Interferometry on a blazed grating ¹

Daigo Tomono

February 4, 2002

Abstract:

Interferometric calculations are done on a flat blazed grating. It is found that to maximize the efficiency,

• Order of diffraction $m$ must be as small as possible.
• Incident angle $\alpha$ and emergent angle $\beta$ must be as close as possible.
• Groove interval $d$ must be as big as possible.

to avoid shadowing. These parameters will be decided from the physical limitation of production of the grating and assembly of the spectrometer.

Secondary peaks beyond the first local minima has 4.7% peak intensity of the primary peak. This is not considered in efficiency calculation.

For the $K$-band, calculated efficiency is more than about 96% throughout the band when $\alpha = \beta$, $m = \pm1$, and $d \sim 22\mu m$. With larger $d$, efficiency increases. When $\alpha$ and $\beta$ are separated, efficiency drops to about 95% for 20 deg difference.

When the slit width is wider than a certain limit, namely, spatial resolution of the instrument is less than the diffraction limit of the telescope, spectral resolution of the spectrometer is limited by the geometrical size of the slit. When the field of view (FOV) of a spatial element is $n^2$ of diffraction limited spatial elements, we need about $n$ times more grating grooves to achieve the same spectral resolving power for a diffraction limited instrument.

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

Interferometric calculation

Interferometry is calculated on a grating with groove interval $d$ and groove angle $\varepsilon$ as in Figure 1.

Figure 1: Geometry of light and blazed grating

Geometry

Incident light from $P_1(x_1,y_1,0)$ with signed angle $\alpha$ comes from distance $r'_1$. Consider detecting emergent light with signed angle $\beta$ at $P_2(x_2,y_2,0)$ distance $r'_2$. Here, light reflected at the point $P(x,y,0)$ is considered. Throughout this report, refractive index of media around the grating is assumed to be 1.

Distance between $P_1$ and $P$ is $r^2 _1 = (x_1 - x)^2 + (y_1 - y)^2$ and distance between $P_1$ and the coordinate origin is $r'^2 _1 = x^2 + y^2$. So, $r_1$ can be approximated as $r_1 \sim r'_1 - (x_1 x - y_1 y)\div r'_1$. Likewise $r_2 \sim r'_2 - (x_2 x - y_2 y)\div r'_2$. Using the incident and emergent angles, $x_1 = r'_1 \sin \alpha$ and $y_1 = r'_1 \sin \beta$. Therefore, light path going through $P$ can be explained as $r_1 \sim r'_1 - (x \sin \alpha + y \cos \alpha)$ and $r_2 \sim r'_2 - (x \sin \beta + y \cos \beta)$.

On the $n$th groove of total $N$ grooves, $x = (n - \frac{N}{2})d - \delta$ and $y = \delta \tan \varepsilon$. Therefore, phase delay $\phi = k\times (r_1 + r_2)$ at $P_2$ of the light from $P_1$ and reflected at $P$ is

$${\frac{\phi(\delta, n)}{k}}$$

$$\sim r'_1 + r'_2 - [\{(n-\frac{N}{2})d - \delta\}(\sin \alpha + \sin \beta) + \delta \tan \varepsilon (\cos \alpha + \cos \beta)]$$ (1)

$$= r'_1 + r'_2 + \frac{N}{2}d(\sin \alpha + \sin \beta)$$

$$- nd(\sin \alpha + \sin \beta) + \delta \frac{1}{\cos \varepsilon} (\sin[\alpha - \varepsilon] + \sin[\beta - \varepsilon])$$ (2)

where $k = 2\pi / \lambda$. Here, we define

$$A = \sin \alpha + \sin \beta$$ (3)

and

$$E= \frac{1}{\cos \varepsilon} (\sin [\alpha - \varepsilon] + \sin [\beta - \varepsilon])$$ (4)

Integration

Using the equations, complex amplitude of the light detected at $P_2$ is calculated from integration over the grating surface. Here we take the shadowing effect into account. As shown in Figure 2, only $d'$ of the groove surface reflects the light.

