstein that a multi-layer coating of symmetrical construction can be replaced by a single-layer coating of equivalent refractive index according to the theory of equivalent coating.

Such a property may be applied to a three-layer coating of the construction substrate - .lambda. /4 - .lambda./2 - .lambda. /4 - medium. If the layer adjacent the substrate is replaced by a three-layer coating of equivalent refractive index and the refractive index of the substrate-adjacent layer of the new three-layer is selected to be the most suitable one for the substrate, then there may be provided an anti-reflection coating which will never be affected by the refractive index of the substrate. This is disclosed in U.S. Pat. No. 3,432,225 (1969) and U.S. Pat. No. 3,565,509 (1971).

Nowadays, with the diversified usages of photographic lenses, the developments of optical instruments, the adaptation of photosensitive materials for a wider band and the specialized usages of such photosensitive materials, it has become essential to reduce the reflection factor over a wide range from near ultraviolet to near infrared. In order to provide an anti-reflection coating with such a property, it would be insufficient to express the refractive index of each layer in a three-layer coating of the form substrate - .lambda./4 - .lambda./2 - .lambda./4 - medium simply by using the concept of alternate layers, inasmuch as variations in the refractive indices of the existing filming materials with respect to wavelength must at least be taken into consideration. An anti-reflection coating will now be considered in terms of its properties as a multi-layer coating, with the construction of such coating regarded as a fundamental periodic layer. For the convenience of description, the wavelength range from a non-transmissive band to the next non-transmissive band is referred to as the periodic width of the fundamental periodic layer.

Here it is assumed that the uppermost layer of the fundamental periodic layer which is adjacent the medium is composed of a filming material having a lowest possible refractive index, such as magnesium fluoride (MgF.sub.2), lithium fluoride (LiF) or cryolite (Na.sub.3 AlF.sub.6), and that an intermediate layer adjacent the uppermost layer is composed of a filming material such as zirconium oxide (ZrO.sub.2), titanium oxide (TiO.sub.2) or scandium oxide (Sc.sub.2 O.sub.3).

Generally, the periodic width of the fundamental periodic layer may be improved by increasing the thickness of such layer. With this method, however, performance is superior for perpendicular incident rays but inferior for oblique incident rays. That is, the angular characteristic is deteriorated.

In this case, the periodic width of the fundamental periodic layer could be increased by slightly increasing the thickness of the layer if the refractive index whose wave number range (1-.sigma., 1+.sigma.) is in the vicinity of .sigma. = 0.3 - 0.35 could be greatly increased or decreased with respect to the refractive index in the center range (the "center" means that wave number equals 1).

SUMMARY OF THE INVENTION

The present invention aims at improving the anti-reflection coating of the conventional type of the form substrate -.lambda./4 - .lambda./2 - .lambda./4 - medium to increase the periodic band width of the fundamental periodic layer.

According to the present invention, there is provided a non-absorbing, substantially colorless, multi-layer anti-reflection coating for use on a substrate having an index of refraction of 1.43 to 2.00 which comprises a first layer of a low-index filming material deposited on the substrate and having a thickness of less than .lambda./4, a second layer of a high-index filming material deposited on the first layer and having a thickness of less than .lambda./4, a third layer of a low-index filming material deposited on the second layer and having a thickness approximately between 5.lambda./16 and 7.lambda./16, a fourth layer of a high-index filming material deposited on the third layer and having a thickness of less than .lambda./4, a fifth layer of a low-index filming material deposited on the fourth layer and having a thickness of less than .lambda./4, a sixth layer of a high-index filming material deposited on the fifth layer and having a thickness of more than .lambda./2, and a seventh layer of a low-index filming material deposited on the sixth layer and having a thickness of .lambda./4, wherein .lambda. is a selected wavelength of near ultraviolet range to near infrared range.

