JPH09197458A - Image processor - Google Patents

Abstract

PROBLEM TO BE SOLVED: To make it possible to execute parallel operation even if the number of elements of vectors is increased and the number of the vectors to be operated is increased by disposing a second lens between a lens array and an input vector display element. SOLUTION: The luminous fluxes reading out the reference vector of a liquid crystal spatial light modulator(SLM) 201 as a reference vector array display section are made incident on the SLM 204 as an input display vector display section and the input vector is read out. The luminous fluxes are condensed by each component to the position of a detector 206 by the third lens 205 and the light quantity thereof is detected and is subjected to inner produce operation. Here, the second lens 203 is disposed between the lens array 202 and the SLM 204. As a result, even if the position of the lens array 202 is apart from a central axis 210, the second lens 203 acts so that the main light passing the lens array 202 arrives at the central axis 310 on its imaging plane and, therefore, the influence of the aberrations by the distance between the optical axes thereof can be greatly lessened.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to an image processing apparatus, and more particularly to an apparatus for obtaining a product of a vector and a matrix in parallel at high speed using light.

[0002]

2. Description of the Related Art As a conventional optical computing device capable of computing the product of a two-dimensional vector and a matrix, for example, "SELF-ORGANIZIN
G OPTICAL NEURAL NETWORK FOR UNSUPERVISED LEARNIN
G "OPTICAL ENGINEERING / 1990 / Vol.29, No.9, pp.1107-11
Some are proposed in 13. This apparatus uses an optical system configured as shown in FIG. 21, and has a two-dimensional input vector x = {x _ij : i = 1, ..., A; j = 1, ..., Having a × b elements.
b} and a two-dimensional reference vector m _kl = {m _klij : i = 1, ..., a; j = 1, ..., b} having a × b elements in the same manner as M × N two-dimensional Dot product with matrix arranged in an array, Has been realized.

For example, specifically, the reference vector m
_kl is a 4 × 4 element (a =
4, b = 4), and 6 × 6 of them (M =
6, N = 6) are arranged and displayed on the spatial light modulator (hereinafter abbreviated as SLM) 101. Moreover, as shown in FIG. 2B, the input vector x has 4 × 4 elements (a = 4, b = 4), and S
It is supposed to be displayed on the LM 103. In this case, SLM10
The information read by the reading light 106 from 1 is M × N
The lens array 102 superimposes the i, j component of the reference vector m _kl for each k, l on the corresponding i, j component of the input vector x displayed on the SLM 103 arranged on the image plane. Is imaged as. The light flux in which the information of the input vector x is superimposed on the information of the reference vector m _kl is further condensed by the lens 104 at corresponding positions on the M × N light receiving arrays 105 for each reference vector. This corresponds to the fact that the product of each i, j component is calculated where the reference vector m _kl and the input vector x are superposed, and that the product of the multiplications is added at the portion where light is collected. Therefore, it is understood that the amount of light detected by the light receiving array 105 is information proportional to the calculation result of the above formula (1).

[0004]

For example, in order to perform a large amount of information calculation in parallel, when the input vector x has a large number of elements (the number of a and b) or the reference vector m _{kl to be} calculated. It is necessary to increase the number of (number of M, N).
In the above-mentioned conventional example, as shown in FIG. 21, in order to increase the number of elements (the number of a and b) of the two-dimensional vector to be calculated, the size of the image displaying the vector may be increased. , by increasing the size of the image, in SLM101 displaying a reference vector m _kl, the distance d from the central axis 109 of the position and the optical system that displays the reference vector m _kl is increased, optical by this The amount of aberration generated in the system becomes large, and the number of elements of many vectors cannot be handled.

Specifically, the focal length of the lens array: 10 mm, the size displaying the reference vector m _kl : vertical and horizontal dimensions 2 ×
2 mm Number of reference vectors: M × N = 10 × 10 Size displaying input vector x: Vertical and horizontal dimensions 10 ×
The optical system whose optical path is shown in FIG.
The optical paths are shown only for 2 and 5. ), The position of the reference vector m _kl in the reference vector array (SLM 101) is the farthest position (the position. The amount of deviation is 12.7 mm).
3 (T represents the tangential direction and R represents the radial direction). In the worst case, MTF5
It can be seen that the 0% resolution is about 0.5 lines / mm, which is considerably low. Therefore, a large number of vectors cannot be calculated due to the low resolution, and therefore a large amount of calculation cannot be performed. Note that FIG. 24 is an aberration diagram showing spherical aberration (a) in the horizontal direction, spherical aberration (b) in the vertical direction, and astigmatism at that position.

