JPH04237009A - Zoom lens - Google Patents

Abstract

PURPOSE:To obtain large power variation and to shorten the overall lens length by composing the lens of four lens groups which have specific refracting power and specifying the lens constitution and movement conditions of the respective lens groups accompanying power variation. CONSTITUTION:This zoom lens has a 1st group with negative refracting power, a 2nd group with positive refracting power, a 3rd group with positive refracting power, and a 4th group with negative refracting power in order from the object side. When the power is varied from the wide-angle end to the telephoto end, the 2nd and 4th groups move toward the object side, the interval between the 1st and 2nd groups becomes minimum on the telephoto end, and the 3rd group is so moved as to make the current image plane position constant. Then an inequality I holds. In the inequality I, O2 is the distance from the 1st lens surface of the 2nd group to the object-side principal point of the 2nd group and f2 is the focal length of the 2nd group.

Description

[Detailed description of the invention]

0001

[Industrial Application Field] The present invention relates to a compact, high-power zoom lens suitable for lens-shutter cameras, video cameras, etc. In particular, the present invention relates to a compact, high-power zoom lens suitable for lens-shutter cameras, video cameras, etc. ) This relates to a highly portable zoom lens that is designed to shorten the distance.

0002

2. Description of the Related Art Recently, with the miniaturization of lens shutter cameras, video cameras, etc., there has been a demand for compact zoom lenses with short overall lens lengths. Among these, the present applicant has developed a relatively small zoom lens that includes a standard angle of view (photographing angle of view 2ω = 47 degrees and a focal length of about 50 mm when converted to a 35 mm still camera).
This method has been proposed in Japanese Patent Application No. 214, Japanese Patent Laid-Open No. 64-72114, etc.

[0003] According to this publication, there are three lens groups in order from the object side: a first group with a negative refractive power, a second group with a positive refractive power, and a third group with a negative refractive power. This disclosure discloses a so-called three-group zoom lens with a variable power ratio of approximately 2, in which all groups are moved toward the object side under certain conditions to change the power from the wide-angle end to the telephoto end.

Furthermore, Japanese Patent Laid-Open No. 64-88512 has four lens groups having negative, positive, positive, and negative refractive powers in order from the object side, and these four lens groups are independently moved toward the object side. A zoom lens with a variable power ratio of about 3 has been proposed.

[0005]

[Problem to be Solved by the Invention] Generally, in a zoom lens, if the refractive power of each lens group is strengthened, the amount of movement of each lens group to obtain a predetermined variable power ratio will be reduced, and the overall length of the lens can be shortened. High magnification is possible. However, if you simply strengthen the refractive power of each lens group, aberration fluctuations will increase as you change the magnification, which makes it difficult to obtain good optical performance over the entire magnification range, especially when aiming for a high zoom ratio. There is a point.

[0006] The present invention is based on the applicant's earlier Japanese Patent Application Laid-open No. 63-27.
The zoom lens proposed in JP-A No. 1214 and JP-A No. 64-72114 is improved and consists of four lens groups as a whole, and the zoom lens proposed in JP-A 64-88512 is a variable power zoom lens. The purpose of the present invention is to improve the method and provide a compact zoom lens that has a high zoom ratio of about 7 while reducing the overall length of the lens, and has high optical performance over the entire zoom range.

[0007]

[Means for Solving the Problems] The zoom lens of the present invention includes, in order from the object side, a first group with negative refractive power, a second group with positive refractive power, a third group with positive refractive power, and a third group with negative refractive power. It has a fourth group with refractive power, and when changing power from the wide-angle end to the telephoto end, the second and fourth groups move toward the object side, and the distance between the first and second groups is the same as that at the telephoto end. The third group is moved to keep the image plane position constant, and the distance from the first lens surface of the second group to the object-side principal point of the second group is O2 ( When measuring towards the image side, it is positive; when measuring towards the object side, it is negative).
When the focal length of the second group is f2, -1.25<O2
/f2<-0.18...(1) It is characterized by satisfying the following condition.

In particular, in the present invention, when changing the magnification from the wide-angle end to the telephoto end, the second group and the fourth group move independently and linearly toward the object side, and the third group moves convexly toward the image plane side. The first lens group is characterized in that it moves with a locus of , and that the first group moves non-linearly toward the object side when changing the magnification from the wide-angle end to the telephoto end.

