Abstract

The optical matrix formalism is applied to find optical parameters such as focal distance, back and front focal points, principal planes and the object image equation for a thick spherical lens immerse in two optical media of different indexes of refraction. Then, the formalism is applied to systems compound of two, three and N thick lenses in cascade, immersed in three, four and N+1 optical media of different indexes, respectively. It is found that a simple Gaussian equation is enough to relate object and image distances no matter the number of lenses. This formalism is validated trough a simple optical matrix model of the human eye.

1. Introduction

Feynman, in his lecture book ¹, stated that for an optic system composed of an arbitrary number of thick lenses immersed in air, two principal planes can be calculated, respect to which it is possible to treat mathematically this system as an idealized thin lens. As a direct consequence, a simple Gaussian equation is enough to relate object and image position of this configuration. This feature was demonstrated in ². The objective of this work is to shown that this claim is extensive to an arbitrary optic system of thick lenses immersed in a succession of different optical media (i.e., with dissimilar index of refraction). It is used the matrix formalism to find optical parameters such as focus, back and front focal distances, principal planes and the Gaussian equation that relates object and image distances for the case of a thick lens made with a glass of index n_a, which is immersed between two optical media of index n₁ and n₂. Then, a two lenses system is treated, the second lens having a glass with index n_b and a third media of index n₃ to the right of this lens. Formally identical equations that describe focal points and principal planes were obtained for this case. Also, it is shown that a simple Gaussian equation relates object and image distances for this system. This analysis was extended to an optical system composed by N thick lenses in cascade in a succession of different optical media and identical results were found. As a consequence, a simple Gaussian equation relates image and object distances for this system.

Finally, this formalism is validated by developing a simple matrix model for the human eye, where a known optical parameter for this organ (back focal distance) was found with very good accuracy.

2. Matrix Optics

The optical matrixes used in this analysis, i.e., the refraction and the displacement matrixes, can be found in ³. These matrixes are deduced for the paraxial approximation, as a consequence, all our results are valid just in that case. Here, the results are shortly derived just for the sake of clarity. Then, application of these matrixes to develop the expressions for spherical thick lenses are shown. It is adopted the classical signal convention, which states that object distances to the left (right) of some reference interface will be positive (negative), image distances to the right (left) of such interface will be positive (negative). Regarding the curvature radios of surfaces, they will be positive (negative), if they were convex (concave). This convention is usually adopted in all specialized literature concerning the subject of optic.

Suppose an optical ray that makes an angle α₁ with respect to horizontal direction z (from here on, the optical axis of the system), is propagated in an optical medium with a refraction index n₁. Then, it hits a spherical medium with refraction index n_a at a height y on the surface, measure from the optical axis, suffers refraction and changes the angle with respect to the optical axes to α₂, see Figure 1.

In Figure 1, ‘R’ is the radium of the sphere. The equation that describes refraction is:

(1)

In Eq. (1), if ‘R’ tends to infinity, it reduces to Snell law (in paraxial form), as expected.

The Eq. (1) can be expressed in a matrix form, by defining the vector (nα, y):

(2)

Applying the operation of matrix product, the result corresponding to the n_aα_a component reproduces Eq. (1). The other term, y = y₁, accounts the obvious fact that no change in height is verified at the point of refraction.

We describe now the change of height, i.e., in ‘y’ coordinate of the optical ray as it propagates in an optical medium of index n_a. See Figure 2.

As the ray propagates a distance d measured over the optical axis in the medium with refraction index n_a, the change in y coordinate is given by:

(3)

The paraxial approximation implies α1 << 1, so, tanα1~ α1, Eq. (3) becomes:

(4)

Now, Eq. (4) can be written in matrix form as:

(5)

Performing the matrix product, the expression that corresponds to the element ‘y₂’ reproduces Eq. (4), while the result corresponding to the component n₂α₂ reproduces the Snell law, in paraxial form, at the interface between media.

3. Thick Lenses between Two Optical Media, Focal Points and Principal Planes

The spherical thick lens immersed between two different optical media, consist in two refractive spherical surfaces of radios R₁ and R₂, separated a distance d over the optical axis, is shown in Figure 3.