The high-index filming material may be one of ZrO.sub.2, TiO.sub.2, Nd.sub.2 O.sub.3, CeO.sub.2, TaO.sub.3, Ti.sub.2 O.sub.3, Pr.sub.6 O.sub.11, Ta.sub.2 O.sub.3.sup.. Pr.sub.6 O.sub.11 and InO.sub.2, and the low-index filming material is one of MgF.sub.2, SiO.sub.2, Na.sub.3 AlF.sub.6 and LiF. The low-index filming materials of the first, third, fifth and seventh layers are identical and the high-index filming materials of the second, fourth and sixth layers are identical.

The low-index filming material may preferably be MgF.sub.2, and the high-index filming material may preferably be ZrO.sub.2.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will become fully apparent from the following detailed description thereof taken in conjunction with the accompanying drawings, in which:

FIG. 1 is a diagram showing the manner in which may be provided for the wave number 0.7 by substituting for the substrate-adjacent layer in a conventional three-layer coating of the form substrate - .lambda./4 - .lambda./2 - .lambda./4 - medium;

FIG. 2 is a similar diagram for the wave number 1.3;

FIG. 3 is a diagram showing the manner in which improvement may be provided for the wave number 0.7 by substituting for the intermediate layer in said three-layer coating;

FIG. 4 is a similar diagram for the wave number 1.3;

FIG. 5 is a graph illustrating the spectral characteristics provided by a five-layer coating of the form substrate - .lambda./4 - .lambda./4 - .lambda./4 - .lambda./2 - .lambda./4 - medium and having the numerical data as given in Table I; and

FIG. 6 is a graph illustrating the spectral characteristics provided by a seven-layer coating of the present invention having the numerical data as given in Table II.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

The principles underlying the present invention will first be described in detail.

When the refractive index of the uppermost layer adjacent the medium in a coating of the conventional type substrate - .lambda./4 - .lambda./2 - .lambda./4 - medium is considered to be limited to physico-chemically stable materials such as MgF.sub.2, LiF, cryolite, etc., the intermediate layer has nothing to do with the overall reflection factor in the center wavelength range, that is, the intermediate layer becomes an "absent-layer," and therefore the refractive index of the substrate-adjacent layer can be determined by determining the residual reflection factor to be left in the center wavelength range and by determining the refractive index of the substrate.

Further, the refractive index of the intermediate layer, which has so far been irrelevant, can be readily obtained by determining the refractive index of the substrate-adjacent layer determined by the center wavelength range; by determining the refractive index of the uppermost layer adjacent the medium; and by determining the residual reflection factor allowed in the marginal wavelength range. This is disclosed in detail by French Pat. No. 1,005,866 (1952).

In order to further enhance the anti-reflection effect of the anti-reflection coating of the described type which is determined by the above-described method, there are the following two alternative methods typically in the case of the reflection factor for the wave number range of (0.7, 1.3) (.sigma. = 0.3).

1. To replace the substrate-adjacent layer by a layer of a higher refractive index than that of the intermediate layer and having a thickness of .lambda./4 + m.lambda./2, where m=1, 2, 3 and so on.

2. To replace the intermediate layer by a layer of a lower refractive index than that of the uppermost layer adjacent the medium and having a thickness of .lambda./2 + m.lambda. , where m=1, 2, 3 and so on.

The implementation of these two methods will be discussed with reference to FIGS. 1 to 4.

In these figures, the coating employs the basic form substrate - .lambda./4 - .lambda./2 - .lambda./4 - medium in which the refractive indices are 1.4, 2.0 and 1.52 for the uppermost layer, the intermediate layer and the substrate, respectively. The refractive index of the substrate is herein assumed to be 1.52 but the refractive index may be less than 1.52.

FIG. 1 shows the implementation of the method (2) for the wave number 0.7; FIG. 2 shows the implementation of the method (2) for the wave number 1.3; FIG. 3 shows the implementation of the method (1) for the wave number 0.7; and FIG. 4 shows the implementation of the method (1) for the wave number 1.3.

In these figures, reflection is represented according to the vector expression.

