Abstract

A previous paper [2] presented an analysis of a class of microstructured optical surfaces in two dimensions, in which a classification of the microstructures was obtained (regular and anomalous) and a concept of 2D ideal microstructures was introduced. In this paper the study of those microstructured optical surfaces is extended to three dimensions with rotational symmetry. As a starting point, non-microstructured rotational optical systems in the First Order Approximation are also classified as point-spot type and ring-spot type, with remarkable perfect particular cases. This classification is also extended to the case in which ideal microstructured rotational surfaces are used, for both regular and anomalous type. The case of perfect ring-spot type system with an odd number of rotational, anomalous, ideal microstructures enables the definition of an anomalous aplanatic system that has direct application for mixing spatially and angularly the light emitted by several sources.

©2007 Optical Society of America

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References

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  1. R. Winston, J.C. Miñano, and P. Benítez, Nonimaging Optics, (Elsevier, 2005)
  2. P. Benítez, J. C. Miñano, and A. Santamaría, “Analysis of microstructured surfaces in two dimensions,” Opt. Express 14, 8561-8567 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-19-8561
    [Crossref] [PubMed]
  3. M. Born and E. Wolf, Principles of Optics, (Pergamon, Oxford, 1975).
  4. W.A. Parkyn and D, Pelka, “Compact non-imaging lens with totally internal reflecting facets”, in Nonimaging Optics: Maximum Efficiency Light Transfer, Roland Winston, ed., Proc. SPIE 1528, 70-81, (1991)
    [Crossref]
  5. R.K. Luneburg, Mathematical theory of Optics, (U. California, Berkeley, 1964), chapter IV
  6. D. Korsch, Reflective Optics, pg. 27, (Academic, New York, 1999)

2006 (1)

1991 (1)

W.A. Parkyn and D, Pelka, “Compact non-imaging lens with totally internal reflecting facets”, in Nonimaging Optics: Maximum Efficiency Light Transfer, Roland Winston, ed., Proc. SPIE 1528, 70-81, (1991)
[Crossref]

Benítez, P.

Born, M.

M. Born and E. Wolf, Principles of Optics, (Pergamon, Oxford, 1975).

Korsch, D.

D. Korsch, Reflective Optics, pg. 27, (Academic, New York, 1999)

Luneburg, R.K.

R.K. Luneburg, Mathematical theory of Optics, (U. California, Berkeley, 1964), chapter IV

Miñano, J. C.

Miñano, J.C.

R. Winston, J.C. Miñano, and P. Benítez, Nonimaging Optics, (Elsevier, 2005)

Parkyn, W.A.

W.A. Parkyn and D, Pelka, “Compact non-imaging lens with totally internal reflecting facets”, in Nonimaging Optics: Maximum Efficiency Light Transfer, Roland Winston, ed., Proc. SPIE 1528, 70-81, (1991)
[Crossref]

Pelka, D,

W.A. Parkyn and D, Pelka, “Compact non-imaging lens with totally internal reflecting facets”, in Nonimaging Optics: Maximum Efficiency Light Transfer, Roland Winston, ed., Proc. SPIE 1528, 70-81, (1991)
[Crossref]

Santamaría, A.

Winston, R.

R. Winston, J.C. Miñano, and P. Benítez, Nonimaging Optics, (Elsevier, 2005)

Wolf, E.

M. Born and E. Wolf, Principles of Optics, (Pergamon, Oxford, 1975).

Opt. Express (1)

Proc. SPIE (1)

W.A. Parkyn and D, Pelka, “Compact non-imaging lens with totally internal reflecting facets”, in Nonimaging Optics: Maximum Efficiency Light Transfer, Roland Winston, ed., Proc. SPIE 1528, 70-81, (1991)
[Crossref]

Other (4)

R.K. Luneburg, Mathematical theory of Optics, (U. California, Berkeley, 1964), chapter IV

D. Korsch, Reflective Optics, pg. 27, (Academic, New York, 1999)

R. Winston, J.C. Miñano, and P. Benítez, Nonimaging Optics, (Elsevier, 2005)

M. Born and E. Wolf, Principles of Optics, (Pergamon, Oxford, 1975).

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Figures (8)

Fig. 1.
Fig. 1. Coordinate system for tolerance study of rotational symmetric optical systems

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Fig. 2.
Fig. 2. Locus of the ray directions emitted by P1 (x 1=ε>0, y 1=0, z=0) passing through points P2 on circumference x 2 2+y 2 2=ρ2(β) (a) for point-spot optical systems and (b) for ring-spot optical systems. The difference between both is summarized saying that ring-spot type systems always enclose the origin, while point-spot type never does it.