The references point that will be initially used in the lens are the vertices points V_1a and V_2a, which can be seen in Figure 3 at the intersections of the spherical surfaces with the optical axis. In the same figure, it can be seen a generic optical ray propagating in the first media with index of refraction n₁ with a α₁ angle with respect to the optical axis, which it is refracted by the first surface of the lens, at a height y_1. Then the ray propagates inside the lens (a distance d_a measured over the optical axis) and is refracted by the second surface, leaving the lens with an angle of α₂ at a height y₂ with respect to the optical axis, following further propagation into the n₂ medium. The mathematical relation between angles and heights in the two surfaces can be expressed as:

(6)

An intuitive way to compose/understand Eq. (6) is by visualizing it from right to left: first, we write column matrix corresponding to the input ray, propagating in the n₁ optical medium, hitting the lens with angle α₁ at a height y₁. The following matrix describes the refraction by the first surface with radio R_1a. Then the ray is propagated a horizontal distance d_a inside the lens, which has a refraction index n_a, which is described by the propagation matrix. Finally, the ray is refracted by the second refractive surface, with radius R_2a, in contact with the n₂ medium, described by the last matrix (or the first from left to right, after the equal sign). This gave the resultant optical ray leaving the lens, characterized by α₂ and y₂ (propagating in the n₂ medium).

Performing the matrix product, Eq. (6) results:

(7)

Where the matrix elements are specified by:

(8)

(9)

(10)

(11)

Note that if n₁ = n₂ = 1, i.e., a thick lens immersed in air, the element a₁₂ in the matrix becomes equal to minus the inverse of the focal distance for a thick lens in air as described in ³. Furthermore, if d_a = 0 such element becomes the (negative of the inverse) familiar expression for the focus of an ideal thin lens. From here on, the element a₁₂ will be referred to as -1/f_a.

Now it will be deduced the back focal length, z_b, this is the distance, measured from V_2a, to which a thick lens focalizes a bunch of optical ray incident parallel to the optical axis, as shown in Figure 4.

This concept can also be though in terms of waves. A plane wave front (being the wave fronts perpendicular to the optical rays) is refracted by the lens and a convergent spherical wave is produced. The center of this wave is at back focal point. Some of these fronts are shown in Figure 4.

In order to find the expression of the height y over the optical axis, of an optical ray that has travelled a horizontal distance z in the n₂ medium after leaving the lens, it must be used, in addition to the thick lens matrix, the displacement matrix in the following way:

(12)

Again, it´s convenient to read Eq. (12) from right to left, i.e., an input ray (n₁α₁ y₁) is refracted by the lens, and then propagates a horizontal distance z in the medium with index n₂. Performing the product of the square matrixes, Eq. (12) becomes:

(13)

The expression for y:

(14)

In our particular case α₁ = 0. By definition, the back focal point is that value of z that makes y null for all values of y₁. The result is: z_b = -n₂a₂₂a₁₂ Writing z_b in terms of the matrix elements explicitly:

(15)

Regarding the signal, back focal length is an image point, and the corresponding signal convention should be applied, i.e., if the result is positive, it is localized to the right of V_2a, (as schematically shown at Figure 4, where a positive value of z_b is supposed), if negative, it is localized to the left of V_2a.

Now it can be deduced the front focal length, z_f, that is the distance, measure from V_1a on the optical axis, in which an object point (or a source of spherical divergent wave fronts), must be located in order that a thick lens produce a bunch of parallel rays after refraction (corresponding to plane wave fronts). The situation is pictured in Figure 5.

Now we write the expression to find (n₂α_2, y₂):

(16)

Note the order in the matrix’s product in Eq. (16), indicating that a ray with height y and angle α₁ is propagated a certain distance z (in optical medium n₁), before hitting the lens. Performing the product, analyzing the term corresponding to n₂α₂ element, and imposing the conditions y = 0, and α₂ = 0, it can be easily deduced that the point over the optical axis that corresponds to z_f is:

(17)

Front focal point is an object point and, if positive, should be located to the left of V_1a (as indicated in Figure 5, where that case was supposed). If negative, it should be localized to the right of V_1a.

Now there will be defined and studied the principal planes.

In Figure 6, it is shown the front focal point for a given thick lens, and a generic optical ray departing from it, with an angle α₁, being refracted by the first surface of the lens at a certain height y₁ and then refracted by the second surface at height y₂.