[0009] Also, when changing the magnification from the wide-angle end to the telephoto end,
The second group and the fourth group are always multiplied, and each is adjusted so that the amount of change in the imaging magnification of the second group is larger than the amount of change in the imaging magnification of the fourth group. It is characterized by a set of elements.

0010

DESCRIPTION OF THE PREFERRED EMBODIMENTS FIG. 1 is an explanatory diagram of the paraxial refractive power arrangement of a zoom lens according to the present invention, and FIGS. 2, 3, and 4 are lens sectional views of numerical embodiments 1, 2, and 3 of the present invention, respectively.

In FIGS. 1 to 4, (A) is at the wide-angle end, and (B) is at the wide-angle end.
) indicates the zoom position at the telephoto end. 5 to 7 are various aberration diagrams at the wide-angle end, intermediate, and telephoto end of Numerical Example 1 of the present invention, and FIGS. 8 to 10 are diagrams of various aberrations at the wide-angle end, intermediate, and telephoto end of Numerical Example 2 of the present invention. Various Aberration Diagrams, FIGS. 11 to 13 show various aberration diagrams at the wide-angle end, intermediate, and telephoto end of Numerical Example 3 of the present invention.

In the figure, 1 is the first group with negative refractive power (φ1),
2 is the second group with positive refractive power (φ2), and 3 is the positive refractive power (φ2).
3), the third group and 4 are the fourth group having negative refractive power (φ4). The arrows indicate the movement direction of each lens group when changing the magnification from the wide-angle side to the telephoto side.

Next, the characteristics of the paraxial refractive power arrangement of the zoom lens of this embodiment will be explained.

When the first group with refractive power φ1 and the second group with refractive power φ2 are arranged with principal point spacing e, the composite refractive power φ of the entire lens system is φ=φ1+φ2−eφ1・φ2 (a) becomes. In order to change the composite refractive power φ at this time, that is, to change the magnification, the principal point spacing e may be changed as follows. (a) When the refractive powers φ1 and φ2 are both positive values, the principal point distance e may be increased to decrease the composite refractive power φ, that is, to change it to the telephoto end. (b) When the signs of the refracting powers φ1 and φ2 are different, the principal point spacing e may be decreased in order to reduce the composite refractive power φ, that is, to change it to the telephoto end.

In the zoom lens of this embodiment, when changing the magnification from the wide-angle end to the telephoto end, the second and fourth groups are moved toward the object side as described above, and the air distance between the second and third groups (
The principal point interval) gradually increases and becomes widest at an intermediate zoom position. The air gap is made slightly narrower from the intermediate zoom position to the telephoto end.

That is, the third group is moved so as to have a convex trajectory toward the image plane. As a result, as shown in Figure 1, the paraxial refractive power arrangement of the composite system of the second and third groups corresponds to the case (a) above from the wide-angle end to the intermediate zoom position, and from the wide-angle end to the intermediate zoom position. It effectively changes the magnification up to the intermediate zoom position, making it easy to increase the magnification.

By satisfying conditional expression (1), the zoom lens can be made compact and a high variable power ratio can be achieved.

If the lower limit of conditional expression (1) is exceeded, the overall length of the lens becomes long and the diameter of the front lens increases, which is not good. If the upper limit is exceeded, the movement range of the second lens group becomes narrower, making it difficult to obtain a high zoom ratio of about 7x.

Furthermore, when the refractive power arrangement according to the present invention is adopted, the curvature of field at an intermediate zoom position is effectively prevented from increasing in the negative direction. When changing magnification from an intermediate zoom position to the telephoto end, the air gap between the second and third groups is narrowed to effectively correct aberration fluctuations mainly associated with changing magnification. Furthermore, fluctuations in the image plane due to zooming are also corrected.

Furthermore, as is clear from the refractive power arrangement of each lens group shown in FIG. 1, the zoom lens of this embodiment has a first group with negative refractive power (φ1<0) and a positive refractive power (φ2>0 ) second
The refractive power relationship of the groups and the refractive power relationship between the third group with positive refractive power (φ3>0) and the fourth group with negative refractive power (φ4<0) are as described above at the telephoto end compared to the wide-angle end. This corresponds to case (b).