The circle centered at the base point 0 of the vector represents the range within which the reflection factor R is within 0.3%. In other words, if the end of the composite vector lies within such circle, the overall reflection is within 0.3%. Referring to FIG. 1, the vectors designated by .alpha. show a case where the refractive index of the substrate-adjacent layer in the three-layer coating is varied, and the ends of those vectors are passed by a dashed line p. The vector designated by .beta. shows a case where improvements have been made by assuming 3 .lambda./2 and order of 1.5 (1.50 - 1.59) for the thickness and refractive index of the intermediate layer, respectively. Similarly, in FIG. 2, the solid line vectors commencing at 0 represent the conventional anti-reflection coating of the form substrate - .lambda./4 - .lambda./2 - .lambda./4 - medium. The dashed line p shows a case where the refractive index of the substrate-adjacent layer is varied, and it is seen in this case that the residual reflection factor R' is much greater than 0.3%. The vector designated by .beta. shows a case where the refractive index of the intermediate layer is varied, and in this case the residual reflection factor R.sub.opt is approximately 0.3%. Also similarly, in FIG. 3, the dotted line s shows a case where the substrate-adjacent layer has a refractive index of approximately 2.5 and a thickness of 3 .lambda./4. This is also the case with FIG. 4. In both of FIGS. 3 and 4, the residual reflection factor R.sub.opt can be made approximately 0.3%.

Although the two methods are available as described above, it will be seen that the improvement provided by the method (1) is preferable in view of the possible deterioration of the angular characteristic resulting from the increase in the thickness of the coating.

When, according to the method (1), the reflection in the center range is eliminated in a coating of the form substrate - 3.lambda./4 - .lambda./2 - .lambda./4 - medium, there is established the relation as suggested by A.F. Turner:

n.sub.3.sup.2 = n.sub.1.sup.2 n.sub.s (1),

where n.sub.1 is the refractive index of the uppermost layer, n.sub.3 is that of the substrate-adjacent layer and n.sub.s is that of the substrate.

Where some residual reflection factor R is left in the center range, the equation (1) will become:

(1-.sqroot.R/1+.sqroot.R) n.sub.3.sup.2 = n.sub.1.sup.2 n.sub.s (1'),

which determines the refractive index of the substrate-adjacent layer in the center range. If, for example, the refractive indices of the substrate, the uppermost layer and the intermediate layer are 1.52, 1.39 and 2.0, respectively, then it will be seen that the lowermost layer has a refractive index of 2.5 and a thickness of 3.lambda./4 in the marginal range of wave number (0.7, 1.3) and has a refractive index of 1.67 and a thickness of 3.lambda./4 in the center range.

In a coating of the form substrate - .lambda./4 - .lambda./2 - .lambda./4 - medium, this means that the .lambda./4 layer adjacent the substrate may be replaced by a layer of the above-described refractive index and a thickness of 3.lambda./4.

Nevertheless, among the existing filming materials which are physico-chemically stable, there can be found no material whose refractive index is greatly variable with the great wavelength variation as shown in the foregoing example (such as a material having a refractive index of 1.67 in the center range and a refractive index of 2.5 in the marginal range).

On the other hand, in the case previously described, it is theoretically possible to satisfy the required great variation of the refractive index for the wavelength by utilizing the great variation in the equivalent refractive index expressed according to the theory of equivalent coating introduced by Herpin and in the equivalent refractive index expressed by the non-transmissive band. In such case, the fact that the refractive indices in the visible marginal range and in the near ultraviolet and near infrared ranges are variable with respect to the refractive index in the center wavelength range must be taken into account, and it is desirable that the spectral characteristics represented by the equivalent thickness and the equivalent refractive index should have a symmetrical property about the center wavelength in the wavelength range. This means that an assembly must be formed by a symmetrical coating having a thickness substantially equivalent to p .lambda./4 where p is an integer.

From such a theory, it is possible to satisfy the above-described property by a construction employing a combination of layers of p .lambda./4 where p is an integer and to overcome the limitations existing in the conventional form substrate - .lambda./4 - .lambda./2 - .lambda./4 - medium.

