A-Omega calculation on the slit micro-lens

Daigo Tomono

July 26, 2002

Abstract:

A$\Omega$ change by the micro-lens to be integrated into the spectrograph entrance slit is considered. From paraxial calculations, the effective A$\Omega$ depends upon distance $s$ between the fiber exit and the micro-lens. A$\Omega$ change is minimum with $s=r_{slit}/\theta_{core}$ when $r_{slit}$ is radius of the imaged slit and $\theta_{core}$ is the light cone angle of the light from the fiber measured in the air. In the case, the effective A$\Omega$ becomes $(r_{core}/r_{slit}+1)^2$ times larger.

For the smallest A$\Omega$, focal length of the micro lens is $f = w_{slit} / (2 \arcsin N\!A)$ with numerical aperture of light $N\!A$ (with $N\!A = \sin \theta_{core}$) and pseudo slit width $w_{slit}$ (with $w_{slit} = 2r_{slit}$). If the lens is directly attached to the fiber exit, its length should be $L_{lens} = w_{slit} / (2 n_{lens} \arcsin N\!A)$ and surface curvature radius be $c_{lens} = w_{slit}\times(n_{lens} - 1) / (2 \arcsin N\!A)$. Moreover, lens radius is $r_{lens} = r_{core} + f \times \arcsin N\!A$ and focal ratio is $F = 1/[2\arcsin N\!A \times (D_{core}/w_{slit} - 1)]$.

Postscript version of this report is also available: slitlens.ps.gz.

Introduction

On considerations of spectrometer design¹, it was made clear that we can not afford for A$\Omega$ enlarged by the sparse core area surrounded by the cladding layers in the pseudo slit. Table 1 shows an example of the K-band spectrometer. The design assumes 50 $\mu$m diameter fiber cores and 100 $\mu$m interval to align the fibers, and the 100 $\mu$m interval is imaged on 2 pixels of the FPA. As this is a paraxial calculation, lens diameter of the camera optics is practically larger than 290 mm, which is too big to be really constructed.

A micro-lens array on the spectrometer entrance slit is considered here to minmize the A$\Omega$ required for the spectrometer.

Table 1: An example of K-band spectrometer design

fiber FOV	0.2 arcsec
N.A. of light	two times bigger to allow FRD
fiber core diameter	50 $\mu$m
fiber cladding diameter	100 $\mu$m
resolution power	4000
grating diffraction order	2
groove interval	12.6 $\mu$m
central wavelength	2.2 $\mu$m
width of a spectral element on FPA	37 $\mu$m
incident angle $\alpha$	-4.6 deg
emergent angle $\beta$	25.4 deg
collimator focal length	1035 mm
camera focal length	382 mm
incident beam width	321 mm
grating width	322 mm
emergent beam width	290 mm

Basic assumptions

A micro-lens array is assumed to be attached directly onto the fiber exits. The fiber has a core radius of $r_{core}$ in $\mu$m, with the scrambled telescope pupil is imaged on the exit. The micro-lens images the scrambled telescope focal plane onto an image plane of radius $r_{slit}$ in $\mu$m, which is the assumed cladding diameter. The paraxial micro-lens is assumed to be placed $s$ (in $\mu$m) from the fiber exit, making the image plane $t$ (in $\mu$m) from the lens. Here, effective focal length of the micro-lens $f$ is the same as $t$.

A$\Omega$ of a ray

Figure 1 shows the assumed geometry of the fiber core and the micro-lens. Beam from the core has a light cone angle of $\theta_{core}$ in the air. Output light cone angle is calculated as follows for small $\theta$.

As a collimated beam is focused onto a single point on the image plane, $r_{slit}$ can be calculated as

$$r_{slit} = t \theta_{core}.$$ (1)

The output light cone angle $\theta_{slit}$ is calculated tracing light from upper most point and lower most point on the fiber core going in to the lens in the same direction as the optical axis:

$$\theta_{slit} = \frac{r_{core}}{t}$$ (2)

Therefore, A$\Omega$ at the slit for a collimated beam is

$$\mbox{A}\Omega = r_{slit}^2\theta_{slit}^2\pi^2$$ (3)

$$= r_{core}^2\theta_{core}^2\pi^2.$$ (4)

A$\Omega$ of a ray on the fiber eixt is the same as on the pseudo slit.

Figure 1: Geometry of the fiber exit and the micro-lens

Effective A$\Omega$

We have to expand our consideration light from anywhere on the fiber core. Figure 2 shows that the beam direction $\theta_{center}$ of the light depends upon the direction of light from the fiber core.

