JPH0132961B2 - - Google Patents

Description

[Detailed description of the invention]

Plastic Fresnel lenses have advantages over glass lenses in terms of large diameter, light weight, and economy, and are often used as condensing lenses. The commonly used Fresnel lens is
Its Fresnel surface has a planar configuration, and although spherical aberration is well corrected when used as a condensing lens, off-axis aberration cannot be removed, so it is used for image forming purposes. I couldn't do it. Therefore, an invention was developed in Japanese Patent Application Laid-Open No. 53-57054 to curve the Fresnel surface to improve off-axis aberrations.
Publications, Special Publication No. 3775 and Special Publication No. 51-
It is published in Publication No. 14900, etc. However, these inventions have the following drawbacks. That is, in the invention, although the aberration is reduced compared to the conventional Fresnel lens optical system,
Since the correction method is determined experimentally rather than theoretically, it has the disadvantage that it does not quantitatively represent aberrations and cannot be applied when conditions such as object distance change. In the invention, an aberration-free image can be formed only when two sets of object points are on the optical axis, and the image points are also focused on the optical axis, so it is effective as long as it is used as a condensing lens. However, it is not suitable when an aberration-free image is required on any part of the object surface that has a spread. In the invention, an aberration-free image can be formed only when the object point is in a plane perpendicular to the optical axis and on the circumference of one concentric circle centered on the optical axis, The image also becomes a circle concentric with the optical axis. Here, only the rays in the meridian plane are aberration-free, and the spherical rays are not taken into consideration, so it is effective as long as it is used as a condensing lens, similar to the invention of 2. It is unsuitable when an aberration-free image is required anywhere on the surface. In order to obtain an image with little aberration in any off-axis part, which was previously impossible, the present invention determined a theoretical aberration formula for a Fresnel lens, and applied this formula to a lens system composed of two Fresnel surfaces. The aim is to design an aberration-free optical system. First, the aberration theoretical formula regarding the Fresnel surface is derived based on FIGS. 2 to 4 as follows. Each annular refractive surface that makes up a Fresnel lens is
It is assumed that the Fresnel surfaces 1 have a radius centered on the center of curvature, that is, they are arranged on a generatrix surface on which each refracting surface is arranged. The light ray emerging from the object point O is refracted at the refraction point q at the incident height h, and the refracted ray reaches the image point O'. The refracting surface at a refractive point q at an arbitrary height is a fixed point c
If we consider q to be part of a spherical surface of radius r centered on q
A straight line 2 passing through and c becomes a normal line perpendicular to the refractive surface. c and are on a straight line and form a coaxial system. In particular, r at h=0 is called the paraxial radius of curvature, and this is expressed as r ₀ . The distances from the vertex A of the refractive surface to the object point O and to the image point O' are s and s', and those regarding the paraxial ray are