Generally, however, it is not so simple to find the filming materials whose refractive indices are suitable as the respective layer required to provide said construction.

Such a problem may be satisfied by a method which will be described hereinafter. By introducing a characteristic matrix IM which expresses the electric and magnetic fields in the interior of the coating in terms of the matrix, the electric field in the multi-layer coating may be expressed by the product of the characteristic matrices in the respective layers forming the coating. The nature of such characteristic matrices makes the refractive index symmetrical to some extent, but a multi-layer coating which is symmetrical about its thickness can be replaced by a certain equivalent single-layer coating within a range which satisfies the relation given below. If suffixes a, b and c are used to represent the substrate-adjacent layer, the intermediate layer and the medium-adjacent layer in an ordinary three-layer coating, then ##SPC1##

where g.sub.k = 2 .pi. n.sub.k d.sub.k /.lambda..sub.s .sup.. .lambda..sub.s /.lambda.,

where k represents a, b or c, n.sub.k is the refractive index, d.sub.k is the thickness of the coating, and j is the imaginary number unit (.sqroot.-1).

If there is a very little difference between the magnitudes of n.sub.a and n.sub.c, the following rewriting may be possible:

n.sub.a = n.sub.c (1 + .DELTA.n/n.sub.c)

If the .DELTA. n/n.sub.c is of the order of 0.05, ##SPC2##

This may further be simplified as follows, by multiplying the respective matrices by each other: ##SPC3##

where

H * = 2.pi. N*D*/.lambda..sub.s .sup.. .lambda..sub.s /.lambda.

If N is the equivalent refractive index of the symmetrical three-layer coating,

N* .apprxeq. N(1 + .DELTA.n/n.sub.c) (4)

where the sign .apprxeq. means approximately equal to. Such N* is referred to as the pseudo-equivalent refractive index of a pseudo-symmetrical three-layer coating. Further, let ND be the equivalent thickness of the three-layer coating and N*D* be the pseudo-equivalent thickness of the pseudo-symmetrical three-layer coating. Then, there is established the relation:

N*D* .apprxeq. ND (5)

these relations mean that the degree of freedom can be increased in the combination of the limited existing filming materials which are physico-chemically stable. It will thus be seen that, even in one and the same filming material, the degree of freedom of the expression of the equivalent refractive index can be increased by utilizing the difference in refractive index arising from the control of such factors as the degree of vacuum and temperature. As is apparent especially from formula (4) above, it can be uniformly increased or decreased by .DELTA.n/n.sub.c .times. 100(%) for the respective wavelengths. In the marginal wavelength range as shown in FIGS. 3 and 4, the refractive index greatly different from that in the center wavelength range can determine the construction of a three-layer coating which satisfies the required conditions by relating it with the non-transmissive band in such marginal wavelength range. In the symmetrical three-layer coating forming a 3.lambda./4 layer, let n.sub.u and n.sub.v be the refractive indices of the substrate-adjacent layer and the next layer and d.sub.u and d.sub.v be the thickness of three layers, respectively. Then, under the condition that

{2n.sub.u d.sub.u + n.sub.v d.sub.v = 3.lambda./4

n.sub.u d.sub.u = n.sub.v d.sub.v

there is obtained and equation:

.vertline.cos g.sub.H .vertline. = n.sub.u /(n.sub.u + n.sub.v) (6),

where g.sub.H = 2.pi.(n.sub.u d.sub.v /.lambda..sub.s)(.lambda..sub.s /.lambda..sub.H),

where .lambda..sub.s /.lambda..sub.H = .sigma. and .lambda..sub.H represents a wavelength in the wavelength range given as 1.25<.sigma. and 0.7>.sigma.. Thus, g.sub.H shows the relation between n.sub.u and n.sub.v by equation (6). Meanwhile, the value of the ideal refractive index of the substrate-adjacent layer is obtained by the equation (1'), considering the residual reflective index R within the center wavelength range. The equivalent refractive index N to the ideal refractive index may be expressed, as follows:

N=n.sub.u .sqroot.[n.sub.u n.sub.v sin Zg.sub.u cos g.sub.v + (n.sub.v.sup.2 cos.sup.2 g.sub.u - n.sub.u.sup.2 sin.sup.2 g.sub.u) sin g.sub.v /n.sub. u n.sub.v sin Zg.sub.u cos g.sub.v + (n.sub.u.sup.2 cos.sup.2 g.sub.u - n.sub.v.sup.2 sin.sup.2 g.sub.u) sin g.sub.v ] (6')

The values of n.sub.u and n.sub.v are determined from the above two relations (6) and (6') or from the relation (6), (6') and (4). There is thus provided a five-layer anti-reflection coating of the form substrate - .lambda./4 - .lambda./4 - .lambda./4 - .lambda./2 - .lambda./4 - medium, which is a wide-band anti-reflection coating effective for a wider range than the conventional three-layer anti-reflection coating of the form substrate - .lambda./4 - .lambda./2 - .lambda./4 - medium.

Some examples of the numerical data are given in the Table I below, where n.sub.A represents the refractive index of the medium (air), and n.sub.1 to n.sub.5 represent the refractive indices of the successive layers beginning with the uppermost or first layer, and n.sub.s represents the refractive index of the substrate.

Table I

a b medium n.sub.A = 1.0 n.sub.A = 1.0 .lambda./4 n.sub.1 = 1.39 n.sub.1 = 1.39 .lambda./2 n.sub.2 = 2.0 n.sub.2 = 2.1 .lambda./4 n.sub.3 = 1.62 n.sub.3 = 1.54 .lambda./4 n.sub.4 = 1.58 n.sub.4 = 1.39 .lambda./4 n.sub.5 = 1.64 n.sub.5 = 1.58 substrate n.sub.s = 1.58 n.sub.s = 1.74

The spectral characteristics provided thereby are illustrated in FIG. 5 in which the ordinate is the refractive index and the abscissa is wave length. The above forms the basic form of the present invention.

However, such basic form would encounter the problems which will be described hereinafter. Firstly, there are not always to be found filming materials required to provide the most suitable construction for several different refractive indices of the substrate. Moreover, even if such combination was selected, the use of four to five different filming materials would raise inconveniences in practice and also, it would be difficult to maintain a sufficient physico-chemical stability in these filming materials.

To overcome such difficulties, a combination of only two stable filming materials may be used by utilizing the fact that such combination can express any desired refractive index between the values of the refractive indices of the two materials.

According to the present invention, once the equivalent refractive index and equivalent thickness of the symmetrical coating are determined by the theory of equivalent coating, the thickness of the respective thin layers are primarily determined by that equivalent thickness so that no more room is left to consider the dispersion of the refractive index. However, the variation in the refractive index can be brought into consideration by introducing the asymmetry with respect to the thickness. For example, in the case of a three-layer coating of (n.sub.a,n.sub. a d.sub.a)(n.sub.b,n.sub. b d.sub.b)(n.sub.(,n.sub. c d.sub.c), where n.sub.a d.sub.a, etc. are optical thicknesses, asymmetry may be introduced with the aid of a variation .DELTA.d which is derived from the relation that d.sub.a =d.sub.c .+-..DELTA.d. Of course, .DELTA.d=0 means the provision of a symmetrical coating.

From this fact, it follows that the aforesaid basic form of five-layer coating can be expressed in terms of a seven-layer type of coating by using two different types of physico-chemically stable materials.

An example of this will be shown below.