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Fig. 3.
Fig. 3. (a) Point-spot type (at bottom) and ring-spot type (at top) slices of a parabolic reflector. (b) Ray traces obtained from an off-axis point source, shown as a yellow sphere in (a).

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Fig. 4.
Fig. 4. In the First Order Optics framework, (a) point-spot system when α=0 (aplanatic), in which the spot is point-like and its center independent of β (so the whole system produce sharp imaging) (b) ring-spot system when pC = 0, in which the spot is a ring centered on the optical axis and its radius α varies with β.

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Fig. 5.
Fig. 5. Regular and anomalously deflecting microstructures in two dimensions.

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Fig. 6.
Fig. 6. The TIR lens is an example of anomalous structured surface. When considered with finite facet size (i.e., out of the infinitesimal limit) such anomalism translates in the function ρ (β) being discontinuous, with its envelope having positive derivative but ρ’(β) < 0 in each facet.

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Fig. 7.
Fig. 7. (a) The use of anomalous microstructures allows the conception of devices that produce a well collimated ring pattern for chips placed off-axis, and thus color mixing collimators. (b) The introduction of a small angle holographic diffuser eliminates the central dark region, if desired.

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Fig. 8.
Fig. 8. (a) Far field pattern produced by an off-axis chip in a specific color mixing device, which performs close to an anomalous aplanatic system. (b) Horizontal and vertical cross sections of that pattern produced by this device for a single off-axis light source showing that FWHM = 5° (chip is modelled as a Lambertian square with side = 0.25 mm, chip center being set at a distance of 0.5 mm to the optical axis; the optics diameter is 50 mm).

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Equations (75)