In this way, when changing the magnification from the wide-angle end to the telephoto end, the composite system of the first and second groups, especially the The imaging magnification of the second lens group is always multiplication, and each element is set so that the amount of change in the imaging magnification at this time is greater than that of the other lens groups.

Similarly, the distance between the principal points of the third and fourth groups is made shorter at the telephoto end than at the wide-angle end, so that the composite system of the third and fourth groups, especially the imaging magnification of the fourth group, is reduced. is always multiplied.

In this way, the second group and the fourth group both increase the magnification when changing the magnification from the wide-angle end to the telephoto end, making it easy to increase the magnification of the entire lens system. In particular, as described above, the amount of change in the imaging magnification of the second group is larger than the amount of change in the imaging magnification of the fourth group, thereby effectively achieving a high variable magnification.

Contrary to the present invention, if the amount of change in the imaging magnification of the fourth group becomes larger than the amount of change in the imaging magnification of the second group, the amount of movement of the fourth group increases, and the amount of movement of the fourth group increases. This is not good because the lens effective diameter of the fourth group increases and the brightness (F number) of the lens system on the telephoto side becomes darker.

Furthermore, the refractive power of the fourth group becomes too strong, which increases positive distortion at the wide-angle end, further increases the negative Petzval sum, and tends to overcorrect the curvature of field, which is not good.

When changing the magnification from the wide-angle end to the telephoto end, the zoom lens according to this embodiment moves the first group non-linearly in the object side direction independently of the other lens groups, as shown in FIG. It has been moved to satisfy the following conditions.

As described above, in this embodiment, when changing the magnification from the wide-angle end to the telephoto end, by moving each lens group toward the object side while satisfying the above-mentioned conditions, a zoom ratio of 7 and a high zoom ratio can be achieved. This effectively shortens the overall lens length at the wide-angle end while ensuring the same. In other words, the lens has a refractive power arrangement in which the overall length of the lens is short on the wide-angle side and long on the telephoto side.

In addition, in the present invention, in order to effectively obtain a high zoom ratio, the i-th
When the amount of movement of the group is Mi (the direction toward the image plane is positive) and the amount of change in focal length of the entire system is Δf, 0.3<|M2/Δf|<0.9 (2 )0
．． 3<|M4/Δf|<0.9 (3) is satisfied.

If the upper limits of conditional expressions (2) and (3) are exceeded and the amount of movement of the second and fourth groups increases, the entire lens system increases; If the amount of movement of the groups becomes too small, the amount of change in the value of the principal point interval e in the above-mentioned equation (a) of the second and fourth groups, which contribute the most to zooming, will decrease, ensuring the desired zoom ratio. It's not good because it becomes difficult.

In this embodiment, in order to obtain high optical performance over the entire screen, each lens group is preferably configured as follows.

It is preferable that the first group has at least one negative lens and at least one positive lens, and that an air lens having a convex shape on the object side is formed in the first group. It is particularly advantageous to configure the second group to have at least two positive lenses and at least one negative lens closest to the object side, in order to satisfactorily correct spherical aberration over the entire zoom range. Further, when the Abbe number of the material of one of the two positive lenses is ν2P, it is preferable to satisfy the following condition: ν2P>50 (4).

If Conditional Expression (4) is not satisfied, fluctuations in longitudinal chromatic aberration mainly increase during zooming, which is not good.

In order to correct aberrations, it is preferable to arrange the diaphragm at an arbitrary position in the second group and move it integrally with the second group as the magnification changes.

At this time, the negative lens in the second group is arranged to face the diaphragm, and the refractive index of the material of the negative lens is N2n.
When 1.75<N2n, it is preferable to satisfy the condition (b).

If conditional expression (b) is not satisfied, the Petzval sum increases in the negative direction and the curvature of field becomes overcorrected. Particularly in this embodiment, it is preferable that the Abbe number ν2P of the material of the positive lens in the second group and the refractive index N2n of the negative lens in the second group be 60 <ν2P (c)
1.8<N2n It is preferable to set as shown in (d).

In this embodiment, the diaphragm is placed between the second group and the third group instead of being placed in the lens system of the second group,
It may be moved independently of the second lens group as the power is changed, and this is preferable since it is possible to reduce fluctuations in the F number caused by the power change.

The fourth group is preferably configured to have at least one positive lens with a convex surface facing the image plane side and at least one negative lens with a concave surface facing the object side.