$$U(P_2) \propto \sum_{n=0}^{N}\int_{0}^{d'} \exp[i\phi(\delta, n)] d\delta$$ (5)

$$\propto \sum_{n=0}^{N} \exp[-ikndA] \times \int_{0}^{d'} \exp[ik\delta E] d\delta$$ (6)

$$\propto \frac{1-\exp[-ikNdA]}{1-\exp[-ikdA]} \times \frac{1}{ikE}(\exp[ikd'E] - 1)$$ (7)

Figure 2: Shadowing of the incident or emergent light on different grating shapes.

(a)	(b)

Normalizing at $A=0$ and $E=0$, intensity of the light detected at $P_2$ is

$$I(P_2) = \frac{1}{N^2}\frac{1-\cos[kNdA]}{1-\cos[kdA]} \times \frac{2}{(kd'E)^2} (1-\cos[kd'E])$$ (8)

Grating function

The first term of Equation 8, which we call grating function is from the periodical sum over the grooves of the grating which has peaks at $kdA = 2\pi m$ so

$$m\lambda = d(\sin\alpha + \sin\beta)$$ (9)

When the incident angle $\alpha$ and diffraction order $m$ is known, emergent angle $\beta$ can be calculated as

$$\beta = \arcsin[\frac{m\lambda}{d} - \sin\alpha]$$ (10)

where $m$ is an integer of the diffraction order.

On the other hand, when we define difference of incident and emergent angles,

$$\Delta \equiv \alpha - \beta$$ (11)

$\alpha$ and $\beta$ is calculated as follows with fixed $m$:

$$\alpha \mbox{,} \beta = \arcsin\left[ \frac{ \frac{m\lamb... ...os\Delta)-(\frac{m\lambda}{d})^2}} {2(1+\cos\Delta)} \right]$$ (12)

Derivation is described in Appendix A

Slit function

The second term of Equation 8, which is called slit function, is the diffraction from the grooves. This peaks at $E=0$ or

$$\alpha + \beta = 2 \varepsilon$$ (13)

Distance from the peak to the first minima is almost at the same interval that the first term has for the peaks on $A$.

Spectrometer with a narrow entrance slit

When the entrance slit is narrow enough, Equation 8 can be used to asses properties of the output spectra. As shown below, the slit can be treated narrow and $I(P_2)$ can be treated as instrumental profile when difference of incident light direction (half width) $\delta\alpha$ is smaller than $\sim \lambda / (2Nd\cos[\alpha_0])$ where $\alpha_0$ is the center direction of the incident light. In Figure 3, left-most curve shows normalized profile of $I(P_2)$ at $kdA\sim 0$ and $kd'E\sim 0$. Horizontal axis is change of $kNd\sin\beta$ from the center of the spectra.

Spectral resolution

For $N\gg 1$, spectral resolution of the grating is calculated as follows. First, we have a peak at $A_1 = \lambda_1 m / d$ for a wavelength $\lambda_1$. Near the peak, there are local minima at $A_2 = A_1 \pm \lambda_1 / (Nd)$. We define spectral resolution when the next spectral element of $\lambda_2$ is at the local minima $A_2$. Namely, for the $m$th order, we have a peak for $\lambda_2$ at $A_2 = \lambda_2 m / d$. Thus, the wavelength difference $\Delta\lambda = \vert\lambda_2 - \lambda_1\vert$ or

$$\Delta\lambda = \lambda / (mN)$$ (14)

or

$$R \equiv \frac{\lambda}{\Delta\lambda} = mN$$ (15)

Secondary peaks

Beyond the local minima, there are second local maxima near $A'$ that makes $\cos[kdNA'] = -1$ beyond the first local minima. For the peak of $m$th order diffraction of $\lambda$, $A' = \lambda/d \times (m + 3/N)$. The position $A'$ corresponds to the peaks for $\lambda' = \lambda + 3 \Delta\lambda$.