The present invention has been made in view of such problems of the prior art, and an object thereof is to eliminate the above-mentioned drawbacks of the prior art and to reduce the number of elements of a vector (the number of a and b). An object of the present invention is to provide an image processing apparatus suitable for an optical operation device capable of performing parallel operation even if the number of vectors to be operated (the number of M and N) is increased.

[0007]

An image processing apparatus of the present invention which achieves the above object, includes a reference vector array display section for displaying a reference vector array representing characters, numbers, symbols, etc. in one dimension or two dimensions. A light source for reading the reference vector array, a lens array arranged in a one-to-one correspondence with the reference vector array, and a light flux emitted from the light source and passing through each of the reference vector arrays and the lens array can be simultaneously incident. A second lens having an entrance pupil of various sizes, and an input vector display unit arranged on the image plane for the reference vector array display unit related by the lens array and the second lens, and displaying an input vector. And each reference vector of the reference vector array, and the light flux that has passed through the input vector display unit is condensed for each reference vector of the reference vector array. A third lens for the third
A detector for detecting the light flux condensed by the lens is provided, and the calculation between the vectors is performed by superposing each reference vector of the reference vector array and the input vector.

In this case, it is desirable that the second lens comprises at least one negative lens and at least two positive lenses. Further, it is desirable that the second lens be a retrofocus type lens system.

Further, it is desirable that the position of the lens array is near the front focal point of the second lens or farther from the object side on the object side.

Hereinafter, the reason why the above structure is adopted in the present invention and the operation thereof will be described. Figure 1 is 4x4
An input vector (a = 4, b = 4) with 3 elements,
An example of a vector matrix calculation with a reference vector array in which reference vectors having the same number of elements as this are arranged in an array of 6 vertically and 6 horizontally (M = 6, N = 6) will be shown. Here, an electric address type liquid crystal spatial light modulator (SLM) 201 is used as a reference vector array display unit, and an electric address type liquid crystal spatial light modulator (SLM) 204 is similarly used as an input vector display unit.

As shown in FIG. 2A, the SLM 201 displays 6 × 6 reference vectors having 4 × 4 elements in an array, and the SLM 204 displays as shown in FIG. 2B. Display an input vector with 4x4 elements. S
The reference vector array displayed on the LM 201 is read by the reading light 207 and is imaged by the lens array 202 and the second lens 203 so that the respective patterns are overlapped at the position where the SLM 204 is placed. In this case, the individual reference vectors m _kl displayed in an array form have their centers on the central axis 210, and the individual components i, j of the reference vector m _{kl have} the individual components i, j of the input vector x. Arranged correspondingly.

Therefore, the light flux from which the reference vector of the SLM 201 is read out enters the SLM 204, and the SLM 20
The input vector x displayed in 4 is read. SLM204
The light flux emitted from the SLM 201 and the information displayed on the SLM 201
It is a superposition of the information displayed on the LM204, and _mijkl x
_ij is calculated. Therefore, if the third lens 205 condenses the components of k and l at the position of the detector 206 and the amount of light is detected by the detector 206, the above (1)
This means that the inner product of the expression was calculated.

At this time, by using the second lens 203 between the lens array 202 and the input vector display element 204, as shown in the optical path in FIG.
Is separated from the central axis 210, the chief ray passing through the lens array 202 is not reflected by the central axis 210 on the image plane 204.
Since the second lens 203 acts so as to reach, the influence of aberration due to the distance between the optical axes can be greatly reduced. Therefore, the resolution is greatly improved, and the number of vectors (the number of a and b) that can be handled and the number of matrices (M,
It is also possible to increase the number of N).

When it is desired to increase the number of matrices (the number of M and N), it is necessary to increase the number of lens arrays 202. For that purpose, the second lens 203 is required to be a lens having an extremely large aperture. For this reason, it is particularly difficult to correct spherical aberration, coma and the like. Therefore, at least one negative lens and two positive lenses are required.