Similarly, let s ₀ and s ₀ ' be the distances to the aperture 3 and to the image 3' of the aperture, and let z ₀ and z ₀ ' be the distances related to the paraxial rays. Furthermore, let the refractive indices of the object space and the image space be n, n', the interplanar spacing be d', and the object point O
The angle where the ray of light intersects with the optical axis x is u, the image point
Let u′ be the angle at which the refracted ray toward O′ intersects the optical axis x,
Let the size of the object be y and the size of the image be y'. The sign of the distance is positive when measured in the same direction as the traveling direction of the ray with respect to the vertex A of the refracting surface, and the sign of the distance is positive when measured upward from the optical axis x for the incident height h. The angle is measured by the smallest angle from the ray to the normal or from the ray to the optical axis x, and is positive if the direction is the same as the rotation of the clock's hands. Auxiliary amount of a spherical lens system The auxiliary amount of a spherical lens system, which is obtained from the tracing results of paraxial rays and is generally used, is as follows, where the surface number is i, Δ(1/ns) _i = 1/n _i s _0i −1/ n _i ′s′0 _i Q _si = n _i (1/r _0i −1/s _0i )=n′ _i (1/r _0i −1/s
′ _0i ) Q _xi = n _i (1/r _0i −1/z _0i )=n′ _i (1/r _0i −1/z
′ _0i ) L _i =Q _xi −Q _si = n _i (1/s _0i −1/z _0i )=n′ _i (1/s′ _0i −1/z
′ _0i ) ε _i = (h ₁ /h _i ) ² 1/Q _si δ _i = _i 〓 ² d′ _i-1 /n _i h _i /h ₁ h _i-1 /h ₁ τ _i =ε _i +δ _i P _i =1/r _0i (1/n′ _i −1/n _i ). In addition to these shall be. In a spherical lens system, there is the following relational expression, where H is the incident height of the ray passing through the center of the aperture, that is, the principal ray, at the refraction point q. H _i h _i L _i =H ₁ h ₁ L ₁ (1-1) H _i /H ₁ = h _i /h ₁ (1+L ₁ δ _i ) (1-2) H _i Q _xi =H ₁ Q _si h _i /h ₁ (1+L ₁ τ _i ) (1-3) From these equations, Q _si /L _i =1/ε _i (1/L ₁ +δ _i ) (1-4) Q _xi /L _i =1/ ε _i (1/L ₁ +τ _i ) (1-5) Q _si /Q _zi =1+L ₁ δ _i /1+L ₁ τ _i (1-6) Considering the above formula, the aberration-free design of the present invention is as follows. The expressions for each aberration of the Fresnel lens system are as follows. Spherical aberration The imaging formula for paraxial rays is n/s ₀ −n′/s′ ₀ = n−n′
/r ₀ (2-1) If the distances from the refraction point q to the object point O and the image point O' are p and p', and the angle of incidence and angle of refraction are i and i', respectively, then according to the sine law, sini/s−r ₀ = sinθ/−p sini′/s′−r ₀ = sinθ/
−p′ p/s=1−h ² /2s (1/r−1/s)−Δr/s s′/p′=1+h ² /2s′(1/r−1/s′)−Δr/ s
′ Therefore, n(1/r ₀ −1/s)=n′(1/r ₀ −1/s′)p/s
s'/p' = n' (1/r ₀ -1/s') [1-h ² /2s (1/r-1
/s)+h ² /2s'(1/r-1/s')+(1/s-1/s')Δr] Since the second and subsequent terms in brackets are correction terms, n(1/ r−1/s)≒Q _s , then n/s−n′/s′=n−n′/r ₀ +h ² /2Q ² _s Δ(1/ns
) − Qs (1/s − 1/s′) Δr If the spherical aberration is Δs, Δs′, s=s ₀ +Δs, s′=
s′ ₀ +Δs′ ∴ n/s ² / ₀ Δs−n′/s′ ₀ ² Δs′= −h ² /2Q ² _s [Δ(1/ns)−(P−)] (2−2) Refraction If there are k surfaces and Δs ₁ = 0, then (2−
2) Adding the sides of the equation, Δs′ _k = s′ _0k /2n′ _k h ⁴ / ₁ /h ² / _kk 〓 ¹ (h _i /h ₁ ) ⁴ Q ² _si [Δ(1/ns) _i −(P−) _i ] where (h _i /h ₁ ) ⁴ Q ² _si [Δ(1/ns) _i −(P−) _i ]=A _i (
2-3), the condition that there is no spherical aberration is _i 〓 ¹ A _i = 0 comatic aberration In Fig. 2, let the straight line passing through the off-axis object point and the spherical center c, and the line between the axis and the r ₀ plane. Let the height to the intersection be h ₀ . Furthermore, the ray of light emitted from the object point reaches the image point ' after being refracted at a height q from the axis.