Near the secondary peaks, from $kdA = 2\pi m$, grating function is

$$I \sim \frac{1}{N^2} \frac{1-\cos[kdNA']}{1-\cos[kdA']}$$ (16)

$$= \frac{1}{N^2} \frac{2}{1-\cos[3\pi/N]}$$ (17)

$$\sim \frac{1}{N^2} \frac{2}{1/2(3\pi/N)^2}$$ (18)

$$\sim 4.5 \%$$ (19)

assuming $N \gg 3\pi$.

Exact positions of secondary peaks can be obtained by taking differential of the grating function. Position and value of the peaks weakly depends on $N$. The secondary peaks are nearer to the primary peak and has intensity about 4.7 % of the primary peak when $N$ is more than about 1000.

Spectrometer with a wide entrance slit

Geometrical limit of spectral profile

When the entrance slit is wide, Equation 8 has to be integrated over the entrance slit, assuming that the light coming to different locations on the entrance slit is not coherent. This results in spectral resolution not being geometrically limited rather than being diffraction limited as for a narrow entrance slit.

When the direction of the collimated light spreads from $\alpha_1 \equiv \alpha_0-\delta\alpha$ to $\alpha_2 \equiv \alpha_0+\delta\alpha$, Equation 8 has to be integrated from $\alpha_1$ to $\alpha_2$ and profile of the spectra can be assessed.

$$J(P_2) \equiv \int_{\alpha_1}^{\alpha_2} I(P_2) \mbox{ d}\alpha$$ (20)

$$= \int_{A_1}^{A_2} I(P_2) \cos\alpha \mbox{ d}A$$ (21)

$$= \int_{A_1}^{A_2} \frac{1}{N^2}\frac{1-\cos[kNdA]}{1-\cos[kdA]} \frac{2}{(kd'E)^2} (1-\cos[kd'E]) \cos\alpha \mbox{ d}A$$ (22)

with $A_{1,2} \equiv \sin\alpha_{1,2} + \sin\beta$. We can assume $\delta\alpha\ll 1$ (slit width is small, if not narrow) and $kd'E\ll 1$ (the spectrometer is used around the Bragg's condition). In the case, $\cos\alpha \sim \cos\alpha_0$ and $1-\cos[kd'E]\sim(kd'E)^2/2$. When the grating is used in the $m$th order, $A$ can be replaced with $A' \equiv A - 2\pi m/(kd)$. Near the $\alpha$ and $\beta$ of spectra we are observing, $A'\ll 1$. Therefore, $1-\cos[kdA] = 1-\cos[kdA'] \sim (kdA)^2/2$. Here, we define $A'_{1,2} = A_{1,2} - 2\pi m/(kd) \sim \sin \alpha_0 + \sin\beta - 2\pi m/kd \pm \cos[\alpha_0] \delta\alpha$ and the integration becomes

$$J(P_2) \sim \cos\alpha_0 \int_{A'_1}^{A'_2} \frac{2(1-\cos[kNdA'])}{(kNdA')^2} \mbox{ d}A'$$ (23)

$$= \frac{\cos\alpha_0}{kNd} \int_{x_0-\delta x}^{x_0+\delta x} \frac{2(1-\cos x)}{x^2} \mbox{ d}x$$ (24)

$$\propto \int_{x_0-\delta x}^{x_0+\delta x} \frac{1-\cos x}{x^2} \mbox{ d}x$$ (25)

with $x_0\equiv kNd\times(\sin\alpha_0 + \sin\beta - 2\pi m/kd)$ and $\delta x\equiv kNd\times\cos[\alpha_0] \delta\alpha$. The equation cannot be integrated analytically.

Figure 3 shows results from numerical integration of Equation 26 with different $\delta x$. For $\delta x$ smaller than $\sim \pi$, width of the spectra is comparable with that for $\delta x = 0$. On the other hand, for $\delta x > \pi$, the profile gets half of the center at $x_0 \sim \delta x$.

