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Two stage linear Fresnel lenses
Solar Energy Vol. 33, No. I, pp. 35-39, 1984
0038-092X/84 $3.00 + .00 Pergamon Press Ltd.
Printed in the U.S.A.
TWO STAGE LINEAR FRESNEL LENSES E. M. KRITCHMAN Elscint Ltd., Advanced Technology Center, P.O.B. 5258, Haifa 31-051, Israel (Received 8 September 1981; accepted 11 July 1983)
Abstract--A new design of a second stage reflective element, which is closely related to the earlier trumpet configuration, is presented. The implementation for linear Fresnel lenses is derived, leading to high solar concentration with both convex and flat lenses. 1. INTRODUCTION Fresnel lenses as well as concave reflectors may fit well the low cost requirements of a solar concentrator since they may consist of thin optical material. However, both suffer from large optical aberrations (including random errors of their optical surfaces) which decrease their concentration capability and make desirable therefore the incorporation of second stage elements to increase it back. The case of Fresnel lenses is easier since the second stage, being installed between the primary concentrator and the absorber, does not shade any part of the primary as happens in the reflective primary case. Hence the reflective surface of the second stage may be designed in this case to just connect the edges of the primary concentrator with those of the absorber, where the entire value of the second stage may be exercised. In the following sections a new design of ideal linear second stage element is discussed, which is closely related to the earlier trumpet element[l], and some applications to primary Fresnel lenses are presented and computed. This analysis may obviously imply to the 3D case by rotating the linear concentrator about its optical axis. This, however, is not just straight forward since an optimization against additional loss of skew-rays should be done.
y
plane
focot fi
" il
A
/c'/~'
t
Fig. 1. The two stage concentrator (only the left side of the second stage is illustrated), with an exit aperture CC'. at B, B' are required to be redirected directly on the opposing edges C', C of the second stage exit, which has been already shown to exactly coincide with the trumpet's exit [2]. Starting from point A, that requirement may be satisfied by a hyperbolic wall in the Oh rectangular axes system with the points B and C' being its loci. Obviously, from the intersection J with line A'B and lower, the mirror does not see the virtual source edge rays any more and instead it would be required to aim the bundle of rays that originate at A', toward the exit edge C'. It follows that this lower part conforms to an elliptical shape in the system O e with points A ' and C ' being its loci. Obviously, in order to have a symmetric concentrator about the optical axis, the left edge point C of the exit should be equally spaced from O as C'.
2. DESIGN CONSIDERATIONS
Given a primary concentrator A A " and a segment B B ' behind it (Fig. 1) which includes all its transmitted radiation (usually B B ' is defined by the smallest waist of the focused rays); the trumpet, which includes two reflective walls that stretch from A and A ' to the B B ' plane, has been designed to redirect rays that may propagate in the direction of B or B ' to C ' and C, respectively. The bundle of rays that are originally directed onto the segment B B " has been notated as the virtual source of the second stage and the rays that may be focused at either B or B ' ~ t h e edge rays of the virtual source. Obviously, according to the trumpet requirements, any edge ray of the virtual source would reach the exit aperture C C ' of the trumpet after an infinite number of reflections but any other ray that is originally included in the virtual source would reach it after a finite number of reflections if any. The present design is a modification of the trumpet in the sense that the edge rays of the virtual source
3. MATHEMATICAL REPRESENTATION
The left side hyperbolic curve may be represented by (xh/ah) 2 -- yh2/(fh 2 -- ah2) = 1
(1)