For angles and other quantities, use the symbols shown. From formula (2-2), n/S ² / ₀ Δs−n′/s′ ₀ ² Δs′=−h/− ² /2Q ² _s Δ
(1/ns) +(h/-+h ₀ ) ² /2Q ² _s (P-) If cosθ≒1-h ² / ₀ /2r ² / ₀ , then Δ=Δs-h ² / ₀ /2 s ₀ −r ₀ /r ² / ₀ Similarly, Δ′=Δs′−h ² / ₀ /2 s′ ₀ −r ₀ /r ² / ₀ ∴ n/s ² / ₀ Δ−n′/s′ ₀ ² Δ ′ =n/s ² / ₀ Δs−n′/s′ ₀ ² Δs′ −nh ² / ₀ /2 s ₀ −r ₀ /s ² / ₀ r ² / ₀ +n′h ² / ₀ /2 s
′ ₀ −r ₀ /s′ ₀ ² −r ² / ₀ = −h ² /2Q ² _s [Δ(1/ns)−(P−)] +h ₀ Q ² _s (P−)−h ² / ₀ /2Q ² _s (3-1) The lateral magnification β is β=y′/y=s′−r ₀ /s−r ₀ =np′/n
'p Here, p=s[1-h/- ² /2s (1/r-1/s)+Δr/s
] ≒s ₀ (1+h ² / ₀ /2nr ₀ Q _s +Δs/-/s ₀ ) (1-h
/− ² /2ns ₀ Q _s + Δr/s ₀ ) p′≒s′ ₀ (1+h ² / ₀ /2n′r ₀ Q _s +Δs/−′/s′ ₀ )
(1−h/− ² /2n′s′ ₀ Q _s +Δr/s′ ₀ ) ∴ β=ns′ ₀ /n′s ₀ [1+h/− ² /2Q _s Δ(1/ns
) + h ² / ₀ / 2Q _s P − (Δs/−/s ₀ −Δs/−′/s′ ₀ )−(1/s ₀ −
1/s′ ₀ )Δr] β ₀ =ns′ ₀ /n′s ₀ (1/s ₀ −1/s′ ₀ )Δr=(h+h ₀ ) ² /2Q _s (P−
) From this, β/β ₀ =1+h/- ² /2Q _s [Δ(1/ns)-(P-
)] -h ₀ Q _s (P-) + h ² / ₀ / 2Q _s - (Δs/-/s
₀ − Δs/−′/s′ ₀ ) Q _x −Q _s = L, then Δs/−/s ₀ −Δs/−′/s′ ₀ = 1/L(n/s ² / ₀
Δ−n′/s′ ² / ₀ Δ′) −Δs/−/z ₀ −s ₀ +Δs/−′/z′ ₀ −s′ ₀ ∴ β/β ₀ =1+h/− ² /2 Q _x Q _s /L[Δ(1/ns
)−(P− P)] −h ₀ Q _x Q _s /L(P−)+h ² / ₀ /2 Q _x Q _s /L +Δs/−/z ₀ −s ₀ −Δs′/z′ ₀ − s′ ₀ (3−2) If there are multiple Fresnel surfaces, − _i = θ _i −θ _i-1 −′ _i-1 β _i = n _i s′ _i /n′ _{i s} iTherefore, h/ ― _i /s _i =θ _i −θ _i-1 +1/β _i-1 n _i-1 /n′ _i-1 h
/- _i-1 /s _i-1Here , if h/- ₁ /s ₁ = h ₁ /s ₁ , β _i,k = β _i β _i+1 ......β _k ≒ n _i / n′ _k s′ _k /s _i h _i /h _{k h i} /s′ _i =h _i+1 /s _i+1 θ _i =y _i /s _i −r _0iFrom these, the following results are derived. _i = h _i [(s _i /s _i −r _0i −1)y _i /h _i +(1/h _i −1/
h _i-1 )y _i +(1/h _i-1 −1/h _i-2 )y _i-1 …−(s ₁ /s ₁ −r ₀₁
−1) y ₁ /h ₁ +1] From Lagrange Helmholtz's theorem, n ₁ u ₁ y ₁ = n _i u _i y _i Therefore, (s _i /s _i −r _0i −1) y _i /h _i = −n ₁ u ₁ y ₁ /h ² / ₁ ε _i (1/h _i −1/h _i-1 )y ₁ ≒−n ₁ u ₁ y ₁ /n _i h _i-1 h _i d′ _{i- 1And} if we set n ₁ y ₁ /s ₁ = w ₁ , then _i = h _i / h ₁ [h ₁ − (ε ₁ − τ _i ) w ₁ ] h _0i = −h ₁ w ₁ / h _i