A ray from the center with maximum divergence from the optical axis goes through the lens at

$$x = s\theta_{core}$$ (5)

and reaches the image plane at

$$r_{slit} = x + t \theta_{center}.$$ (6)

Therefore,

$$\theta_{center} = \theta_{core} \left(1-\frac{s\theta_{core}}{r_{slit}}\right).$$ (7)

Beam divergence $\theta_{max}$ from the image plane is

$$\theta_{max} = \max\left\vert \theta_{center} \pm \theta_{slit}\right\vert$$ (8)

$$= \max\left\vert \theta_{core} \left(1-\frac{s\theta_{core}\mp r_{core}}{r_{slit}}\right) \right\vert$$ (9)

$$= \left\{ \begin{array}{ll} \theta_{core}\times(r_{core}/r_{slit}+1... ...core}/r_{slit}) & \mbox{for } s > r_{slit}/\theta_{core} \\ \end{array}\right.$$ (10)

Figure 2: Geometry of the fiber exit and the micro-lens

Lens radius

When packing the fibers and micro-lens assemblies into the slit, the micro-lens radius $r_{lens}$ determines the slit length when it is bigger than the cladding size and the image size $r_{slit}$. From geometry,

$$r_{lens} = r_{core} + s\theta_{core}$$ (11)

and effective radius of the optics $r_{eff}$ is defined as

$$r_{eff} = \max(r_{lens}, r_{slit})$$ (12)

$$= \left\{\begin{array}{ll} r_{core} \times \left( 1 + s\theta_{core... ... & \mbox{for } s < (r_{slit} - r_{core}) / \theta_{core} \\ \end{array}\right.$$ (13)

Effective A$\Omega$

Effective A$\Omega$ of the light from the slit imaged by the micro-lens is calculated from $r_{eff}$ and $\theta_{max}$. Change factor $k$ of A$\Omega$ at the fiber output (A$\Omega_{core}$) and at the slit (A$\Omega_{slit}$) is defined and calculated as follows:

$$k \equiv \sqrt{\frac{\mbox{A}\Omega_{slit}}{\mbox{A}\Omega_{core}}}$$ (14)

$$= \frac{r_{eff}\theta_{max}}{r_{core}\theta_{core}}$$ (15)

$$= \left\{\begin{array}{ll} 1+r_{slit}/r_{core}-s\theta_{core}/r_{co... ...r_{core}/r_{slit}-1+s\theta_{core}/r_{slit}) & (s_2 < s) \\ \end{array}\right.$$ (16)

for

$$s_1 = \frac{r_{slit} - r_{core}}{\theta_{core}}$$ (17)

$$s_2 = \frac{r_{slit}}{\theta_{core}}$$ (18)

Figure 3 shows the three functions. $k$ has two minima at $s$ = $s_1$ and $s_2$ with the values of

$$k(s_1) = 2$$ (19)

and

$$k(s_2) = \frac{r_{core}}{r_{slit}} + 1$$ (20)

Because $r_{core}$ is smaller than $r_{slit}$ for our purpose, minimum value of $k$ is at $s_2$. Therefore, our micro-lens should be designed so that $s=r_{slit}/\theta_{core}$.

Figure 3: Effective A$\Omega$ calculated for $\theta _{max} = 0.1$, $r_{core}$ = 25 $\mu$m, and $r_{slit}$ = 50 $\mu$m.

Practical considerations about effective A$\Omega$

Practically we will have a large number of fibers on the entrance slit. In the case, change of effective A$\Omega$ also depends upon the number of fibers we have to handle.

When $N$ fibers with core radius $r_{core}$ and emergent light cone angle of $\theta_{core}$ are considered, simple sum of A$\Omega$ of the light is

$$\mbox{A}\Omega_{orig} = N \times \pi r_{core}^2 \times \pi \theta_{core}^2.$$ (21)

First the fibers with cladding radius $r_{clad}$ are considered to be simply lined up on the slit. In the case, we have to extend the acceptance field of view of the spectrograph. With an on-axis system, the field of view have to be a circular with a radius of $N\times r_{clad}$. In the case effective A$\Omega$ of the light that the spectrograph has to accept is

$$\mbox{A}\Omega_{f,c} = \pi (N r_{clad})^2 \times \pi \theta_{core}^2.$$ (22)

When an off-axis system is designed, the smallest field of view can be a rectangle of $N\times r_{clad}$ by $r_{core}$. In the case effective A$\Omega$ is

$$\mbox{A}\Omega_{f,r} = \pi (N r_{clad} r_{core}) \times \pi \theta_{core}^2.$$ (23)

With a micro lens array with $s=r_{slit}/\theta_{core}$, effective A$\Omega$ of the circular field of view becomes

$$\mbox{A}\Omega_{l,c} = \pi (N r_{eff})^2 \times \pi \theta_{max}^2$$ (24)

$$= \pi N^2 (r_{core}+r_{slit})^2 \times \pi \theta_{core}^2 \left( \frac{r_{core}}{r_{slit}} \right)^2.$$ (25)

For the rectangular field case,

$$\mbox{A}\Omega_{l,r} = \pi (2 N r_{eff} \times 2 r_{slit}) \times \pi \theta_{max}^2$$ (26)

$$= 4 N r_{core} \left( 1 + \frac{r_{slit}}{r_{core}} \right) r_{slit} \times \pi \theta_{core}^2 \left(\frac{r_{core}}{r_{slit}}\right)^2$$ (27)

Dividing with A$\Omega_{orig}$ changes of effective A$\Omega$ are calculated as in Table 2.