Further, the position of the lens array 202 is set to the second position.
If the lens 203 is arranged near the front focus position of the lens 203 or farther from the second lens 203 than that, as shown in FIG. 3, the principal rays 301, 302, 3 of the light flux incident on the image plane are formed.
03 does not become a large divergent light flux, so the third lens 2
It is necessary to increase the lens diameter of the third lens 205 or to arrange it near the image formation surface without making the light flux incident on the beam 05 incidentally larger than the size of the image formed on the SLM 204 (D in the figure). Therefore, the aberration correction by the third lens 205 becomes easy.

On the other hand, as shown in FIG. 4, the second lens 20
By increasing the distance f _b from the final surface of 3 to the image forming surface 204-1 by using a reverse telephoto type or the like, for example, the beam splitter 404 or the like is formed on the second lens 203 and the image forming surface 20.
It becomes possible to insert between 4-1, and as shown in FIG. 4, it is possible to simultaneously perform the operation of the expression (1) on the two input vectors x _ij1, x _ij2 .

[0017]

BEST MODE FOR CARRYING OUT THE INVENTION Next, Embodiments 1 to 4 of the image processing apparatus of the present invention will be described. The specifications of each embodiment shown here are the same as in the case of FIG. 22, and are as follows. Working wavelength: 656.28 nm Focal length of lens array: 10 mm Focal length of second lens: 50 mm Size of matrix array: vertical and horizontal dimensions 2 × 2 mm Number of matrix arrays: k × l = 10 × 10 Size of input vector x _ij Size: vertical and horizontal size 10 × 10 mm Size of light receiving array: vertical and horizontal size 1.3 × 1.3 mm In the following optical path diagram, the one corresponding to three reference vectors is shown.

An optical path diagram of the first embodiment of the present invention is shown in FIG.
Numerical data will be described later, but the MTF for the image height ratio of -1,1,0 when the reference vector is at the position farthest from the central axis as shown in FIG. 5 and the aberration is largest is shown in FIG. FIG. 13 is an aberration diagram showing spherical aberration (a) in the lateral direction, spherical aberration (b) in the longitudinal direction, and astigmatism (c).

In this embodiment, lens array 202
Is a radial type gradient index lens, the second lens 203 is a three-lens configuration lens, and the third lens 205 is also a three-lens configuration lens.

The MT of the conventional imaging optical system shown in FIG.
F, when compared with the aberration diagram shown in FIG. 24, it can be seen that the performance is greatly improved. Further, the spot diagram at the position on the final surface is as shown in FIG. 17, and it is clear that the luminous flux is contained in a certain area and the vector matrix calculation is performed.

An optical path diagram of the second embodiment of the present invention is shown in FIG.
Numerical data will be described later. Similarly, the MTF for the image height ratio of -1,1,0 in the image formation by the lens array 202 and the second lens 203 when the reference vector is placed at the position shown in FIG. Fig. 10 (T in the tangential direction,
The radial direction is shown as R. ), A horizontal spherical aberration (a), a vertical spherical aberration (b), and an astigmatism (c) are shown in FIG.

In this embodiment, the lens array 202
Is a radial type gradient index lens, the second lens 203 is a five-element lens, and the third lens 205 is a two-element lens.

From the MTF of FIG. 10 and the aberration diagrams of FIG. 14, it can be seen that the resolution is greatly improved as compared with the three-element configuration of the first embodiment. Further, the spot diagram at the position on the final surface is as shown in FIG. 18, and it is clear that the luminous flux is contained in a certain area and the vector matrix calculation is performed.

An optical path diagram of the third embodiment of the present invention is shown in FIG.
Numerical data will be described later. Similarly, the MTF for the image height ratio of -1,1,0 in the image formation by the lens array 202 and the second lens 203 when the reference vector is placed at the position shown in FIG. In Fig. 11 (T is the tangential direction,
The radial direction is shown as R. ), A horizontal spherical aberration (a), a vertical spherical aberration (b), and an astigmatism (c) are shown in FIG.

In this embodiment, the lens array 202
Is a radial type gradient index lens, the second lens 203 is a 7-element lens, and the third lens 205 is a 2-element lens.

From the MTF of FIG. 11 and the aberration diagrams of FIG. 15, it can be seen that the resolution is improved more than that of the five-element structure of the second embodiment. Similarly, the spot diagram at the position on the final surface is as shown in FIG. 19, and it is clear that the luminous flux is contained in a certain area and the vector matrix calculation is performed.