Figure 3: Numerical integration of Equation 26. From left to right, curves show results for $\delta x$ of 0, 2, 4, 6, 8, 10, and 12. Results are normalized at $x_0 = 0$.

From the definition, $\delta x \sim \pi$ corresponds to a threshold angular half width of the slit

$$\delta\alpha_{th} \sim \pi/(kNd\cos[\alpha_0]) = \lambda/(2Nd\cos[\alpha_0]).$$ (26)

If the half width of the direction of the incident light onto the grating is more than $\delta\alpha_{th}$, spectral resolution of the spectrometer is not limited by diffraction of the grating, but limited by geometrical width of the slit.

Relation with system $A\Omega$

When the instrument is spatially diffraction limited at $\lambda_0$, field of view (FOV) of a spatial element on a telescope with entrance pupil diameter of $D_{tel}$ is in the order of $\lambda_0/D_{tel}$. Thus, the system $A\Omega$ becomes $\sim \lambda_0^2\pi^2/16$. On a seeing limited instrument, FOV of a spatial element is more than for the diffraction limited instrument. Here, we assume to observe FOV $n^2$ times larger than that for a diffraction limited instrument. In the case, system $A\Omega$ becomes $\sim n^2\lambda_0^2\pi^2/16$.

When the diameter of the grating is $D_{gr}$, beam width of incident light becomes $D_{gr}\cos[\alpha_0]$. Considering only about the beam in the dispersion direction, solid angle of the light going into the grating is $\Omega_{gr} \sim n^2\lambda_0^2\pi/(4 D_{gr}^2\cos[\alpha_0])$. Therefore, from $D_{gr} = Nd$, half width of the incident beam direction is

$$\delta\alpha \sim n\lambda_0/(2 D_{gr}\cos[\alpha_0]) \sim n \delta\alpha_{th}.$$ (27)

For a seeing limited instrument, spectral resolution is limited by the geometrical image size of the entrance slit, rather than the diffraction of the grating.

Spectral resolution

When the spectral resolution of the spectrometer is geometrically limited, spectral resolving power is calculated as follows. Half width $\delta\beta$ of the slit image for a certain wavelength in the emergent light from the grating is

$$\delta\beta = \left\vert \frac{\mbox{ d}\beta}{\mbox{ d}\alpha}(A=const.) \right\vert \times \delta\alpha$$ (28)

$$\sim \frac{\cos[\alpha_0]}{\cos\beta} \times \delta\alpha$$ (29)

Therefore, from Equation 9, wavelength difference becomes

$$\Delta\lambda = \frac{d\cos\beta}{m} \times 2\delta\beta$$ (30)

$$\sim \frac{2d\cos[\alpha_0] \delta\alpha}{m}$$ (31)

where $\Delta\lambda$ is the full width. Therefore spectral resolving power becomes

$$R \equiv \frac{\lambda}{\Delta\lambda}$$ (32)

$$\sim \frac{m\lambda}{2d\cos[\alpha_0] \delta\alpha}$$ (33)

$$\sim \frac{mD_{gr}}{nd} \times \frac{\lambda}{\lambda_0}$$ (34)

$$\sim \frac{mN}{n} \times \frac{\lambda}{\lambda_0}$$ (35)

Comparing this with Equation 15, resolving power of the spectrometer is degraded by a factor of $1/n\cos[\alpha_0]$ for $\lambda \sim \lambda_0$. As a consequence, we need $n\cos[\alpha_0]$ times more grooves for a seeing limited spectrometer to achieve the same spectral resolving power as for a diffraction limited spectrometer.

$A\Omega$ consideration

Spectral resolution of the spectrometer is considered from conservation of $A\Omega$. With telescope aperture diameter $D_{tel}$, width of the slit on the sky $\theta_{tel}$, square root of system $A\Omega$ is $D_{tel}\times\theta_{tel}$.