If the height from the optical axis of the intersection of the incident ray and the entrance pupil on the first surface of Q _si is ₁ , then h ₁ = ₁ = ₁ −n ₁ H/- ₁ /s ₁ L ₁ +Q _zi w ₁ / If Q _s1 L ₁ s ₁ is large enough, the second term disappears, so if _i = h _i / h ₁ [ ₁ + (1/L ₁ + τ _i ) w ₁ ] Δ ₁ = 0, then (1-5 ), and according to equation (2-3), equation (3-2) is β/β ₀ =1+1/2 _k 〓 ¹ (1/L ₁ +τ _i ) [ ₁ + (1/L ₁ +τ _i ) w ₁ ] ² A _i +w _1k 〓 ¹ 1/ε _i (1/L ₁ +τ _i ) [ ₁ + (1/L ₁ + τ _i ) w ₁ ] (P-) _i +w ² / ₁ /2 _k 〓 〓 ¹ (1/ L ₁ +τ _i ) _i −Δs/−′ _k /z′ _0k −s′ _0k (3−
3) In equation (3-1), if Δ _i =0,
Δ′ _i is calculated, but this aberration is (i+1)
After being enlarged by the vertical magnification α α _i+1,k = n′ _i /n′ _k (s′ _0k /s′ _0i h _i /h _k ) ² , at the last image plane, Δ _{′
, i} = Δ′ _i α _i+1,k . The aberration throughout the entire system is Δs/-' _k /z' _0k -s' _0k = g/2 _k 〓 ¹ [ ₁ + (1/L ₁ + τ _i )w ₁ ] ² A _i +w ₁ g _k 〓 ¹ 1 /ε _i [ ₁ + (1/L ₁ +τ _i )w ₁ ] (P-) _i +w ² ₁ g/2 _k 〓 ¹ _iwhere g=1/n' _k (h ₁ /h _k ) ² s ′ ₀ ² / _k /z′ _0k −s
′ _0k If the terms that do not include ₁ are a and b, then (3-3)
The formula is β/β ₀ = H/− ² / ₁ /2 _k 〓 ¹ (1/L ₁ + τ _i ) A _i + ₁ w _1k 〓 ¹ (1/L ₁ + τ _i ) ² A _i + ₁ w _1k 〓 ¹ (1/L ₁ +τ _i ) (P−P/−/ε) _i +α− Δs/−′ _k /z′ _0k −s′ _0k (3−4) Δs′ _k /z′ _0k −s′ _0k =H ² / ₁ / 2g _k 〓 ¹ A _i + ₁ w ₁ g [ _k 〓 ¹ (1/L ₁ + τ _i ) A _i + _k 〓 ¹ (P-P/ε) _i ] + b Now _k 〓 ¹ ( 1/L ₁ +τ _i ) A _i =− _k 〓 ¹ (P−P/−/ε) _i =0 (3−5), then Δ′k=(H/− ₁ /2g _k 〓 ¹ A _i + b) (z′ _0k − s′ _pk ), and Δ′ _k is equal to ± ₁ . Furthermore, in equation (3-4), for ± ₁
If the average value of y′ _k is ′ _k , β/β ₀ =y/−′ _k /y′ _0k =Δs/−′ _k /s′ _0k −z′
_0k + α In other words, ′ _k is a linear function of Δ′ _k , and the slope of the straight line y′ _0k / (s′ _0k −z′ _0k ) is approximately equal to the slope of the principal ray, so the ray flux is the principal ray It has symmetry around . Therefore (3-
Equation 5) is a condition for removing coma aberration. Astigmatism If one annular zone is considered to be a part of a spherical surface with a radius r centered at a fixed point c, the spherical beam emitted from the object point O _s lies on a straight line _s c due to its rotational symmetry.