Table 2: Change of A$\Omega$ at the spectrograph entrance slit. $N$ of fibers with core radius of $r_{core}$ and cladding radius of $r_{clad}$ are assumed. For micro lens array a design with $s=r_{slit}/\theta_{core}$ is adopted with radius (half width) of the imaged slit of $r_{slit}$.

	circular FOV	rectangular FOV
Simple fibers	$$N \left(\frac{r_{clad}}{r_{core}}\right)^2$$	$$\frac{4}{\pi} \frac{r_{clad}}{r_{core}}$$
Micro lens	$$N \left(1+\frac{r_{core}}{r_{slit}}\right)^2$$	$$\frac{4}{\pi} \left(1+\frac{r_{core}}{r_{slit}}\right)$$

It is better to integrate a micro lens array on the spectrograph entrance slit if the pesudo slit can be wide enough:

$$\frac{r_{slit}}{r_{core}} > \frac{r_{core}}{r_{clad}-r_{core}}.$$ (28)

With typical values of $r_{core} = 25 \mu\mbox{m}$ and $r_{clad} = 50 \mu\mbox{m}$, $r_{slit}$ should be $> 25 \mu\mbox{m}$ if we are to integrate a micro lens array.

Lens design with the smallest effective A$\Omega$

From Equation 20, the spectrometer get smallest effective A$\Omega$ ( $r_{core}/r_{slit} + 1$ times bigger than the input light) when $s=r_{slit}/\theta_{core}$. From Equation 1, $t = r_{slit}/\theta_{core}$. Therefore, to minimize effective A$\Omega$, $s = t$ in Figures 1 and 2.

From geometry, focal length of the micro lens $f$ is the same as $t$, and defined by

$$f_{lens} = \frac{w_{slit}}{2 \theta_{core}}$$ (29)

$$= \frac{w_{slit}}{2 \arcsin N\!A}$$ (30)

where $\theta_{core}$ is the light cone angle of light in the air with numerical aperture $N\!A$ we have to accept in the slit width $w_{slit} = 2 \times r_{slit}$.

We are thinking of micro lens directly attached to the fiber exit. In the case, lens length $L_{lens}$ has to be made longer from $s$ according to its refractive index $n_{lens}$. Therefore,

$$L_{lens} = s \times n_{lens}$$ (31)

$$= \frac{w_{slit}\times n_{lens}}{2 \arcsin N\!A}$$ (32)

and from the geometry, the lens surface curvature radius becomes

$$c_{lens} = f (n_{lens} - 1)$$ (33)

$$= \frac{w_{slit}\times(n_{lens} - 1)}{2 \arcsin N\!A}$$ (34)

Moreover, from Equation 11, lens radius becomes

$$r_{lens} = r_{core} + f_{lens} \theta_{core}$$ (35)

$$= r_{core} + f_{lens} \times \arcsin N\!A$$ (36)

and focal ratio of the lens becomes

$$F = \frac{f_{lens}}{2\times r_{lens}}$$ (37)

$$= \frac{w_{slit}}{2 \arcsin N\!A \times (w_{slit}+D_{core})}$$ (38)

with core diameter of $D_{core} = 2\times r_{core}$.

Design example

From the equations above, an example of a slit micro-lens is designed. Parameters are shown in Table 3. Actual calculations are done in the directory gin-an:tomono:~/Design02/0508_MicroLens/ with a ruby script shown in Table 4. A zemax simulation for the design is shown in Figure 4.

Table 3: An example of K-band spectrometer design

Input micro-lens
Pupil image diameter $D_{pupil}$	50/1.4 $\mu$m
Fiber FOV $\theta_{FOV}$	0.2 arcsec
Telescope diameter $D_{tel}$	8 m
Lens diameter $D_{inlens}$	300 $\mu$m
Refractive index	1.44
Lens length	1988.98 $\mu$m
Curvature radius	607.74 $\mu$m
Focal ratio	9.55
Light cone angle	0.1086 rad
Output micro-lens
FRD parameter (output/input cone angle)	1.4
Pseudo slit width	200 $\mu$m
Fiber core diameter	50 $\mu$m
Refractive index	1.44
Lens length	947.13 $\mu$m
Curvature radius	289.40 $\mu$m
Lens diameter	250 $\mu$m
Focal ratio	2.63
Light cone angle	0.1520 rad

Figure 4: ZEMAX simulation of the slit micro-lens shown in Table 3

Table 4: A ruby script to calculate the micro-lens geometries

Revision history

2002.2.13.

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• First release

2002.5.8.

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• Error in Equation 13 corrected
• Section 8 added

2002.5.14.

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• Section 8 revised
• Section 8.1 added

2002.7.25.

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• Section 7 added

About this document ...

A-Omega calculation on the slit micro-lens

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Footnotes

... design¹

e.g. tomono@gin-an:/Design01/011229_K_Spec