Grating width defined from system $A\Omega$ and camera focal length

On the focal plane array (FPA), image of the slit has to be $D_{img}$. Thus, with width of the light direction conversing to the FPA $\theta_{img}$, focal ratio of the spectrometer camera is

$$F_{cam} \equiv \frac{1}{2 \tan(\theta_{img}/2)}$$ (36)

$$\sim \frac{1}{\theta_{img}}$$ (37)

$$= \frac{D_{img}}{D_{tel} \theta_{tel}}$$ (38)

Therefore, with focal length of spectrometer camera $f_{cam}$ diameter of beam emerging from the grating is

$$D_{out} = \frac{f_{cam}}{F_{cam}} = \frac{D_{tel} \theta_{tel} f_{cam}}{D_{img}}$$ (39)

and grating width is

$$D_{gr} = \frac{D_{out}}{\cos\beta} = \frac{D_{tel} \theta_{tel} f_{cam}}{D_{img} \cos\beta}$$ (40)

Camera focal length defined from spectral resolution

Size of the image of the slit on the FPA is adjuste to be the same as the size of a spectral element dispersed by the grating. From Equation 9, width of the beam direction of a spectral element $\Delta\lambda$ is

$$\delta\beta = \frac{m \Delta\lambda}{d\cos\beta}$$ (41)

Therefore,

$$D_{img} = f_{cam}\times\delta\beta = \frac{f_{cam} m \Delta\lambda}{d\cos\beta}$$ (42)

Putting Equation 43 into Equation 41,

$$D_{gr} = \frac{D_{tel} \theta_{tel} d}{m \Delta\lambda}$$ (43)

and number of grating grooves illuminated by the beam is

$$N = \frac{D_{gr}}{d} = \frac{D_{tel} \theta_{tel}}{m \Delta\lambda}$$ (44)

Comparison with spectral and spatial resolution

With spectral resolution $R \equiv \lambda/\Delta\lambda$, number of grooves becomes

$$N = \frac{D_{tel} \theta_{tel} R}{m \lambda}$$ (45)

Moreover, if slit FOV $\theta_{tel}$ is $n$ times that of the diffraction limit of the telescope $\sim \lambda/D_{tel}$,

$$N = \frac{n R}{m}$$ (46)

Design of a seeing limited spectrograph

Sometimes, we need to adjust dispersion so that a spectral element is dispersed onto a disired size $\Delta w$, with the image size $\Delta i$ of the slit being different. For this kind of requirements, design steps are considered here.

Requirements

First we define required spectral resolving power $R_{req} \equiv \lambda/\Delta/\lambda$ with $\Delta\lambda$ being full width of wavelenght for a spectral element. Moreover, groove interval $d$ of the grating and diffraction order $m$, difference of incident angle and emergent angle $\Delta$ have to be defined. Also, pixel size of focal plane array and required spatial resolution defines spectral element $\Delta w$ and spatial magnification of the spectrometer $M$.

Incidecnt angle $\alpha_0$ and emergent angle $\beta$ is calculated by Equation 12. Because overall size of the spectrometer is proportional to $\cos\beta$, the solution with larger $\vert\beta\vert$ is better than with larger $\vert\alpha_0\vert$.

Focal lengths

From Equation 9, wavelength dependency of dispersing direction is calculated as

$$\Delta \beta = \frac{m}{d\cos\beta} \Delta\lambda$$ (47)

With camera focal length $f_{cam}$, $\Delta w = f_{cam}\times\Delta\beta$. Therefore, camera focal length has to be

$$f_{cam} = \frac{Rd\cos\beta}{m\lambda} \times \Delta w.$$ (48)

From spatial magnification, focal length of the collimator becomes

$$f_{col} = \frac{1}{M} \times f_{cam}.$$ (49)

Naturally, width of the slit image is

$$\Delta i = \frac{\cos[\alpha_0]}{\cos\beta} M \Delta s$$ (50)

which can be calculated from Equation 9 with $\lambda$ fixed.