where xh, Yh are coordinates in Oh system, and ah, fh 35
E. M.
36
KRITCHMAN
are the hyperbola parameters (Fig. 1). In the coordinates x, y of system O, this equation transforms to (X "k- d)2/ah 2 - y2/(fh2 -- ah ~) = 1
(2)
where d is the displacement of Oh from O. Similarly the elliptic portion may be represented in x e, y ~ coordinates of system Oe as
for all the remaining parameters becomes almost straightforward, as fh and d may be derived from mutual addition and subtraction of eqns (5) and (6), f~ from eqn (13), ~ from eqn (12), a h from eqn (2), a, from eqn (11), and xj, )) from eqns (8) and (9). Equation (10) becomes then a dependent equation. In the region of concern {f?/x.~,f?/y~ ~} ~ 1
(XetG) ~ -- ye2(f¢ 2 -- ae 2) = I
(15)
(3)
where fi and G are the ellipse parameters, or in x, y coordinates
which implies to high solar concentration, a may be approximated from eqn (14) to a ~/x/1
(16)
+ yA:/XA 2
[(x - a) cos ~ + y sin ~ - L ] E / G z - [ - ( x - a ) x sin • + y cos ~]~/(ff - G 2) = 1
(4)
where ct is the angle between line A ' C ' and the x axis. The values of the 7 unknown parameters a, ah, fh, ao fe, d and ct along with the coordinates xj, yj of the joint point J of the two sections, may be derived from the solution of the following 9 equations L+d=L a+d=A
and the explicit expressions for the rest of the parameters become d = ~ - a)/2 (17) (18)
fh = Oc~ + a ) / 2
= arctan [ya/( - XA -- a)]
( 19)
(5)
fe = ~/YA z + (xA + a)2/2
(6)
ah ~ x / ( x A + d ) 2" Fh2/[(XA + d) 2 + yA2],
(20)
(21)
{f,2/XA2.f~2/yA2 } .~ 1 ( x A + d)2/ah 2 -- yA2/(fh 2 -- ah 2) = 1
(7) a~ ..~ ~/fe 2 + 2afe'(l + COS ~), ~2/XA2,f.Z/yA2 } ~ 1 (22)
which represents the requirement that the edge of the primary (xA, YA) is on the second stage wall
and
(xj + d)21ah 2 -- yj2lG ~ - ah 2) = 1
(8)
Xj.•
y / ( x j + f 3 = Y A I ( - - XA +f~)
(9)
E'G
~/L2G 2 + (I - G).(ah 2 + G52)1/(I - G) - d
-
(23) and
where
[(x s - a ) cos • + yj sin cc - f~]2/a]
- [ - (xj - a )
x sin ~ + yj cos ~]2/@2 _ ae 2) = 1
G -~ ah2yA2/[(fh 2 -- ah2)'(-- XA + fv) 2]
(10) and
the later three which represent the three requirements of the joint point J. ( - 2 a cos
ct
--fe)2/ae 2 - (2a sin ct)2/~ 2 - ae2) = 1 (11)
Figure 2 illustrates a representative calculation of I
which characterises point C, and the additional relations tan ct = Y A / ( - x A -- a )
(12)
(24)
yj = (xj + f ~ ) " Y A / ( - - xa + f , ) .
I
I
,~- 3
\ \\~'~
hyperbolic
"S
~ _\ " ~
section
.=_o 2
~.~~,~
I
I
°V,
and
\ \ elliptic
2fe = LV~2 + (XA + a)2] I/2.
(13) ~
f -
As has already been shown (2),
°l -4
",, - 3
I -2
I I -1
0
x (10-3fr0cti0n of f) 2a = A ' B
- AB
(14)
Consequently, the analytical solution of eqns (5)-(13)
Fig. 2. The very low region of the second stage, computed for a spherical primary of a radius curvature R = f and L = 0.05f.
Two stage linear Fresnel lenses the former equations. The primary was considered as a spherical two dimensional thin tens of curvature radius R that is equal to its focal length f, and YA= 0.7f; f~ =f/200 was assumed and a consequent concentration of C = 200 was obtained. Only the lower part of the second stage, along with the respective trumpet curve is illustrated. 4. CONCENTRATION
If a lens concentrates all the radiation that it receives from a given acceptance angular field + 00 onto a segment of 2f~ at its focal plane, its concentration C1 is equal to C i - - --XA/f~.
(26)
Combining eqns (25) and (26), the overall concentration C of the two stage concentrator may be found as
C = C," C2 = ~
+ y2/f~.
of the edge rays of the acceptance field outside of the exit aperture off~ = f t a n 00; otherwise, the radiation from the acceptance field would be confined to that exit, but as instead of eqn (28) we would get then XA2 + yA2 >f2.
(30)
Equation (27) would result in an impossibly greater concentration than the ideal. Unfortunately the spherical Fresnel lens that transmits its radiation into an exit of f~ = f t a n 0o may be obtained only with infinite refractive index n[5]; at finite n it would become more convex, decreasing thereby the concentration, as discussed in paragraphs (b) and (c).