An optical path diagram of the fourth embodiment of the present invention is shown in FIG.
Numerical data will be described later. Similarly, the MTF for the image height ratio of -1,1,0 in the image formation by the lens array 202 and the second lens 203 when the reference vector is placed at the position shown in FIG. In Figure 12, (T is the tangential direction,
The radial direction is shown as R. ), A lateral spherical aberration (a), a vertical spherical aberration (b), and an astigmatism (c) are shown in FIG. 16.

In this embodiment, the lens array 202
Is a radial type gradient index lens, the second lens 203 is a reverse telephoto type lens having a four-element configuration, and the third lens 205 is
Uses a three-lens structure. Second lens 203
By using the reverse-telephoto type for the, it is possible to increase the distance from the final surface of the second lens 203 to the surface 204 displaying the input vector without significantly reducing the resolution, and by using a beam splitter or the like. Operation with a plurality of input vectors is possible. Further, the spot diagram at the position on the final surface is as shown in FIG. 20, and it is clear that the luminous flux is contained in a certain area and the vector matrix calculation is performed.

Numerical data of Examples 1 to 4 will be shown below. Symbols are other than those mentioned above, r ₁ , r ₂ ... Are radii of curvature of each lens surface, d ₁ , d ₂ ... Interval of n ₁ ,
n ₂ ... Is the refractive index of the wavelength of 656.28 nm of each lens. Note that r ₀ is the SLM 201, d ₀ is the distance between the SLM 201 and the first surface of the lens array 202, and d _i is the distance between the final surface of the third lens 205 and the surface of the detector 206. The radial gradient index lens is described as "GRIN". The wavelength 65 of this radial type gradient index lens
The refractive index distribution at 6.28 nm is expressed by the following equation (2).

N (r) = N ₀₀ + N ₁₀ r ² + N ₂₀ r ⁴ + N ₃₀ r ⁶ + N ₄₀ r ⁸ + ... (2) Here, r is the light of the radial type gradient index lens. Radial distance from the axis, N (r) is the refractive index at that radius r, N ₀₀
Is the refractive index on the optical axis, and N ₁₀ , N ₂₀ , N ₃₀ , and N ₄₀ are second-order, fourth-order, sixth-order, and eighth-order refractive index distribution coefficients, respectively.

Example 1 r ₀ = ∞ d ₀ = 8.758664 r ₁ = ∞ d ₁ = 4 "GRIN" r ₂ = ∞ d ₂ = 10 r ₃ = 55.84989 d ₃ = 9.439465 n ₁ = 1.513855 r ₄ = -54.6892 d ₄ = 4.439465 r ₅ = -30.33959 d ₅ = 4.439465 n ₂ = 1.64209 r ₆ = -77.1591 d ₆ = 4.439465 r ₇ = 87.92463 d ₇ = 9.439465 n ₃ = 1.513855 r ₈ = -47.78966 d ₈ = 37.904817 r ₉ = ∞ d ₉ = 15 r ₁₀ = 167.40974 d ₁₀ = 9.5 n ₄ = 1.618773 r ₁₁ = -64.29427 d ₁₁ = 10 r ₁₂ = -995.84663 d ₁₂ = 9.5 n ₅ = 1.689552 r ₁₃ = -58.80883 d ₁₃ = 20 r _{14 = 28.37048 d 14 = 9.5 n 6} = 1.64209 r 15 = 78.57573 d i = 14.976265 "GRIN" n 00 = 1.6461 n 10 = -0.1304 × 10 -1 n 20 = 0.4749 × 10 -3 n 30 = -0.1619 × 10 - ³ N ₄₀ = 0.7672 × 10 ^-9 .