In the present invention, the second group and the fourth group may be moved integrally during zooming, which is preferable because the lens barrel can be simplified.

In addition, in the present invention, where fi is the focal length of the i-th group and fw is the focal length of the entire system at the wide-angle end, 1.5<|f1/fw|<3.0...
...(5) 0.9< f2/fw <2.5
・・・・・・・・・(6)2 <|f4/fw|
<4.0 It is preferable to satisfy the condition (7) in order to effectively secure the desired variable power ratio and to downsize the entire lens system.

If the upper limits of conditional expressions (5), (6), and (7) are exceeded and the refractive power of each lens group becomes too weak, the amount of movement of each lens group to obtain the desired variable power ratio increases. However, the entire lens system becomes larger, which is not good.

If the lower limit of conditional expression (5) is exceeded and the refractive power of the first group becomes too strong, aberration fluctuations will increase when focusing with the first group.

If the refractive power of the second group that performs a magnification change action becomes too strong beyond the lower limit of conditional expression (6), the Petzval sum increases in the positive direction, and the image plane is corrected over the entire magnification range. This is not good because it becomes insufficient (under).

If the lower limit of conditional expression (7) is exceeded and the refractive power of the fourth group becomes too strong, the Petzval sum increases in the negative direction, contrary to conditional expression (6), and the image is distorted over the entire magnification range. This is not good because the surface becomes over-corrected.

In the present invention, in particular, in order to reduce the size of the entire lens system, when the shortest distance of the back focus in the entire zoom range is defined as bf.min, 0.13<bf.min/fw<0.7. ...
...(8) It is preferable to set the refractive power and lens configuration of each lens group so as to satisfy the following condition.

If the upper limit of conditional expression (8) is exceeded, the entire lens system will become larger; if the lower limit is exceeded, the fourth group will be too close to the image forming plane, and dust, etc. in the fourth group will be exposed to the photosensitive surface. It's not good because it comes in the picture.

In the present invention, in order to effectively correct inward coma flare and barrel distortion mainly caused by downward rays on the wide-angle side, at least one lens surface of the first group has positive refraction toward the lens periphery. It is preferable to provide an aspheric surface with a shape that increases the power or weakens the negative refractive power.

Furthermore, in order to correct the inward coma caused by the upward rays on the telephoto side, at least one lens surface of the third or fourth group has a positive refractive power that becomes weaker toward the lens periphery or a negative refractive power that becomes weaker toward the lens periphery. It is best to provide an aspheric surface with a shape that increases refractive power.

It is better to focus on the first group, but the fourth group
It may be performed in groups or in a third group. Further, only a specific area, for example, a close distance, may be focused by the fourth group or the third group. This is preferable because it enables closer focusing and prevents the diameter of the front lens from increasing.

In this embodiment, it is desirable that the following conditional expression is further satisfied in relation to conditional expression (1).

-1.25<O2/fw<-0.25
...(1)-a Exceeding the lower limit of formula (1)-a not only increases the overall length but also increases the diameter of the front lens, which is undesirable. If the upper limit is exceeded, the movable range of the second lens group becomes narrower, making it difficult to obtain a zoom ratio of about 7x.

Next, numerical examples of the present invention will be shown. In the numerical examples, Ri is the radius of curvature of the i-th lens surface from the object side, Di is the thickness and air gap of the i-th lens from the object side, and Ni and νi are the curvature radius of the i-th lens from the object side, respectively. These are the refractive index and Abbe number of glass.

Table 1 shows the relationship between each of the above-mentioned conditional expressions and the numerical values in the numerical examples. Numerical Example 1 F=28.8 ~194.0 FN
O=1:4.1 ~9.2 2ω=73.
8° ~ 12.7° R 1 = 1102.99
D 1 = 1.80 N 1 = 1.
77250 ν 1= 49.6 R
2= 29.12 D 2= 3.98