(25)
For small 00,f~ is usually small and then eqn (16) may be used to derive the expression for the additional concentration C2 that may be brought about by the second stage element Cz - f / a ~ x / ~ + yA2/XA2.
37
(b) Aplanatic Fresnel lens As opposed to continuous lenses, Fresnel lenses do not have to obey Abbe sine law, and aplanatic lenses might still be designed even with a finite n albeit they become more convex than spherical. The best truncation point of these lenses for maximum concentration is, according to eqn (27), where XA2+ yA 2 is maximum, which occurs just at the top of the lenses. If we regard a more reasonable location, e.g. YA = 0.7f which corresponds to xA = - 0.55f for n = 1.49 (5), we get from eqns (27) and (28)
(27)
C = .,/0.552 + 0.72/tan 00 = 0.89/tan 00 ~ 0.89CI (31)
The following paragraphs will exploit the later relation for some known Fresnel lens configurations, all of which have a smooth surface at the sun side.
where the second stage contribution may be found from eqn (26) to be (?2 = 1.62.
(a) The ideal case According to Luque[3] a two stage Fresnel lens concentrator for small values of 00 may be ideal only if the lens profile is spherical with a radius equal to the lens focal length f. In this case the edge point A would satisfy the sphere relation
(c) Color corrected lens When the unavoided chromatic dispersion of available optical material is taken into consideration, the lens must become even more convex, given it should still concentrate its radiation into the segment of f, = f t a n 00. For a color corrected lens xA = - 0.34f at n = 1.49 and YA= 0.7f[6], and hence
xA2 + yA2 = f t .
C = x/0.0342 + 0.72/tan O0~ 0.78C/
(28)
Assuming no optical aberrations, the size of the image of the half acceptance field at the focal plane is f~ = f t a n 00.
(29)
Combining eqns (27)-(29), the two stage concentration C may be derived to be
(32)
which is higher by C2 = 2.3 than the single stage concentration. (d) Flat lens A fiat Fresnel lens shows coma and chromatic aberrations which cause the perceived rays to be transmitted outside of the exit of f~ = f t a n 00. As shown in the Appendix, the enlarged half exit aperture which includes all the radiation becomes
C = 1/tan O0
f~ = (0 cos q5 + An sin ~b)(xA2 +fl)/0rx/1 -- n 2 sin 2 ~ ) which is indeed approximately equal to the ideal limit C~ = 1/sin 00. The same result would imply for an aplanatic continuous lens, where its second principal surface (that must be spherical in order to obey the sine law of Abbe [4]) replaces the Fresnel lens surface. It follows that neither the second principal surface of a continuous lens, nor the Fresnel lens shape may be less convex than that sphere without dispersing part
(33) where 4~ is the apex angle of the outest groove of the lens and An is the half chromatic dispersion of the optical material in the relevant wavelengths. Substituting eqn (33) into eqn (27) leads to C = (1~Co + 1/C,)-'
(34)
38
E.M. KRITCHMAN
where Co = (D
"00"cos ~)-1,
C. = (D
.An .sin ~)-'
and O = x/[1 + (2f/No)-21/[l - (n sin ~b)2]. In addition, eqn (26) may be rearranged as
(72 = x/1 + ( 2 f / N o ) 2.