The output ray is now projected back until intersect the projection of input ray. Doing this for every possible α₁ angle of those optical rays departing of z_f being refracted by the thick lens, it is obtained a plane perpendicular to the optical axis, at a certain distance x from z_f. This plane is called principal plane and will be very useful in order to simplify the equations relating object and image distances. To find the distance x, the angle α₁ in the paraxial approximation can be expressed as:

(18)

Then x is:

(19)

To find y₂ we write, from Eq. (7):

(20)

Writing the expression for α₁ in the paraxial approximation and rearranging, Eq. (19) takes the form:

(21)

Using the explicit form for f_a, i.e., Eq. (9), in Eq. (21) and after a little algebra, it is easily shown that:

(22)

It can be also generated another plane, using the concept of back focal point in an analogous way, and it can be found that the distance between this second plane and z_b is n₂f_a, see Figure 7.

These two planes are called first principal plane, from here on 1PP, and second principal plane (2PP), see Figure 8.

It should be noted that not always the principal planes will be inside the lens, as shown in Figure 8. Depending on the specific lens data, they could be outside the lens.

It is convenient to localize the 1PP and 2PP with respect to the V_1a and V_2a vertices points, respectively; these distances are called h₁ and h₂ (see Figure 8), and their expressions are:

(23)

(24)

From the Eq. (23), it can be seen that distance h₁ obeys the following signal convention: if positive (negative), it is located to the right (left) of V_1a. For h₂, (see Eq. (24)) if positive (negative), h₂ must be located to the right (left) of V_2a.

4. Object-Image Equation for Thick Lens

Now, it will be deduced the equation that relates object and image distances for a thick lens. Consider an object, with height h_o to a distance s_o from V_1a in an optical medium with index n₁. An image will be formed by the thick lens at distance s_i, measured with respect to V_2a, and height h_i, in an optical medium with index n₂ as depicted in Figure 9.

The optical rays from object points will propagate a distance s_o over the optical axis, they are refracted by the thick lens and propagate a distance s_i, to the corresponding image point. In mathematical terms:

(25)

Where the matrix A_ij represent the thick lens, whose elements were already specified in Eqs (8-11).

Performing the operation indicated in Eq. (25), we arrive to:

(26)

To mathematically found the image condition, consider an object point, for instance, the one at the top of the object. This point is a font of spherical divergent waves in media n₁ (some of these wave fronts are shown in Figure 9). The lens, to form an image, should convert this divergent wave front into a convergent one, whose convergence point is the image corresponding to the aforementioned object point. From Eq. (26), the expression corresponding to h_i:

(27)

The height of a specific image point should not depend on the angle α₀. Therefore, the image condition is:

(28)

Applying Eq. (28) in Eq. (27):

(29)

We explore Eq. (29) when d_a = 0, i.e., a thin lens. In that case, the matrix elements become: a₁₁ = 1, a₁₂ = -1/f_a, (with f_a now being the focal distance for a thin lens), a₂₁ = 0 e a₂₂ = 1. With these values for the matrix elements, Eq. (29) becomes the Gaussian formula for thin lenses. Furthermore, making n₁ = n₂ = 1, it is obtained the well-known Gaussian expression for thin lenses immersed in air.

Now, in order to write Eq. (29) in a simpler way, a transformation over the object and image coordinates is introduced:

(30)

(31)

Introducing Eqs. (30) and (31) in Eq. (29), and rearranging:

(32)

From Eq. (32) is easy to see that, in order to obtain an expression that mathematically be identical to the Gaussian formula for thin lens, the following conditions should be imposed:

(33)

(34)

(35)

Solutions to Eqs. (33) and (34) in terms of o and i give:

(36)

(37)

Equations (33) and (34) make Eq. (35) be fulfilled (see the appendix). Replacing the corresponding elements of the lens matrix in Eq. (33) and Eq. (34), and resolving for o and i we obtain:

(38)

(39)

These expressions, when substituted in Eq. (30) and Eq. (31), determine that the distances s_o` and s<sub>i</sub>` should be measure with respect to the 1PP and 2PP respectively, in order to obtain a Gaussian type lens equation. See Figure 10:

In this way, Eq. (32), with the conditions indicated in Eqs. (33-35) and the solutions for o and i indicated in Eqs. (38-39), becomes:

(40)

Making d_a=0, i.e., a thin lens, the two principal planes coalesce into one (which is the idealized thin lens), and in that case, s_o = s´_o and s_i = s´_i.