R3=30.24
D3=5.50 N2=1.68
893 ν 2= 31.1 R 4=
50.89 D 4 = variable

R5=54.35D
5 = 2.96 N 3 = 1.49700
ν 3 = 81.6 R 6 = -267
．． 62 D 6= 0.09

R7= 26.69 D7=
4.00 N 4=1.48749
ν 4= 70.2 R 8= 63.4
0D8=0.09

R9= 18.39 D9=
4.72 N5=1.48749
ν5=70.2 R10=48.06
D10=3.30

R11=(aperture) D11=2.52

R12=-703.30
D12=1.26 N6=1.8340
0 ν 6= 37.2 R13=
16.78 D13= 1.18

R14= 24.12 D1
4=2.83 N 7=1.48749
ν7=70.2 R15=-123
．． 77 D15= variable

R16= 49.06 D16=
2.50 N 8=1.53172
ν 8= 48.9 R17= -61.27
D17=0.07

R18=-107.95 D18=2.
03 N 9=1.78590 ν
9=44.2 R19=59.74
D19=0.32
R2
0= 111.87 D20= 1.97
N10=1.56732 ν10=
42.8 R21=-621.42
D21 = variable
R22=
45.30 D22= 3.86
N11=1.68893 ν11=31.
1 R23=-105.41 D23
= 1.31
R24=-40
．． 69 D24= 1.34 N
12=1.71299 ν12=53.8
R25=-127.98 D25=
3.70
R26=-21.7
3 D26= 1.67 N13
=1.69680 ν13= 55.5
R27=-135.01

Numerical Example 2 F=28.8 ~194.0 FN
O=1:4.1 ~9.2 2ω=73.
8° ~ 12.7° R 1 = -1588.68
D 1 = 1.80 N 1 = 1.
77250 ν 1= 49.6 R
2= 29.62 D 2= 4.76

R3=31.37
D3=5.49 N2=1.68
893 ν 2= 31.1 R 4=
55.84 D 4 = Variable

R5=41.15D
5 = 3.46 N 3 = 1.48749
ν 3= 70.2 R 6= −42
2.32 D6= 0.09

R7= 22.09 D7=
5.07 N4=1.51633
ν 4=64.1 R 8=81.
17 D 8 = 0.09

R9= 22.71 D9=
3.57 N5=1.51633
ν5=64.1 R10=53.49
D10=3.39

R11=(aperture) D11= 1.94

R12=-171.87
D12=1.39 N6=2.0224
4 ν 6= 29.1 R13=
17.97 D13= 0.91

R14=24.83D1
4=2.98 N 7=1.56732
ν7=42.8 R15=-12
3.12 D15= variable

R16= 153.86 D16=
2.24 N8=1.51742
ν8=52.4 R17=-42.13
D17=0.11

R18=-60.28 D18=0.
78 N 9=1.78590 ν
9=44.2 R19=86.88
D19=0.44
R2
0=97.24 D20=1.77
N10=1.56732 ν10=
42.8 R21=-221.80
D21 = variable
R22=
40.51 D22= 5.66
N11=1.68893 ν11=31.
1 R23= -35.72 D23
= 0.99
R24=-2
6.42 D24= 1.34 N
12=1.71299 ν12=53.8
R25=-193.52 D25=
3.77
R26=-23.
84 D26= 1.67 N13
=1.69680 ν13= 55.5
R27=-400.66

Numerical Example 3 F=28.8 ~194.0 FN
O=1:4.1 ~9.2 2ω=73.
8° ~ 12.7° R 1 = -555.76
D 1 = 1.47 N 1 = 1.
80400 ν 1= 46.6 R
2= 26.88 D 2= 3.20

R3=29.16
D3=4.20 N2=1.80
518 ν 2= 25.4 R 4=
50.39 D 4 = variable

R5=34.65D
5 = 4.04 N 3 = 1.48749
ν 3 = 70.2 R 6 = -180
．． 30D6=0.07

R7= 18.38 D7=
5.26 N4=1.48749
ν 4= 70.2 R 8= 89.2
2 D 8 = 0.07

R9= 19.87 D9=
3.68 N5=1.51633
ν5=64.1 R10=52.82
D10=2.38

R11=(aperture) D11=0.92

R12=-145.45
D12=1.11 N6=2.0224
4 ν 6= 29.1 R13=
15.36 D13= 1.10

R14 = 23.41 D1
4=2.30 N 7=1.51454
ν7=54.7 R15=121
．． 66 D15= variable

R16= 117.94 D16=
2.58 N 8=1.58144
ν 8 = 40.8 R17 = -30.58
D17=1.17

R18=-39.14 D18=0.
67 N 9=1.77250 ν
9=49.6 R19=74.29
D19=0.13
R2
0=51.54 D20=2.35
N10=1.58144 ν10=
40.8 R21=-74.41
D21 = variable
R22=
41.01 D22= 4.13
N11=1.72825 ν11=28.
5 R23=-87.39 D23
= 2.94
R24=-22
．． 38 D24= 1.07 N
12=1.71299 ν12=53.8
R25=-257.20 D25=
1.92
R26=-44.1
0 D26= 1.34 N13
=1.69680 ν13= 55.5
R27=-739.08