3. A. Luque, Appl. Opt. (1981). 4. M. Born and E. Wolf, Principles of Optics, pp. 166 169. Pergamon Press, New York (1970). (35) t 5. E. M. Kritchman, A. A. Friesem and G. Yekutieli, Highly concentrating Fresnel lenses. Appl. Opt. 18, 2688 (36) (1979). 6. E. M. Kritchman, Color-corrected Fresnel lens for solar concentration. Opt. Lett. 5, 35 (1980). 7. J. F. Woodman, Acrylics, Modern Plastics Encyclopedia, Vol. 14. McGraw-Hill, New York (1971). (37) 8. M. Collares-Pereira, A. Rabl and R. Winston, Lensmirror combinations with maximal concentration. Appl. Opt. 16, 2677 (1977). (38)
The value of ~b for different f / N o can be found by solving eqns (46) and (47) of the Appendix; by substituting it in eqns (36)-(38) the two stage concentration C is obtained to be
f/No
c~
Co
C,
C
C2
C/C1
1 0.67
37° 41
99 45
132 52
56 24
2.236 1.414
0.28 0.12
for 00 = 5 mr, n = 1.49, An = 0.005, the two last of which are appropriate for acrylic material[7]. 5. CONCLUSION Both the new second stage and the trumpet may substantially increase the solar concentration of linear Fresnel lenses, and satisfy thereby the needs in many cases, where the relatively poor linear concentrating factor is marginal for the intended application. In addition the combination with Fresnel lenses is especially attractive since the optical constituents may stand for the aperture glazing and insulating walls as well. As obvious from the construction, the new second stage surface would touch less rays than the trumpet and hence is both expected to be slightly more efficient and less sensitive to truncation at the top (which might be desired to reduce cost). In addition its shape may be more easily manufactured than the trumpet as it is less curved near the exit; however, it is not expected to be as ideal in the three dimensional case as the trumpet, in the sense that it might lose some skew rays then. In respect to the other second stage that was suggested earlier [8] it is expected to be even more advantageous in the efficiency domain, as the exit of that element was designed to be above the natural plane (the primary focal plane), dictating thereby more rays to be redirected by the second stage.
Acknowledgement--This work was performed under U.S. Department of Energy Grant DE-FG02-79ETOOO89. REFERENCES
1. R. Winston and W. T. Welford, Geometrical vector flux and some new nonimaging concentrators. J. Opt. Soc. Am. 69, 532 (1979). 2. E. M. Kritchman, Second stage ideal elements. J. Opt. Soc. Am. To be published.
APPENDIX. f, OF A FLAT FRESNEL LENS wITH SMOOTH UPWARD SURFACE The narrowest exit aperture at the focal plane of a flat Fresnel lens that yet includes all the transmitted rays is obviously dictated by a blue ray that comes from the right edge of the acceptance field and is transmitted through the rightmost groove of the lens. Given an input impingment angle 0 relative to the optical axis, the angles fl(0, n) and ?(O,n) of the ray direction inside the groove with the normals to the groove respective surfaces are (Fig. 3) /3(0, n) = arcsin(sin O/n)
(39)
~(0, n) = [3(0, n) + ck
(40)
and
where q~stands for the apex of the groove, and the exit angle 0(0, n) of the ray from the groove is 6(0, n) = arcsin[n sin y(0, n)]
(41)
Combining eqns (39)-(41) and derivating with respect to 0 leads to
dO(O,n)/O0 = cos y(0, n) cos 0/ ~/[1 - n 2sinZ(y(0, n))](1 - sin 20/n 2)
(42)
and
06 (0, n)/O0 l0: 0 = cos q~/1 - n 2 sin 2 4~.
(43)
-~0 i._,,2
I I
Is -ca
"-B
/ -fv
0
rata]
plane
-X A
Fig. 3. A flat Fresnel lens (the rightmost groove magnified).
Two stage linear Fresnel lenses Derivation with respect to n would similarly lead to
otherwise direct the ray towards the on axis stigmatic focal point is accordingly
a6(O, n)/~n = [sin ~(0, n) - n cos 7(0, n)
x sin O/(n2X/1 _ sin 20/n2)]/X/1 _ n2.sin2 y(0, n),
(44)
(45)
(46)
which expresses the usual stigmatic focus requirement at 0 = 0 and the mean refractive index n, where 6(0, n) can be easily derived from eqns (39)-(41) to be 6(0, n) = arcsin(n sin ~).
f~ is obtained from the relation f~ = A6(0, An). (xA2 + f2)/f
itself can be obtained as a solution of the equation, 6(0, n) - ~b --- arctan(-xA/YA)
A~ (0, An) ~ a~ (0, n)l~O Io= o0o + ~ (0, n)l~n Io= = (cos q~00+ An sin ~)/~/1 - n 2 sin 2 ~b. (48)
which at 0 = 0 reduces to 06(0, n)lanlo=o = sin ~/x/1 - n 2 sin 2 ~.
39
(47)
The deviation A6(0, An) from the angle 6(0, n) which would
(49)
which may be easily derived from Fig. 3. Combining eqns (48) and (49), the following approximate expression forf~ is obtained, f~ = (cos ~b00+ An sin ~b)(x~2 +f2)/(f~/1 - n 2" sin 2 ~b)
(50)
where An is the dispersion of the blue wavelength from the mean design refractive index n.