5. Two Thick Lenses Systems

Let´s suppose a system of two thick lenses, characterized by matrixes A and B, and separated a distance d_ab immersed in optical medium with index of refraction n₁ (to the left of lens A), n₂ (between the lenses), and n₃ (to the right of second lens, B). The lens A has characteristics parameters indicated by the matrix elements of Eqs. (8-11)_. For the lens B, with index n_b, thickness d_b, radius R_1b, and R_2b, the matrix elements are:

(41)

(42)

(43)

(44)

The system is shown in Figure 11. Note that we have now four vertices’ points, V_1a, V_2a, V_1b, and V_2b.

We write, for the matrix of the two lenses system:

(45)

The resultant matrix has elements:

(46)

(47)

(48)

(49)

We develop now the element M_ab12 of the compound system using the matrix elements explicitly. As already seen for one single lens, the element “1-2” is related to the focal distance. After an algebra, Eq. (47) becomes:

(50)

Making d_a= d_b = 0, i.e., two thin lenses, Eq. (50) reduces to the well-known expression of the (negative of the inverse) focal distance for two thin lenses separated by a distance d (where f_a and f_b, of course, are reduced to the expressions of focal distances for thin lenses). For convenience, we call this term as -1/f_ab and this will be justified later on.

We can determinate the back focal length for this system, proceeding in an analogous way as we did for the single thick lens (see Eq. (12)), i.e.:

(51)

Writing the expression for y:

(52)

For back focal point, α₁=0, and we must find the value of z that makes y null for all values of y₁:

(53)

Writing M_ab12 as -1/f_ab:

(54)

It is easy to see that this expression is formally identical to that find to describe z_b for a single thick lens, Eq (15), where n₂ is now conveniently substituted by n₃, index of refraction to the right of the two-lens system taken as a whole.

Also, we can find the front focal length for the compound system of thick lenses, using a reasoning analogous to that used in Eq. (16). The result is:

(55)

Once more, we observe that the expression is formally identic to the corresponding front focal length for a single thick lens, Eq. (17).

The next step is to find the principal planes for the two thick lens system and find the relation with the focal distance, f_ab. See Figure 12.

In the Figure 12 above, we find the 2PP for the two thick lens system using the property of back focal point, z_b, projecting forward the input ray (corresponding to α₁) and projecting back the output ray (corresponding to α₃). Then, the angle α₃ (in the paraxial approximation):

(56)

Then, we can write for x:

(57)

The relation between α₃, y₃ and α_1, y₁ is find from the expression:

(58)

From Eq. (58) we obtain, for α₃:

(59)

By of the properties of z_b, α₁ =0, and using Eq. (56) in Eq. (59):

(60)

From which is trivial obtain for the distance x:

(61)

The minus signal indicates that, as this distance is formally measure having z_b as origin, it will be pointing to the left (right) if f_ab is positive (negative).

Working with the front focal point, z_f, it is possible to find the 1PP for the two thick lenses system, and demonstrate that the value of the distance from z_f to this plane is n₁f_ab. These results are formally identic with those found for a single thick lens immersed in two media and justified the identification of M_ab12 with the negative of the inverse of the focal distance for the two thick lenses system.

The next steps now are to find the distances of the 1PP and 2PP, h₁ and h₂, which will be measured with respect to the outermost vertices points of the system, i.e., V_1a e V_2b respectively, and also determinate the image condition.

In order to find the distances of the principal planes respect to the outermost vertices’ points, we write:

(62)

(63)

Where Eqs. (54) and (55) were used to express front and back focal distances for the two thick lens system. As before, h₁ and h₂ follow the signal convention for object and images, respectively.