(Table-1)

[0053]

According to the present invention, in a zoom lens consisting of four lens groups having a predetermined refractive power, by setting the movement conditions and lens configuration of each lens group during zooming as described above, the zoom ratio can be increased to 7. It is possible to achieve a zoom lens that has high optical performance over the entire zoom range, has a high zoom ratio, and has a short overall lens length.

[Brief explanation of the drawing]

FIG. 1 is an explanatory diagram of the paraxial refractive power arrangement of the zoom lens of the present invention.

FIG. 2 is a cross-sectional view of a lens of Numerical Example 1 of the present invention.

FIG. 3 is a cross-sectional view of a lens of Numerical Example 2 of the present invention.

FIG. 4 is a cross-sectional view of a lens of Numerical Example 3 of the present invention.

FIG. 5 is a diagram of various aberrations at the wide-angle end of Numerical Example 1 of the present invention.

FIG. 6 is a diagram showing intermediate aberrations of Numerical Example 1 of the present invention.

FIG. 7 is a diagram of various aberrations at the telephoto end of Numerical Example 1 of the present invention.

FIG. 8 is a diagram of various aberrations at the wide-angle end of Numerical Example 2 of the present invention.

FIG. 9 is a diagram showing intermediate aberrations of Numerical Example 2 of the present invention.

FIG. 10 is a diagram of various aberrations at the telephoto end of Numerical Example 2 of the present invention.

FIG. 11 is a diagram of various aberrations at the wide-angle end of Numerical Example 3 of the present invention.

FIG. 12 is a diagram showing various intermediate aberrations of Numerical Example 3 of the present invention.

FIG. 13 is a diagram showing various aberrations at the telephoto end of Numerical Example 3 of the present invention.

[Explanation of symbols]

Claims (6)

[Claims] Claim 1: A first group having negative refractive power in order from the object side,
It has a second group with positive refractive power, a third group with positive refractive power, and a fourth group with negative refractive power, and when changing power from the wide-angle end to the telephoto end, the second group and the fourth group moves toward the object side, the distance between the first and second groups becomes the smallest at the telephoto end, and the distance between the first and second groups becomes the smallest at the telephoto end.
The group is moved so that the image plane position at that time is constant, and the distance from the first lens surface of the second group to the object side principal point of the second group is O2 (correct when measured toward the image plane side). , when measured toward the object side, it is negative), and when the focal length of the second group is f2, -
A zoom lens that satisfies the following condition: 1.25<O2/f2<-0.18. [Claim 2] When changing the magnification from the wide-angle end to the telephoto end,
The second group and the fourth group independently move linearly toward the object side,
2. The zoom lens according to claim 1, wherein the third group moves with a convex trajectory toward the image plane. [Claim 3] When changing the magnification from the wide-angle end to the telephoto end,
3. The zoom lens according to claim 2, wherein the first group moves non-linearly toward the object side. [Claim 4] When changing the magnification from the wide-angle end to the telephoto end,
The second group and the fourth group are always multiplied, and each is adjusted so that the amount of change in the imaging magnification of the second group is larger than the amount of change in the imaging magnification of the fourth group. The zoom lens according to claim 1, characterized in that the elements are set. [Claim 5] When changing the magnification from the wide-angle end to the telephoto end,
The amount of movement of the i-th group is Mi, and the amount of change in focal length of the entire system is Δ
2. The zoom lens according to claim 1, wherein the zoom lens satisfies the following conditions: 0.3<|M2/Δf|<0.9 0.3<|M4/Δf|<0.9. 6. When the focal length of the i-th group is fi and the focal length of the entire system at the wide-angle end is fw, 1.5<|f
6. The zoom lens according to claim 5, wherein the zoom lens satisfies the following conditions: 1/fw|<3.0 0.9<f2/fw<2.5 2<|f4/fw|<4.0.