0038-092X/84 $3.00 + .00 Pergamon Press Ltd.
Printed in the U.S.A.
TWO STAGE LINEAR FRESNEL LENSES E. M. KRITCHMAN Elscint Ltd., Advanced Technology Center, P.O.B. 5258, Haifa 31-051, Israel (Received 8 September 1981; accepted 11 July 1983)
Abstract--A new design of a second stage reflective element, which is closely related to the earlier trumpet configuration, is presented. The implementation for linear Fresnel lenses is derived, leading to high solar concentration with both convex and flat lenses. 1. INTRODUCTION Fresnel lenses as well as concave reflectors may fit well the low cost requirements of a solar concentrator since they may consist of thin optical material. However, both suffer from large optical aberrations (including random errors of their optical surfaces) which decrease their concentration capability and make desirable therefore the incorporation of second stage elements to increase it back. The case of Fresnel lenses is easier since the second stage, being installed between the primary concentrator and the absorber, does not shade any part of the primary as happens in the reflective primary case. Hence the reflective surface of the second stage may be designed in this case to just connect the edges of the primary concentrator with those of the absorber, where the entire value of the second stage may be exercised. In the following sections a new design of ideal linear second stage element is discussed, which is closely related to the earlier trumpet element[l], and some applications to primary Fresnel lenses are presented and computed. This analysis may obviously imply to the 3D case by rotating the linear concentrator about its optical axis. This, however, is not just straight forward since an optimization against additional loss of skew-rays should be done.
y
plane
focot fi
" il
A
/c'/~'
t
Fig. 1. The two stage concentrator (only the left side of the second stage is illustrated), with an exit aperture CC'. at B, B' are required to be redirected directly on the opposing edges C', C of the second stage exit, which has been already shown to exactly coincide with the trumpet's exit [2]. Starting from point A, that requirement may be satisfied by a hyperbolic wall in the Oh rectangular axes system with the points B and C' being its loci. Obviously, from the intersection J with line A'B and lower, the mirror does not see the virtual source edge rays any more and instead it would be required to aim the bundle of rays that originate at A', toward the exit edge C'. It follows that this lower part conforms to an elliptical shape in the system O e with points A ' and C ' being its loci. Obviously, in order to have a symmetric concentrator about the optical axis, the left edge point C of the exit should be equally spaced from O as C'.
2. DESIGN CONSIDERATIONS
Given a primary concentrator A A " and a segment B B ' behind it (Fig. 1) which includes all its transmitted radiation (usually B B ' is defined by the smallest waist of the focused rays); the trumpet, which includes two reflective walls that stretch from A and A ' to the B B ' plane, has been designed to redirect rays that may propagate in the direction of B or B ' to C ' and C, respectively. The bundle of rays that are originally directed onto the segment B B " has been notated as the virtual source of the second stage and the rays that may be focused at either B or B ' ~ t h e edge rays of the virtual source. Obviously, according to the trumpet requirements, any edge ray of the virtual source would reach the exit aperture C C ' of the trumpet after an infinite number of reflections but any other ray that is originally included in the virtual source would reach it after a finite number of reflections if any. The present design is a modification of the trumpet in the sense that the edge rays of the virtual source
3. MATHEMATICAL REPRESENTATION
The left side hyperbolic curve may be represented by (xh/ah) 2 -- yh2/(fh 2 -- ah2) = 1
(1)