A five-layer coating of the form substrate - .lambda./4,n.sub.1 - .lambda./4, n.sub.2 - .lambda./4,n.sub.3 - .lambda./2,n.sub.4 - .lambda./4,n.sub.5 - medium may be made into a nine-layer coating by substituting a symmetrical three-layer coating for the substrate-adjacent layer (.lambda./4,n.sub.1 layer) and the third layer (.lambda./4,n.sub.3 layer), respectively. If the refractive index alone is considered with the thickness neglected, such nine-layer coating may be expressed as: substrate -n.sub.11 -n.sub.12 -n.sub.13 -n.sub.2 -n.sub.31 -n.sub.32 - n.sub.33 - n.sub.4 - n.sub.5 - medium. Here, by establishing the relations that n.sub.11 =n.sub.13 =n.sub.2 =n.sub.32 =n.sub.5 and n.sub.12 =n.sub.31 =n.sub.33 =n.sub.4 and by substituting single layers for the two third and fourth layers (n.sub.13 and n.sub.2) and the two seventh and eighth layers (n.sub.33 and n.sub.4) respectively, the nine-layer coating may be made into a seven-layer coating.

Such seven-layer coating can be realized by employing a low-index material such as magnesium fluoride (MgF.sub.2), lithium fluoride (LiF), silicon oxide (SiO.sub.2) or cryolite and a high-index material such as titanium oxide (TiO.sub.2), zirconium, tantalum oxide (TaO.sub.3), indium oxide (InO.sub.2) or the like.

According to the present invention, each thickness of the first to seventh successive layers beginning with the uppermost medium adjacent layer ranges as follows: ##SPC4##

Table II below shows numerical data where ZrO.sub.2 (refractive index 2.0) is employed as the high-index material and MgF.sub.2 (refractive index 1.39) as the low-index material. In Table II, the thicknesses are all expressed in the unit of standard wavelength (.lambda..sub.s), and the first to seventh layers mean the successive layers beginning with the uppermost layer. The spectral characteristics provided by the combination as shown in Table II are illustrated in FIG. 6.

When a combination of other materials is desired it can readily be provided by expressing the equivalent refractive indices by the use of other materials according to the relation shown in Table II. Also, since the thicknesses of the respective layers for the various types of substrate shown in Table II are linearly correlated with one another, it will be apparent that the same result can be achieved not only for a substrate of optical glass but also for a substrate of single crystal such as CaF.sub.2, MgO or the like or for a substrate of any other refractive index. ##SPC5##

Claims

I claim:

1. A non-absorbing, substantially colorless, multi-layer anti-reflection coating for use on a substrate having an index of refraction of 1.43 to 2.00 comprising:

a first layer of a low-index filming material deposited on the substrate and having a thickness of less than .lambda./4;

a second layer of a high-index filming material deposited on said first layer and having a thickness of less than .lambda./4;

a third layer of a low-index filming material deposited on said second layer and having a thickness approximately between 5 .lambda./16 and 7 .lambda./16;

a fourth layer of a high-index filming material deposited on said third layer and having a thickness of less than .lambda./4;

a fifth layer of a low-index filming material deposited on said fourth layer and having a thickness of less than .lambda./4;

a sixth layer of a high-index filming material deposited on said fifth layer and having a thickness of more than .lambda./2; and

a seventh layer of a low-index filming material deposited on said sixth layer and having a thickness of approximate .lambda./4;

wherein .lambda. is a selected wavelength of near ultraviolet range to near infrared range.

2. A coating according to claim 1, wherein the high-index filming material is one of ZrO.sub.2, TiO.sub.2, Nd.sub.2 O.sub.3, CeO.sub.2, TaO.sub.3, Ti.sub.2 O.sub.3, Pr.sub.6 O.sub.11, Ta.sub.2 O.sub.3.sup.. Pr.sub.6 O.sub.11 and InO, and the low-index filming material is one of MgF.sub.2, SiO.sub.2, Na.sub.3 AlF.sub.6 and LiF.

3. A coating according to claim 1, wherein the low-index filming materials of said first, third, fifth and seventh layers are identical and the high-index filming materials of said second, fourth and sixth layers are identical.

4. A coating according to claim 3, wherein the low-index filming material is MgF.sub.2 and the high-index filming material is ZrO.sub.2.