As shown in Figure 3, the incident height of the principal ray is H, and the intersection angles between the incident ray and q are i and j.
and i' and j' for the refracted rays. Also
Let θ, ω be the angles at which qc, _q , and sc intersect with the optical axis x, The meridional light flux is considered to be two rays passing through adjacent annular zones, and the distance between them is a minute amount dh. If q _n = p _n and q′ _n = p′ _n , then di = du − dθ = [1/rcos (i-j) − 1/p _n cosj] dh di′ = du′ − dθ = [1 /rcos(i-j)-1/p′ _n co
sj′] dh From the relationship ncosidi=n′cosi′di′, n/p _n cosicosj−n′/p′ _n cosi′cosj′ = cos(i−j)/r(ncosi−n′cosi′) and From equation (4-1), n/p _s −n′/p′ _s ≒n/p _n (1−tanitanj) −n′/p′ _n (1−tani′tanj′) (4−2) Now, Let p _s −p _n = δ, p′ _s −p′ _n = δ′, and if δ is small, 1/p _s = 1/p _n −δ/p ² / _n 1/p′ _s = 1/p ′ _n −δ′
/p′ _n ^2From this and equation (4-2), n/p ² / _n δ−n′/p′ _n ² δ′=n/p _n tanitanj −n′/p′ _n tani′tanj′ p _n ≒s ₀ , p′ _n ≒s′ ₀ , tani≒tanu−tanθ, n/s ² / ₀ δ−n′/s′ ₀ ² δ′=H ² Q ² _x (1/ N _x s ₀ −1/
N′ _x s′ ₀ )(4-3) However,

【formula】

[Formula] If the size of the object is y, then Q _x −Q _s = ny/s ₀ H=n′y′/s′ ₀ H From this and equation (4-3), δ/ny ² −δ′ /n′y′ ² = (Q _x /L) ² (1/N _x s ₀ −1
/N' _x s' ₀ ) (4-3') 1/nz ₀ -1/n'z' ₀ = Q _x /Q _s Δ(1/ns) + L/Q _s P
(4-4) From this, 1/N _x s ₀ -1/N' _x s' ₀ = Δ(1/ns) - Q _s /Q _x (P-
). If there are multiple refractive surfaces, add a subscript i and write 1/N _xi s _0i −1/N′ _xi s′ _0i = Δ(1/ns) _i −(P−
) _i +L ₁ ε _i /1+L ₁ τ _i (P-) _i (4-5) Formula (4-3') adds the sides and becomes δ ₁ /n ₁ y ² / ₁ −δ′ _k /n′ _k y' _k ² = _k 〓 ¹ Q ² / _xi /L ² / _i (1/N _xi s _0i -1/N' _xi s' _0i ) If δ ₁ = 0, there is no astigmatism. _δ′k
= 0, so the sum on the right side only needs to be zero. The right side is _k 〓 ¹ (1/L ₁ + τ _i ) ² A _i + _k 〓 ¹ (1/L ₁ + τ _i ) (P-P/-/ε ) _i = 0 (4-6) Distortion aberration The distances from Fresnel surface vertex A to the aperture 3 and its image 3' are z and z' respectively, those due to paraxial rays are z ₀ and z' ₀ , and the spherical aberration is Δz , Δz′, then z=z ₀ +Δz, z′=z′ ₀ +Δz′ If the magnification by the paraxial ray is β ₀ , then β ₀ = (z′ ₀ −s′ ₀ )z ₀ /(z ₀ −s ₀ )z′ ₀ Δz/z ₀ −s ₀ −Δz/z ₀ = Δz/z ² / ₀ (1/s ₀ −1/z ₀ ) Δz/z ₀ −Δz′/z′ ₀ =−1/L(n/z ² / ₀ Δz−n′
/z' ₀ ² Δz') +Δz/z ₀ -s ₀ -Δz'/z _' 0 -s' _0From this and equation (2-2), β=β ₀ {1+H ² Q ² / _x /2L[Δ (1/nz)-(P-)
]− H ₂ /2Q _x } If there are k refracting surfaces, β ₀₁ β ₀₂ ………β _0k = _k 〓 ¹ β _0i , β= _k 〓 ¹ β _0i {1+H ² / ₁ /2 _k 〓 ¹ (H _i /H ₁ ) ² Q ² / _xi / _Li [Δ(1/nz) _i − (P-) _i ] − H ² / ₁ / 2 _k 〓 ¹ (H _i /H ₁ ) ² Q _{xi i} } (5-1) Now, considering equations (1-1) and (1-3), (H _i /H ₁ ) ² Q ² / _xi /L _i [Δ(1/nz) _i −( P-) _i