where xh, Yh are coordinates in Oh system, and ah, fh 35
E. M.
36
KRITCHMAN
are the hyperbola parameters (Fig. 1). In the coordinates x, y of system O, this equation transforms to (X "k- d)2/ah 2 - y2/(fh2 -- ah ~) = 1
(2)
where d is the displacement of Oh from O. Similarly the elliptic portion may be represented in x e, y ~ coordinates of system Oe as
for all the remaining parameters becomes almost straightforward, as fh and d may be derived from mutual addition and subtraction of eqns (5) and (6), f~ from eqn (13), ~ from eqn (12), a h from eqn (2), a, from eqn (11), and xj, )) from eqns (8) and (9). Equation (10) becomes then a dependent equation. In the region of concern {f?/x.~,f?/y~ ~} ~ 1
(XetG) ~ -- ye2(f¢ 2 -- ae 2) = I
(15)
(3)
where fi and G are the ellipse parameters, or in x, y coordinates
which implies to high solar concentration, a may be approximated from eqn (14) to a ~/x/1
(16)
+ yA:/XA 2
[(x - a) cos ~ + y sin ~ - L ] E / G z - [ - ( x - a ) x sin • + y cos ~]~/(ff - G 2) = 1
(4)
where ct is the angle between line A ' C ' and the x axis. The values of the 7 unknown parameters a, ah, fh, ao fe, d and ct along with the coordinates xj, yj of the joint point J of the two sections, may be derived from the solution of the following 9 equations L+d=L a+d=A
and the explicit expressions for the rest of the parameters become d = ~ - a)/2 (17) (18)
fh = Oc~ + a ) / 2
= arctan [ya/( - XA -- a)]
( 19)
(5)
fe = ~/YA z + (xA + a)2/2
(6)
ah ~ x / ( x A + d ) 2" Fh2/[(XA + d) 2 + yA2],
(20)
(21)
{f,2/XA2.f~2/yA2 } .~ 1 ( x A + d)2/ah 2 -- yA2/(fh 2 -- ah 2) = 1
(7) a~ ..~ ~/fe 2 + 2afe'(l + COS ~), ~2/XA2,f.Z/yA2 } ~ 1 (22)
which represents the requirement that the edge of the primary (xA, YA) is on the second stage wall
and
(xj + d)21ah 2 -- yj2lG ~ - ah 2) = 1
(8)
Xj.•
y / ( x j + f 3 = Y A I ( - - XA +f~)
(9)
E'G
~/L2G 2 + (I - G).(ah 2 + G52)1/(I - G) - d
-
(23) and
where
[(x s - a ) cos • + yj sin cc - f~]2/a]
- [ - (xj - a )
x sin ~ + yj cos ~]2/@2 _ ae 2) = 1
G -~ ah2yA2/[(fh 2 -- ah2)'(-- XA + fv) 2]
(10) and
the later three which represent the three requirements of the joint point J. ( - 2 a cos
ct
--fe)2/ae 2 - (2a sin ct)2/~ 2 - ae2) = 1 (11)
Figure 2 illustrates a representative calculation of I
which characterises point C, and the additional relations tan ct = Y A / ( - x A -- a )
(12)
(24)
yj = (xj + f ~ ) " Y A / ( - - xa + f , ) .
I
I
,~- 3
\ \\~'~
hyperbolic
"S
~ _\ " ~
section
.=_o 2
~.~~,~
I
I
°V,
and
\ \ elliptic
2fe = LV~2 + (XA + a)2] I/2.
(13) ~
f -
As has already been shown (2),
°l -4
",, - 3
I -2
I I -1
0
x (10-3fr0cti0n of f) 2a = A ' B
- AB
(14)
Consequently, the analytical solution of eqns (5)-(13)
Fig. 2. The very low region of the second stage, computed for a spherical primary of a radius curvature R = f and L = 0.05f.
Two stage linear Fresnel lenses the former equations. The primary was considered as a spherical two dimensional thin tens of curvature radius R that is equal to its focal length f, and YA= 0.7f; f~ =f/200 was assumed and a consequent concentration of C = 200 was obtained. Only the lower part of the second stage, along with the respective trumpet curve is illustrated. 4. CONCENTRATION
If a lens concentrates all the radiation that it receives from a given acceptance angular field + 00 onto a segment of 2f~ at its focal plane, its concentration C1 is equal to C i - - --XA/f~.
(26)
Combining eqns (25) and (26), the overall concentration C of the two stage concentrator may be found as
C = C," C2 = ~
+ y2/f~.
of the edge rays of the acceptance field outside of the exit aperture off~ = f t a n 00; otherwise, the radiation from the acceptance field would be confined to that exit, but as instead of eqn (28) we would get then XA2 + yA2 >f2.
(30)
Equation (27) would result in an impossibly greater concentration than the ideal. Unfortunately the spherical Fresnel lens that transmits its radiation into an exit of f~ = f t a n 0o may be obtained only with infinite refractive index n[5]; at finite n it would become more convex, decreasing thereby the concentration, as discussed in paragraphs (b) and (c).