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x 2 = ρ ( β ) cos ⁡ ( ϕ + ϕ 0 )
y 2 = ρ ( β ) sin ⁡ ( ϕ + ϕ 0 )
( p 2 * q 2 * ) = ( a − b cos ⁡ ( 2 ϕ ) b sin ⁡ ( 2 ϕ ) c cos ⁡ ϕ b sin ⁡ ( 2 ϕ ) a + b cos ⁡ ( 2 ϕ ) − c sin ⁡ ϕ ) ( x 1 y 1 z )
a = − n 1 2 ( cos ⁡ β ρ ' ( β ) + sin ⁡ β ρ ( β ) )
b = n 1 2 ( cos ⁡ β ρ ' ( β ) − sin ⁡ β ρ ( β ) )
c = n 1 ( sin ⁡ β ρ ' ( β ) )
sin ⁡ β ρ ( β ) ≥ 0 ⇔ a + b ≤ 0
Point − spot type ≡ cos ⁡ β ρ ' ( β ) > 0    Ring − spot type ≡ cos ⁡ β ρ ' ( β ) < 0
Point − spot type ⇔ b > a    Ring − spot type ⇔ b < a
Point − spot type ⇛ − a ≥ ∣ b ∣ ≥ 0 , a ≠ 0    Ring − spot type ⇛ − b ≥ ∣ a ∣ ≥ 0 , b ≠ 0
p 2 * = ε ( a − b cos ⁡ ( 2 ϕ ) )
q 2 * = ε ( b sin ⁡ ( 2 ϕ ) )
p c * = aε    q c * = 0    α = ∣ b ∣ ε
Point − spot type ⇛ ∣ p c * ∣ ≥ α ≥ 0 , p c * ≠ 0    Ring − spot type ⇒ α ≥ ∣ p c * ∣ ≥ 0 , α ≠ 0
α = 0 ⇔ = sin ⁡ β ρ ( β ) = cos ⁡ β ρ ' ( β ) ⇔ ρ ( β ) = f sin ⁡ β
p c * = − ε f
p c * = 0 ⇔ sin ⁡ β ρ ( β ) = − cos ⁡ β ρ ' ( β ) ⇔ ρ ( β ) = f sin ⁡ β
α = ε f sin 2 β
p c = 0 ⇔ sin β ρ ( β ) = cos β ρ′ ( β ) ⇔ ρ ( β ) = f sin β
α = ε f
p 2 = p 2 ( x 1 , y 1 , z ; β , ϕ )
q 2 = q 2 ( x 1 , y 1 , z ; β , ϕ )
p 2 ( x 1 , y 1 , z ; β , ϕ; ) = p 2 x 0 ( β , ϕ; ) x 1 + p 2 y 0 ( β , ϕ ) y 1 + p 2 z 0 ( β , ϕ ) z + …
q 2 ( x 1 , y 1 , z ; β , ϕ ) = q 2 x 0 ( β , ϕ ) x 1 + q 2 y 0 ( β , ϕ ) y 1 + q 2 z 0 ( β , ϕ ) z + …
p 2 x 0 ( β , ϕ ) = [ ∂ p 2 ∂ x 1 ] 0 p 2 y 0 ( β , ϕ ) = [ ∂ p 2 ∂ y 1 ] 0 p 2 z 0 ( β , ϕ ) = [ ∂ p 2 ∂ z ] 0
q 2 x 0 ( β , ϕ ) = [ ∂ q 2 ∂ x 1 ] 0 q 2 y 0 ( β , ϕ ) = [ ∂ q 2 ∂ y 1 ] 0 q 2 z 0 ( β , ϕ ) = [ ∂ q 2 ∂ z ] 0
p 2 q 2 = M 2,0 ( β , ϕ ) ( x 1 y 1 z ) + … M 2,0 ( β , ϕ ) = ( p 2 x 0 ( β , ϕ ) p 2 y 0 ( β , ϕ ) p 2 z 0 ( β , ϕ ) q 2 x 0 ( β , ϕ ) q 2 y 0 ( β , ϕ ) q 2 z 0 ( β , ϕ ) )
p ' 2 = p 2 x ' 1 y ' 1 z ; β , ϕ + φ
q ' 2 = q 2 x ' 1 y ' 1 z ; β , ϕ + φ
( x ' 1 x ' 2 ) = R ( φ ) ( x 1 y 1 )    ( p ' 2 q ' 2 )    R ( φ ) = ( p 2 q 2 )    R ( φ ) = ( cos ⁡ φ sin ⁡ φ − sin ⁡ φ cos ⁡ φ )
( p 2 q 2 ) = R t ( φ ) M 2,0 ( β , ϕ + φ ) R E ( φ ) ( x 1 y 1 z ) + …
R E = ( cos ⁡ φ sin ⁡ φ 0 − sin ⁡ φ cos ⁡ φ 0 0 0 1 )
R t ( φ ) M 2,0 ( β , ϕ + φ ) R E ( φ ) ≡ M 2,0 ( β , ϕ )
M 2,0 β ϕ ≡ R ( ϕ ) M 2,0 β 0 R E t ( ϕ )
x 2 = ρ ( β ) cos ⁡ ( ϕ 0 )
y 2 = 0
y 1 p 1 x 1 y 1 z ; β , 0 − x 1 q 1 x 1 y 1 z ; β , 0 = − cos ⁡ ( ϕ 0 ) ρ ( β ) q 2 x 1 y 1 z ; β , 0
p 1 00 0 ; β , ϕ = n 1 sin ⁡ β cos ⁡ ϕ
q 1 00 0 ; β , ϕ = n 1 sin ⁡ β sin ⁡ ϕ
n 1 y 1 sin ⁡ β + … = − cos ⁡ ( ϕ 0 ) ρ ( β ) ( q 2 x 0 β 0 x 1 + q 2 y 0 β 0 y 1 + q 2 z 0 β 0 z ) + ⋯
q 2 x 0 ( β , 0 ) = q 2 z 0 ( β , 0 ) = 0 q 2 y 0 ( β , 0 ) = − cos ( ϕ 0 ) n 1 sin β ρ ( β )
x 1 = a 1 t p 1 = p 1 ( t , β )