−L _i /Q _{xi i} ] =1/L ₁ ε ² / _i (1+L ₁ τ _i ) ³ [Q _si /Q _xi Δ(1/nz
) _i −(P−) _i −Q _si /Q ² / _xi L _i P _i +(1−Q ² / _si /Q ²
/ _xi )(P-P) _i ]. Furthermore, from equation (4-4) [ ]
Inside is Δ(1/ns) _i −(P−) _i +L _i /Q _xi P _i −Q _si /Q ² / _xi
L _i P _i + (1-Q ² / _si / Q ² / _xi ) (P-) _i = Δ (1/ns) _i - (P-) _i + (L ₁ ε _i /1 + L ₁ τ _i
) ² _i +2(L ₁ ε _i /1+L ₁ τ _i )(P-) _iIn the end, equation (5-1) is β= _k 〓 ¹ β _0i {1+1/2H ² ₁ L ² _1k 〓 ¹ [(1/ L ₁ +τ _i ) ³ A _i +(1/L ₁ +τ _i ) _i +2(1/L ₁ +τ _i ) ² (P-P
/-/ε) _i ]} (5-2) Therefore, the condition for no distortion is _k 〓 ¹ (1/L ₁ + τ _i ) ³ A _i + _k 〓 ¹ (1/L ₁ + τ _i ) _i + 2 _k 〓 ¹ (1/L ₁ +τ _i ) ² (P-P/-/ε) _i = 0 (5-3) Image curvature Let the radii of curvature of the object surface and image be R and R'. Taking ^point A as the origin, the coordinates of ^the off-axis object point, _{image point} O', and refraction point q are: ′, y′) From equation (4-1), the spherical luminous flux is n/p _s −ncosi/r=n′/p′ _s −n′cosi′/r (6-1) cosi≒1−i ² /2, i≒H (1/r ₀ −1/z ₀ ) From these, the left side of equation (6-1) is Do the same for the right side Subtracting the sides, 1/n′R′ _s −1/nR _s = (Q _x /L) ² Δ(1/ns) + P
-(Q _s /L) ² (P -) (6-2) If there are k refracting surfaces, 1/n' _k R' _sk -1/n ₁ R _s1 = 1/L ² / _1k 〓 ¹ (1+L ₁ τ _i ) ² A _i + _k 〓 ¹ P _i +1/L ² / _1k 〓 ¹ (1+L ₁ τ _i /ε _i ) ² [1− (1+L ₁ δ _i /1+L ₁ τ _i ) ² ] × (P −) _i = _k 〓 ¹ (1/L ₁ +τ _i ) ² A _i + _k 〓 ¹ _i +2 _k 〓 ¹ (1/L ₁ +τ _i ) (P-P/-/ε) _i (6-3) Therefore, if R _s1 =∞, the right side of the above equation should be zero in order to avoid truncated field curvature. From equation (4-2), the meridional luminous flux is n/p _n (1-tanitanj)-n/rcosi = n'/p' _n (1-tani'tanj')-n'/rcosi'(6
-Four) Therefore, referring to the case of a spherical luminous flux, the left side of equation (6-4) is Similarly, find the right side of equation (6-4) and subtract each side to get 1/n'R' _n -1/nR _n = (Q _x /L) ² [Δ(1/ns) +
2 (1/N _x s ₀ - 1/N' _x s' ₀ )] +P- (Q _s /L) ² (P-) If there are k refracting surfaces, 1/n from equation (6-2) ′ _k R′ _nk −1/n ₁ R _n1 =1/n′ _k R′ _sk −1/n ₁ R _s1 +2 _k 〓 ¹ (Q _xi /L _i )(1/N _xi s _0i −1/N ′ _xi s′ _0i ) =3 _k 〓 ¹ (1/L ₁ +τ _i ) ² A _i + _k 〓 ¹ _i +4 _k 〓 ¹ (1/L ₁ + τ _i )(P−/ε) _i (6− 5) Therefore, if R _n1 = ∞, the right side of the above equation should be zero in order for there to be no curvature in the meridional image plane. To summarize each aberration explained above, spherical aberration S(1)=ΣA _i =Σ1/ε ² / _i [Δ(1/ns) _i −(P−
) _i ]Meridional curvature of field S(3)=3Σ(1/L ₁ +τ _i ) ² A _i +Σ _i +4Σ(1/L ₁ +