Width of the grating and the beams

From focal length of the collimator, width of the grating is calculated as follows. When the FOV is $n^2$ times larger than that for a diffraction limited instrument, system $A\Omega$ becomes $\sim n^2\lambda_0^2\pi^2/16$. With the slit width $\Delta s$, light disperses with half angle of $n\lambda_0/(2\Delta s)$. Thus, beam width of the incident light would be

$$D_{in} = \frac{n\lambda_0 f_{col}}{\Delta s}$$ (51)

$$= R n \times \frac{d\cos\beta}{m} \times \frac{\lambda_0}{\lambda} \times \frac{\Delta w}{M\Delta s}$$ (52)

From geometry, grating width is

$$D_{gr} = \frac{n\lambda_0 f_{col}}{\Delta s\cos[\alpha_0]}$$ (53)

$$= R n \times \frac{d}{m} \times \frac{\lambda_0}{\lambda} \times \frac{\Delta w\cos\beta}{M\Delta s\cos[\alpha_0]}$$ (54)

And finally, emergent angle is

$$D_{out} = \frac{n\lambda_0 f_{col}\cos\beta}{\Delta s\cos[\alpha_0]}$$ (55)

$$= R n \times \frac{d\cos\beta}{m} \times \frac{\lambda_0}{\lambda} \times \frac{\Delta w\cos\beta}{M\Delta s\cos[\alpha_0]}$$ (56)

From $D_{gr}$, number of grating grooves $N = D_{gr}/d$ becomes

$$N = \frac{R n}{m} \times \frac{\lambda_0}{\lambda} \times \frac{\Delta w\cos\beta}{M\Delta s\cos[\alpha_0]}$$ (57)

With slit image width $\Delta i$ being $M \Delta s \cos[\alpha_0]/\cos\beta$, $N$ is $\Delta w/\Delta i$ times more than that for simple seeing limited spectrographs.

In total, we have to have the grating width $n\times\Delta w/\Delta i$ times bigger than that of diffraction limited spectrographs.

Shadowing

Grating made with an acute angle cutting tool

This section considers the case (a) in Figure 2. When $\varepsilon$ is not zero and $\alpha$ or $\beta$ is not zero, some of the light does not reach the mirror surface of a groove. When the light has the angle $\gamma$ as in Figure 2 (a), $d'$ can be calculated as

$$d' = d \times \frac{\;\cos \gamma \cos \vert\varepsilon\vert\;}{cos(\gamma - \vert\varepsilon\vert)}$$ (58)

where

$$\gamma = \left\{ \begin{array}{ll} 0 & (\alpha\varepsilo... ...\mbox{ and } \beta\varepsilon \geq 0) \\ \end{array} \right.$$ (59)

In a practical grating, we have to have $\varepsilon > 0$ and with $\beta_c$ for the central wavelength $\lambda_c$ of the spectrometer, $\alpha + \beta_c = 2\varepsilon$ to maximize the slit function. Within the requirements, $d'$ is maximum when $\alpha = \beta_c$.

Grating made with a right angle cutting tool

For the case (b) in Figure 2, another geometry has to be considered. In this case,

$$d' = d \times \frac{\;\cos[\vert\varepsilon\vert - \gamma'] \cos \vert\varepsilon\vert\;}{cos\gamma'}$$ (60)

where

$$\gamma' = \left\{ \begin{array}{ll} 0 & ( (\alpha-\varep... ...\beta-\varepsilon)\varepsilon \geq 0) \\ \end{array} \right.$$ (61)

When $\gamma$ is defined as in Equation 60, $\gamma' = \gamma - \vert\varepsilon\vert$ and Equations 61 and 62 are identical to Equations 59 and 60.

Splitting into other orders

The other sources for degradation of efficiency is light splitting into other orders of diffraction. Assuming the grating function as a delta function, intensity of the light in the $m$th order is calculated as follows.