(25)
For small 00,f~ is usually small and then eqn (16) may be used to derive the expression for the additional concentration C2 that may be brought about by the second stage element Cz - f / a ~ x / ~ + yA2/XA2.
37
(b) Aplanatic Fresnel lens As opposed to continuous lenses, Fresnel lenses do not have to obey Abbe sine law, and aplanatic lenses might still be designed even with a finite n albeit they become more convex than spherical. The best truncation point of these lenses for maximum concentration is, according to eqn (27), where XA2+ yA 2 is maximum, which occurs just at the top of the lenses. If we regard a more reasonable location, e.g. YA = 0.7f which corresponds to xA = - 0.55f for n = 1.49 (5), we get from eqns (27) and (28)
(27)
C = .,/0.552 + 0.72/tan 00 = 0.89/tan 00 ~ 0.89CI (31)
The following paragraphs will exploit the later relation for some known Fresnel lens configurations, all of which have a smooth surface at the sun side.
where the second stage contribution may be found from eqn (26) to be (?2 = 1.62.
(a) The ideal case According to Luque[3] a two stage Fresnel lens concentrator for small values of 00 may be ideal only if the lens profile is spherical with a radius equal to the lens focal length f. In this case the edge point A would satisfy the sphere relation
(c) Color corrected lens When the unavoided chromatic dispersion of available optical material is taken into consideration, the lens must become even more convex, given it should still concentrate its radiation into the segment of f, = f t a n 00. For a color corrected lens xA = - 0.34f at n = 1.49 and YA= 0.7f[6], and hence
xA2 + yA2 = f t .
C = x/0.0342 + 0.72/tan O0~ 0.78C/
(28)
Assuming no optical aberrations, the size of the image of the half acceptance field at the focal plane is f~ = f t a n 00.
(29)
Combining eqns (27)-(29), the two stage concentration C may be derived to be
(32)
which is higher by C2 = 2.3 than the single stage concentration. (d) Flat lens A fiat Fresnel lens shows coma and chromatic aberrations which cause the perceived rays to be transmitted outside of the exit of f~ = f t a n 00. As shown in the Appendix, the enlarged half exit aperture which includes all the radiation becomes
C = 1/tan O0
f~ = (0 cos q5 + An sin ~b)(xA2 +fl)/0rx/1 -- n 2 sin 2 ~ ) which is indeed approximately equal to the ideal limit C~ = 1/sin 00. The same result would imply for an aplanatic continuous lens, where its second principal surface (that must be spherical in order to obey the sine law of Abbe [4]) replaces the Fresnel lens surface. It follows that neither the second principal surface of a continuous lens, nor the Fresnel lens shape may be less convex than that sphere without dispersing part
(33) where 4~ is the apex angle of the outest groove of the lens and An is the half chromatic dispersion of the optical material in the relevant wavelengths. Substituting eqn (33) into eqn (27) leads to C = (1~Co + 1/C,)-'
(34)
38
E.M. KRITCHMAN
where Co = (D
"00"cos ~)-1,
C. = (D
.An .sin ~)-'
and O = x/[1 + (2f/No)-21/[l - (n sin ~b)2]. In addition, eqn (26) may be rearranged as
(72 = x/1 + ( 2 f / N o ) 2.
3. A. Luque, Appl. Opt. (1981). 4. M. Born and E. Wolf, Principles of Optics, pp. 166 169. Pergamon Press, New York (1970). (35) t 5. E. M. Kritchman, A. A. Friesem and G. Yekutieli, Highly concentrating Fresnel lenses. Appl. Opt. 18, 2688 (36) (1979). 6. E. M. Kritchman, Color-corrected Fresnel lens for solar concentration. Opt. Lett. 5, 35 (1980). 7. J. F. Woodman, Acrylics, Modern Plastics Encyclopedia, Vol. 14. McGraw-Hill, New York (1971). (37) 8. M. Collares-Pereira, A. Rabl and R. Winston, Lensmirror combinations with maximal concentration. Appl. Opt. 16, 2677 (1977). (38)
The value of ~b for different f / N o can be found by solving eqns (46) and (47) of the Appendix; by substituting it in eqns (36)-(38) the two stage concentration C is obtained to be
f/No
c~
Co
C,
C
C2
C/C1
1 0.67
37° 41
99 45
132 52
56 24
2.236 1.414
0.28 0.12
for 00 = 5 mr, n = 1.49, An = 0.005, the two last of which are appropriate for acrylic material[7]. 5. CONCLUSION Both the new second stage and the trumpet may substantially increase the solar concentration of linear Fresnel lenses, and satisfy thereby the needs in many cases, where the relatively poor linear concentrating factor is marginal for the intended application. In addition the combination with Fresnel lenses is especially attractive since the optical constituents may stand for the aperture glazing and insulating walls as well. As obvious from the construction, the new second stage surface would touch less rays than the trumpet and hence is both expected to be slightly more efficient and less sensitive to truncation at the top (which might be desired to reduce cost). In addition its shape may be more easily manufactured than the trumpet as it is less curved near the exit; however, it is not expected to be as ideal in the three dimensional case as the trumpet, in the sense that it might lose some skew rays then. In respect to the other second stage that was suggested earlier [8] it is expected to be even more advantageous in the efficiency domain, as the exit of that element was designed to be above the natural plane (the primary focal plane), dictating thereby more rays to be redirected by the second stage.