τ _i ) (P-P/-/ε) _i Spherical curvature of field S(4)=Σ(1/L ₁ +τ _i ) ² A _i +Σ _i +2Σ(1/L ₁ +
τ _i ) (P-P/-/ε) _i Distortion S(5)=Σ(1/L ₁ +τ _i ) ³ A _i +Σ(1/L ₁ +τ _i ) _i +2Σ(1/L ₁ +τ _i ) ² (P-P/-/ε) _i Average field curvature S(3)+S(4)/2=2Σ(1/L ₁ +τ _i ) ² A _i +Σ _i +3Σ(1/L ₁ +τ _i )(P-P/-/ε) _i Astigmatism S(3)-S(4)/2=Σ(1/L ₁ +τ _i ) ² A _i +Σ(1/L ₁
+τ _i ) (P-P/-/ε) _i (7-1) Here, if Σ is the surface number from 1 to k, then _k 〓 ¹
This is the meaning of In order to make each aberration zero, the right side of the above equation should be made zero. The condition for removing comatic aberration is Σ(1/L ₁ +τ _i )A _i =−Σ(P−P/−/ε) _i =0(7
-1') If the curved surface is a normal spherical surface, then _i = P _i , so it is immediately obvious that the last term in these equations disappears, giving Seidel's third-order aberration equation. Hereinafter, the third-order aberration formula disclosed by the present invention will be applied to a lens system composed of two Fresnel surfaces. Embodiment: FIG. 1 shows a prior art lens that is composed of two spherical surfaces, each of whose center of curvature coincides with the center of the aperture. This lens has no comatic aberration due to its symmetry with respect to the principal ray, but as can be seen from the tracing results of meridional rays in Table 1, spherical aberration and image curvature are large. In this embodiment, the spherical surfaces of the first and second surfaces are replaced with Fresnel surfaces, and the above-mentioned aberration-free condition is applied to simultaneously eliminate coma aberration and image curvature. Now let the refractive index of the object space with respect to the first surface be n ₁ , the refractive index of the image space with n′ ₁ , those with respect to the second surface be n ₂ ,
n' ₂ , and if the center of each paraxial radius of curvature r ₀₁ and r ₀₂ coincides with the center of the aperture, then for each surface, 1/L ₁ +τ _i =0, (7-1), (7- 1') All terms containing this in equation become zero. In order to have no comatic aberration as a whole system (7-1')
From the equation -Σ(P-P/-/ε) _i = 0, P ₁ - ₁ = -ε ₁ /ε ₂ (P ₂ - ₂ ) (7-2). Also, in order to have no field curvature as a whole system, from the field curvature equation in equation (7-1), Σ=0,
That is, it is sufficient if ₁ + ₂ = 0. Substituting this into equation (7-2), P ₁ − ₁ = −ε ₁ /ε ₂ (P ₂ + ₁ ) If the refractive index at the front and rear of the lens system is the same, then n ₁ =
as n′ ₂ (7-3) If the object is far enough away, s ₀₁ = ∞ and 1/ε ₁ = Q _s1 = n ₁ / r ₀₁ h ₂ / h ₁ = 1-d′ ₁ /s′ ₀₁ = n ₁ / n ₂ (1−n ₁ −n ₂ /n ₁ r _0
2 /r ₀₁ )(7-4) 1/s ₀₂ =1/s' ₀₁ -d' ₁ =-1/n ₂ r ₀₁ /n ₁ -n ₂ -r