Example 2 r ₀ = ∞ d ₀ = 8.875864 r ₁ = ∞ d ₁ = 4 “GRIN” r ₂ = ∞ d ₂ = 11 r ₃ = 67.60799 d ₃ = 4.575753 n ₁ = 1.82898 r ₄ = -69302.38048 d ₄ = 0.1 r ₅ = 25.4335 d ₅ = 8.712737 n ₂ = 1.82738 r ₆ = 54.12257 d ₆ = 1.3433 r ₇ = 65.72241 d ₇ = 7.865766 n ₃ = 1.73244 r ₈ = 14.2744 d ₈ = 8 r ₉ = -22.68205 d ₉ = 8.268915 n ₄ = 1.69297 r ₁₀ = -28.01349 d ₁₀ = 3.320135 r ₁₁ = 33.63214 d ₁₁ = 4.815951 n ₅ = 1.7678 r ₁₂ = -458.20135 d ₁₂ = 0.1 r ₁₃ = ∞ d ₁₃ = 35.116077 r ₁₄ = 122.34031 d ₁₄ = 3 n ₆ = 1.618773 r ₁₅ = -54.48485 d ₁₅ = 13.537462 r ₁₆ = 27.5859 d ₁₆ = 4 n ₇ = 1.64209 r ₁₇ = 115.67648 d _i = 24.282717 “GRIN” N ₀₀ = 1.6461 N ₁₀ = -0.1304 × 10 ^-1 N ₂₀ = 0.4749 × 10 ^-3 N ₃₀ = -0.1619 × 10 ^-3 N ₄₀ = 0.7672 × 10 ^-9 .

Example 3 r ₀ = ∞ d ₀ = 8.575864 r ₁ = ∞ d ₁ = 4 “GRIN” r ₂ = ∞ d ₂ = 10 r ₃ = 43.077 d ₃ = 5.3 n ₁ = 1.82898 r ₄ = 195.422 d ₄ = 0.1 r ₅ = 28.209 d ₅ = 5.57 n ₂ = 1.82738 r ₆ = 42.312 d ₆ = 1.5 r ₇ = 63.924 d ₇ = 1.69 n ₃ = 1.73244 r ₈ = 17.788 d ₈ = 17.7 r ₉ = -16.972 d ₉ = 1.65 n ₄ = 1.83653 r ₁₀ = -52.932 d ₁₀ = 6 n ₅ = 1.69297 r ₁₁ = -24.316 d ₁₁ = 0.1 r ₁₂ = -95.653 d ₁₂ = 5.8 n ₆ = 1.7678 r ₁₃ = -30.655 d ₁₃ = 0.1 r ₁₄ = 88.296 d ₁₄ = 3 n ₇ = 1.79387 r ₁₅ = -363.534 d ₁₅ = 3 r ₁₆ = ∞ d ₁₆ = 71.693597 r ₁₇ = 74.03638 d ₁₇ = 5.5 n ₈ = 1.618773 r ₁₈ = -97.62713 d ₁₈ = 5.5 r ₁₉ = 24.79004 d ₁₉ = 5.5 n ₉ = 1.64209 r ₂₀ = 58.62267 d _i = 21.788148 “GRIN” N ₀₀ = 1.6461 N ₁₀ = -0.1304 × 10 ^-1 N ₂₀ = 0.4749 × 10 ^-3 N ₃₀ = -0.1619 × 10 ^-3 N ₄₀ = 0.7672 × 10 ^-9 .

Example 4 r ₀ = ∞ d ₀ = 8.758664 r ₁ = ∞ d ₁ = 4 “GRIN” r ₂ = ∞ d ₂ = 10 r ₃ = -31.80848 d ₃ = 8.636719 n ₁ = 1.618775 r ₄ = 128.98866 d ₄ = 4 r ₅ = -122.36165 d ₅ = 8.636719 n ₂ = 1.618775 r ₆ = -49.59975 d ₆ = 3 r ₇ = 1049.42976 d ₇ = 18.636719 n ₃ = 1.682485 r ₈ = -68.90477 d ₈ = 0.1 r ₉ = 83.5619 d ₉ = 15.4027 n ₄ = 1.618775 r ₁₀ = -145.91523 d ₁₀ = 70.636719 r ₁₁ = ∞ d ₁₁ = 8.951824 r ₁₂ = 34.78746 d ₁₂ = 3.953527 n ₅ = 1.618775 r ₁₃ = -797.44237 d ₁₃ = 23.947539 r ₁₄ = 31.17273 d ₁₄ = 3.953527 n ₆ = 1.642093 r ₁₅ = -126.19518 d ₁₅ = 8.947539 r ₁₆ = -54.95686 d ₁₆ = 2 n ₇ = 1.7678 r ₁₇ = -73.52657 d _i = 2.858962 “GRIN” N ₀₀ = 1.6461 N ₁₀ = -0.1304 x 10 ^-1 N ₂₀ = 0.4749 x 10 ^-3 N ₃₀ = -0.1619 x 10 ^-3 N ₄₀ = 0.7672 x 10 ^-9 .