To find the object- image equation, we initially measure the object distance, s_o, with respect to V_1a and image distance, s_i, with respect to V_2b, see Figure 13:

The matrix equation relating object-image distances in the two thick lenses system is:

(64)

We see that Eq. (64) is formally identic to Eq. (25), furthermore, it is verified that the image condition is:

(65)

Which is formally identic to Eq. (29), therefore, we can now follow the same reasoning of section 4, Eqs. (30) to (40), and write the image condition for the two thick lenses system in the simple form of a Gauss equation. To achieve this, object and image distances should be measured with respect to the principal planes, and the equation for the two thick lenses system is:

(66)

6. Three and N Thick Lenses, Image Condition

The expressions for a succession of N thick lenses systems immersed in different optical media follow the same general structure. Consider a three thick lenses system, A, B, and C, being the two lenses system A-B, identical to the one described in the last section, and C lens is located to a distance dbc to the right of B lens. Between the lenses B and C, we have the optical medium, with index n₃, and to the right of C lens, there is an optical medium with index n₄. The C lens has matrix elements mathematically identic to those represented in the Eqs. (41) to (44), and characterized by parameters; refraction index n_c, thickness d_c, radius R_1c, and R_2c. The matrix for this system is:

(67)

And the matrix elements:

(68)

All the optical parameters for this system, i.e., front focal point, back focal point, and principal planes, can be obtained from these elements, using the expressions already seen for the two thick lenses system, and substituting the corresponding matrix element by the analogous element belonging to the three lenses system. As before, the focal distance is associated to the M_abc12 element. The Equations (68) can be written in a compact form:

(69)

Where ij represents the index of the matrix.

Finally, we can write the expression for the matrix elements of a system of N thick lenses:

(70)

Where n_ij represents the matrix elements corresponding to the last thick lens of the system (i.e., the last on the right), and d_n-1n, the distance between this last lens and the one immediately at left, N-1. Do not confuse n_ij with the index of refraction of media, n_m. Also, regarding notation, `n´ in subindex stands for the total number of lenses, characterized as `N ` in the text.

The matrix elements of Eq. (70) can be used to find all optical parameters, i.e., back and front focal points and principal planes:

(71)

(72)

(73)

(74)

Also, we can relate object and image distances with a simple Gaussian equation that in this case takes the general expression:

(75)

keeping in mind that the focal distance is related to the M_ab..n,12 element through the expression:

(76)

and the object an image distances should be measures with respect to the 1PP and 2PP, respectively.

7. Optical Matrix Model of the Human Eye

In this section, it is developed a simple matrix model for the eye using the formalism previously developed. There will be used the parameters of the theorical eye model of Gullstrand-Le Grand ⁴. A schematic is shown in Figure 14:

The correspondent parameters are shown in Table 1:

To validate the model developed, it will be calculated the distance d_br, which is the distance between the second lens of the eye (lens B in our model), to the retina, where the real image of an object placed at infinite should be formed considering a healthy and totally relaxed eye. Gradual variation of the index of refraction is not taken account in this model. With the parameters corresponding to the lens A and B, and the media between (and surrounding) them, is obtained the matrix M_ab for the human eye:

From the M_abeye values and the expressions derived in this work, we can calculate all optical parameters of interest for our theoretical model of the eye, which are summarized in Table 2.

We can see from Table 2, the value corresponding to z_b, which, by definition, is the distance measured from the intersection of R_2b with the optical axis, to the place where a real image is formed for an object placed at infinity. The result is in very good agreement with the value of d_br shown in Table 1.

8. Conclusions

In this work, a system of N thick lenses immersed in a succession of different optical media has been studied. Using the optical matrix formalism, and beginning from a single lens immersed between two optical media, there has been determined characteristics parameters, such as focal distance, back and front focal points, principal planes and the equation relating the object and image distances. Such expression takes the simple form of a Gaussian type equation, formally identical to that used to studied the idealized case of thin lenses, when object and image distances are measured with respect to the principal planes. All the results obtained in this work, are reduced to the expressions found in ² when all index of refractions of the optical media between the lens (and also the one before the first lens and the last after the N lens) become 1, i.e., a system immersed in air.

Finally, the formalism is validated trough a simple optical matrix model of the human eye, where a known optical parameter of the eye, the back focal length, was calculated.

References

Appendix

The objective of this appendix is to demonstrate that the following identity is true:

(A1)

Where o and i are given by Eqs. (36) and (37). Substituting those equations in Eq. (A1) we have:

(A2)

After an algebra and further simplifications, Eq. (A2) is reduced to:

(A3)

Substituting the expressions for the matrix elements a_ij, Eq. (A3) becomes:

(A4)

After the usual algebra:

(A5)

The term between brackets in Eq. (A5) is -1/f_a, and Eq. (A1) is demonstrated.