₀₂ Q _s2 = n ₂ (1/r ₀₂ - 1/s ₀₂ ) = n ₂ / r ₀₂ (1/1 - n ₁ -
n ₂ /n ₁ r ₀₂ /r ₀₁ ) (7-5) 1/ε ₂ = (h ₂ /h ₁ ) ² Q _s2 Substituting equations (7-4) and (7-5) into this, we get ∴ ε ₁ /ε ₂ =1−n ₁ /n ₂ +n ₁ /n ₂ r ₀₁ /r _02From this and equation (7-3) When the Fresnel surface is defined in this way, aberrations other than spherical aberration are removed. Figure 5 shows the case of ₁ = ₂ = ∞, (7-6)
From the formula, r ₀₁ = −n ₂ / n ₁ r ₀₂ The meridian tracking calculation result in the above case is shown as the second
As shown in the table, it can be seen that coma aberration and image curvature are eliminated even at large angles of view, and spherical aberration is also sufficiently small. As explained above, a spherical lens composed of two spherical surfaces cannot eliminate field curvature, but by replacing it with a Fresnel surface, coma aberration and field curvature can be simultaneously eliminated, and spherical aberration can be eliminated. In addition, by using plastic materials, lightweight lenses can be obtained and mass production becomes possible.

[Brief explanation of drawings]

Figure 1 shows a conventional spherical lens composed of two spherical surfaces and has no coma, Figure 2 is an explanatory diagram for deriving the aberration theory formula regarding a Fresnel surface, and Figure 3 is an explanatory diagram of coma aberration. , FIG. 4 is an explanatory diagram of astigmatism, and FIG. 5 is a diagram showing an aberration-free Fresnel lens system according to the present invention. 1...Each annular refractive surface of the Fresnel lens, 3...Aperture, 3'...Image of the aperture.

【table】

【table】

【table】

Claims (1)

[Claims] 1. In a lens system in which two Fresnel surfaces are arranged coaxially, each annular refractive surface whose height from the optical axis is h is considered to be a part of a spherical surface whose radius of curvature is r. hand,
The center of each refractive surface is configured to coincide with the center of the diaphragm, and the refractive index of the object space of the first surface is n ₁ , the refractive index of the object space of the second surface is n ₂ , and the Fresnel surface of the first surface The radius is expressed as ₁ , the Fresnel surface radius of the second surface is expressed as ₂ , and the radius of curvature r at h=0 is the paraxial radius of curvature, and the paraxial radius of curvature of the first surface is r ₀₁ and the paraxial radius of curvature of the second surface is When expressed as r ₀₂ , An aberration-free Fresnel lens system that eliminates comatic aberration and field curvature as a whole system by selecting a Fresnel surface radius that satisfies the following.