y 1 = b 1 t    q 1 = q 1 ( t , β )
z 1 = c 1 t r 1 = r 1 ( t , β )
x 2 = ρ ( β ) cos ( ϕ 0 ) p 2 = p 2 ( t , β )
y 2 = 0 q 2 = q 2 ( t , β )
z 2 = z ap r 2 = 1 − p 2 2 ( t , β ) − q 2 2 ( t , β )
d x 1 d p 1 + d y 1 d q 1 + d z 1 d r 1 ≡ d x 1 d p 2 + d y 2 d q 2 + d z 2 d r 2
∂ x 1 ∂ t ∂ x 1 ∂ β ∂ p 1 ∂ t ∂ p 1 ∂ β + ∂ y 1 ∂ t ∂ y 1 ∂ β ∂ q 1 ∂ t ∂ q 1 ∂ β + ∂ z 1 ∂ t ∂ z 1 ∂ β ∂ r 1 ∂ t ∂ r 1 ∂ β = ∂ x 2 ∂ t ∂ x 2 ∂ β ∂ p 2 ∂ t ∂ p 2 ∂ β + ∂ y 2 ∂ t ∂ y 2 ∂ β ∂ q 2 ∂ t ∂ q 2 ∂ β + ∂ z 2 ∂ t ∂ z 2 ∂ β ∂ r 2 ∂ t ∂ r 2 ∂ β
a 1 0 ∂ p 1 ∂ t ∂ p 1 ∂ β + b 1 0 ∂ q 1 ∂ t ∂ q 1 ∂ β + c 1 0 ∂ r 1 ∂ t ∂ r 1 ∂ β = 0 ρ ' ( β ) cos ( ϕ 0 ) ∂ p 2 ∂ t ∂ p 2 ∂ β
∂ p 1 ∂ β a 1 + ∂ q 1 ∂ β b 1 + ∂ r 1 ∂ β c 1 = − ρ ' ( β ) cos ( ϕ 0 ) ∂ p 2 ∂ t
p 1 ( t , β ) = n 1 sin β + [ p 1 x 0 ( β , 0 ) a 1 + p 1 y 0 ( β , 0 ) b 1 + p 1 z 0 ( β , 0 ) c 1 ] t + …
q 1 t β = [ q 1 x 0 ( β , 0 ) a 1 + q 1 y 0 ( β , 0 ) b 1 + q 1 z 0 ( β , 0 ) c 1 ] t + …
r 1 ( t , β ) = n 1 cos β + [ r 1 x 0 ( β , 0 ) a 1 + r 1 y 0 ( β , 0 ) b 1 + r 1 z 0 ( β , 0 ) c 1 ] t + …
p 2 t β = [ p 2 x 0 ( β , 0 ) a 1 + p 2 y 0 ( β , 0 ) b 1 + p 2 z 0 ( β , 0 ) c 1 ] t + …
( n 1 cos ⁡ β ) a 1 + ( − n 1 sin ⁡ β ) c 1 + . . . = ρ ′ ( β ) cos ⁡ ( ϕ 0 ) [ p 2 x 0 β 0 a 1 + p 2 y 0 β 0 b 1 + p 2 z 0 β 0 c 1 ] + . . .
[ n 1 cos ⁡ β + ρ ´ ( β ) cos ⁡ ( ϕ 0 ) p 2 x 0 ( β , 0 ) ] a 1 + [ ρ ´ ( β ) cos ( ϕ 0 ) ⁡ P 2 y 0 ( β , 0 ) ] b 1 + [ − n 1 sin ⁡ β + ρ ´ ( β ) cos ⁡ ( ϕ 0 ) P 2 z 0 ( β , 0 ) ] c 1 = 0
p 2 x 0 β 0 = - n 1 cos ⁡ β ρ ′ ( β ) cos ⁡ ( ϕ 0 )
p 2 y 0 β 0 = 0
p 2 z 0 β 0 = n 1 sin ⁡ β ρ ′ ( β ) cos ⁡ ( ϕ 0 )
M 2,0 β ϕ = R ( ϕ ) ( − n 1 cos ⁡ β ρ ′ ( β ) cos ( ϕ 0 ) 0 n 1 sin ⁡ β ρ ′ ( β ) cos ⁡ ( ϕ 0 ) 0 − n 1 sin ⁡ β ρ ( β ) cos ⁡ ( ϕ 0 ) 0 ) R E t ( ϕ )
n ap ρ z = n 2 ρ z − h 2 ρ 2
v mer 2 = g 2 + r 2 = n 2 ρ z − h 2 ρ 2 = n ap 2 ρ z
v i = ( v i ⋅ u ϕ ) u ϕ + v i , mer v d = ( v d ⋅ u ϕ ) u ϕ + v d , mer
v d ⋅ u ϕ = v i ⋅ u ϕ
v d , mer = F 2 D ( v i , mer , N ; n micro 2 − ( h 2 ρ 2 ) )
n micro 2 − ( h 2 ρ 2 ) = n micro − 1 2 n micro ( h 2 ρ 2 ) + . . .
v d , mer ≈ F 2 D ( v i , mer , N )
v d = F 3 D ( v i , N ) ≈ ( v i ⋅ u ϕ ) u ϕ + F 2 D ( v i − ( v i ⋅ u ϕ ) u ϕ , N )
d E i = dρd g i + dzd r i + dθd h i ≡ d l mer ⋅ d v i , mer + dθd h i
d E d = dρd g d + dzd r d + dθd h d ≡ d l mer ⋅ d v d , mer + dθd h d
d ρ d g i + dzd r i + dθdh ≡ ± ( d ρd g d + dzd r d ) + dθdh
d ρ i d g i + d z i d r i + d θ i dh ≡ ( − 1 ) m ( d ρ d d g d + d z d d r d ) + d θ d dh
d x 1 d p 1 + d z 1 d r 1 + d y 1 d q 1 ≡ − ( d x 2 d p 2 + d z 2 d r 2 ) + d y 2 d q 2
∂ p 1 ∂ β a 1 + ∂ q 1 ∂ β b 1 + ∂ r 1 ∂ β c 1 = − η ˈ ( β ) cos ⁡ ( ϕ 0 ) ∂ p 2 ∂ t

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