Acknowledgement--This work was performed under U.S. Department of Energy Grant DE-FG02-79ETOOO89. REFERENCES
1. R. Winston and W. T. Welford, Geometrical vector flux and some new nonimaging concentrators. J. Opt. Soc. Am. 69, 532 (1979). 2. E. M. Kritchman, Second stage ideal elements. J. Opt. Soc. Am. To be published.
APPENDIX. f, OF A FLAT FRESNEL LENS wITH SMOOTH UPWARD SURFACE The narrowest exit aperture at the focal plane of a flat Fresnel lens that yet includes all the transmitted rays is obviously dictated by a blue ray that comes from the right edge of the acceptance field and is transmitted through the rightmost groove of the lens. Given an input impingment angle 0 relative to the optical axis, the angles fl(0, n) and ?(O,n) of the ray direction inside the groove with the normals to the groove respective surfaces are (Fig. 3) /3(0, n) = arcsin(sin O/n)
(39)
~(0, n) = [3(0, n) + ck
(40)
and
where q~stands for the apex of the groove, and the exit angle 0(0, n) of the ray from the groove is 6(0, n) = arcsin[n sin y(0, n)]
(41)
Combining eqns (39)-(41) and derivating with respect to 0 leads to
dO(O,n)/O0 = cos y(0, n) cos 0/ ~/[1 - n 2sinZ(y(0, n))](1 - sin 20/n 2)
(42)
and
06 (0, n)/O0 l0: 0 = cos q~/1 - n 2 sin 2 4~.
(43)
-~0 i._,,2
I I
Is -ca
"-B
/ -fv
0
rata]
plane
-X A
Fig. 3. A flat Fresnel lens (the rightmost groove magnified).
Two stage linear Fresnel lenses Derivation with respect to n would similarly lead to
otherwise direct the ray towards the on axis stigmatic focal point is accordingly
a6(O, n)/~n = [sin ~(0, n) - n cos 7(0, n)
x sin O/(n2X/1 _ sin 20/n2)]/X/1 _ n2.sin2 y(0, n),
(44)
(45)
(46)
which expresses the usual stigmatic focus requirement at 0 = 0 and the mean refractive index n, where 6(0, n) can be easily derived from eqns (39)-(41) to be 6(0, n) = arcsin(n sin ~).
f~ is obtained from the relation f~ = A6(0, An). (xA2 + f2)/f
itself can be obtained as a solution of the equation, 6(0, n) - ~b --- arctan(-xA/YA)
A~ (0, An) ~ a~ (0, n)l~O Io= o0o + ~ (0, n)l~n Io= = (cos q~00+ An sin ~)/~/1 - n 2 sin 2 ~b. (48)
which at 0 = 0 reduces to 06(0, n)lanlo=o = sin ~/x/1 - n 2 sin 2 ~.
39
(47)
The deviation A6(0, An) from the angle 6(0, n) which would
(49)
which may be easily derived from Fig. 3. Combining eqns (48) and (49), the following approximate expression forf~ is obtained, f~ = (cos ~b00+ An sin ~b)(x~2 +f2)/(f~/1 - n 2" sin 2 ~b)
(50)
where An is the dispersion of the blue wavelength from the mean design refractive index n.