[0035]

As is apparent from the above description, since the present invention can provide an optical system with high resolution, the number of vector components is large and the number of reference vectors to be calculated is large. It is possible to provide an image processing device suitable for a vector matrix operation device capable of performing the above.

[Brief description of drawings]

FIG. 1 is a diagram showing a configuration of an optical system of an example of an image processing apparatus according to the present invention.

FIG. 2 is a diagram showing an example of an element array of a reference vector and an input vector.

FIG. 3 is a diagram showing an optical path in the configuration of FIG.

FIG. 4 is a diagram showing a configuration of an optical system of a modified example of the image processing apparatus of FIG.

FIG. 5 is an optical path diagram of the image processing apparatus according to the first exemplary embodiment.

FIG. 6 is an optical path diagram of the image processing apparatus according to the second embodiment.

FIG. 7 is an optical path diagram of an image processing apparatus of Example 3.

FIG. 8 is an optical path diagram of an image processing apparatus of Example 4.

FIG. 9 is a diagram showing an MTF of the image processing apparatus according to the first embodiment.

FIG. 10 is a diagram showing an MTF of the image processing apparatus according to the second embodiment.

FIG. 11 is a diagram showing an MTF of the image processing apparatus according to the third embodiment.

FIG. 12 is a diagram showing an MTF of an image processing apparatus according to a fourth embodiment.

FIG. 13 is an aberration diagram of the image processing device of Example 1.

FIG. 14 is an aberration diagram of the image processing device of Example 2.

FIG. 15 is an aberration diagram of the image processing device of Example 3.

FIG. 16 is an aberration diagram of the image processing apparatus of Example 4.

FIG. 17 is a diagram showing a spot diagram of the image processing apparatus of the first embodiment.

FIG. 18 is a diagram showing a spot diagram of the image processing apparatus of the second embodiment.

FIG. 19 is a diagram showing a spot diagram of the image processing apparatus of the third embodiment.

FIG. 20 is a diagram showing a spot diagram of the image processing apparatus according to the fourth embodiment.

FIG. 21 is a diagram showing a configuration of an optical system of one conventional image processing apparatus.

22 is a diagram showing an optical path of the optical system in the configuration of FIG.

23 is a diagram showing the MTF of the optical system in the configuration of FIG.

24 is an aberration diagram of the optical system in the configuration of FIG.

[Explanation of symbols]

201 ... Reference vector array display section (liquid crystal spatial light modulator (SLM)) 202 ... Lens array 203 ... Second lens 204, 204-1, 204-2 ... Input vector display section (liquid crystal spatial light modulator (SLM)) 205, 205-1, 205-2 ... Third lens 206, 206-1, 206-2 ... Detector 207 ... Readout light 210 ... Center axis 301, 302, 303 ... Chief ray 404 ... Beam splitter

Claims (4)

[Claims] 1. A reference vector array display unit for displaying a reference vector array representing one-dimensional or two-dimensional characters, numbers, symbols, etc., a light source for reading the reference vector array, and a one-to-one correspondence with the reference vector array. A lens array arranged correspondingly; a second lens having an entrance pupil having a size capable of simultaneously entering the light fluxes emitted from the light source and having passed through each of the reference vector arrays and the lens array; An input vector display unit arranged on an image plane with respect to the reference vector array display unit, which is associated with the second lens, and which displays an input vector;
Each reference vector of the reference vector array, a third lens that focuses the light flux that has passed through the input vector display unit for each reference vector of the reference vector array, and to detect the light flux that is focused by the third lens An image processing apparatus, comprising: a detector of 1), and performing calculation between vectors by superposing each reference vector of the reference vector array and the input vector. 2. The image processing apparatus according to claim 1, wherein the second lens includes at least one negative lens and at least two positive lenses. 3. The image processing apparatus according to claim 1, wherein the second lens is a retrofocus type lens system. 4. The image processing apparatus according to claim 1, wherein the position of the lens array is near the front focus position of the second lens or farther from the second lens on the object side.