First, defining the incident angle $\alpha$, emergent angle $\beta_m$ to the $m$th order is calculated from Equation 10. With

$$E_m = \frac{1}{\cos \varepsilon} (\sin[\alpha - \varepsilon] + \sin[\beta_m - \varepsilon] )$$ (62)

intensity of the emergent light in the $m$th order is

$$I_m = \frac{d'}{d} \frac{2}{(kd'E_m)^2} (1-\cos[kd'E_m])$$ (63)

There is a possibility of light splitting into $-\pi/2 < \beta < \pi/2$. With $m_{min}$ as minimum integer grater than $d/\lambda \times (\sin \alpha - 1)$ and $m_{max}$ as maximum integer less than $d/\lambda \times (\sin\alpha + 1)$, total emergent intensity is $\sum_{m'=m_{min}}^{m_{max}} I_{m'}$. Thus, Efficiency regarding the order split is

$$\frac{I_m}{I_{total}} = \frac{ \frac{d'}{d} \frac{2}{(kd'E... ...} \frac{d'}{d} \frac{2}{(kd'E_{m'})^2} (1-\cos[kd'E_{m'}]) }$$ (64)

Free spectral range

From Equation 10, emergent angle depends on both $\lambda$ and $m$ when $\alpha$ and $d$ are fixed. When taking spectra of a wavelength band from $\lambda_{min}$ through $\lambda_{max}$, we have to be careful not to allow the light from the next orders ($m\pm 1$) into the observed $\beta$. For this, following condition has to be satisfied:

$$\lambda_{max} (m-1) < \lambda_{min} m \mbox{ and } \lambda_{max} m < \lambda_{min} (m+1)$$ (65)

Table 1 shows the allowed $m$ for the NIR bands when we want to take spectra of the whole band at once.

Table 1: Allowed orders of diffraction with spectra not overlapping.

Band	$\lambda_{min}$($\mu m$)	$\lambda_{max}$($\mu m$)	allowed orders
$J$	1.13	1.37	-5 $<$ $m$ $<$ 5
$H$	1.50	1.80	-5 $<$ $m$ $<$ 5
$K$	2.01	2.42	-5 $<$ $m$ $<$ 5

Model calculations of grating efficiency

Figures 4 - 9 shows the results from calculation in the Excel file for J-band and K-band gratings.

It turned out that:

• Lower order has higher efficiency.
• Better efficiency can be achieved when $\alpha$ and $\varepsilon$ have the same sign.
• Incident angle and emergent angle should be as near as possible.
• Grating constant should be as big as possible.

All the results indicates that we get better efficiency with bigger optics.

Figure 4: Efficiency of the J-band grating plotted for emergent angle. $d:5.83\mu m$, $\varepsilon :13.4 deg$ optimized for $\lambda :1.25 \mu m$ with $\alpha - \beta = \pm 45 deg$.

Figure 5: Efficiency of the J-band grating plotted for wavelength. $d:5.83\mu m$, $\varepsilon :13.4 deg$ optimized for $\lambda :1.25 \mu m$ with $\alpha - \beta = \pm 45 deg$.

Figure 6: Slit function for different orders with $\alpha - \beta = +45 deg$ and $-45 deg$. Squares and circles indicates the integer orders which angles light of $1.25 \mu m$ actually emitted. For the $\alpha - \beta = +45 deg$ configuration, integer order exists in the wing of the primary peak of slit function, which makes the efficiency worth.

Figure 7: Change of efficiency in different orders for K-band grating. $d = 22 \mu m$, $\Delta = -45 deg$, $\lambda = 2 - 2.5 \mu m$

Figure 8: Change of efficiency in different $\Delta = \alpha - \beta$ for the K-band grating. $d = 22 \mu m$, $m = 1$.

Figure 9: Efficiency dependence on groove interval $d$ for the K-band grating. $\Delta = -30 deg$, $m = 1$.

Derivation of $\alpha$ and $\beta$ from $\Delta$

Equation 12 is derived to satisfy the following conditions.

$$\left\{ \begin{array}{ll} \Delta = \alpha - \beta \\ \frac{m\lambda}{d} = \sin\alpha + \sin\beta \\ \end{array}\right.$$ (66)

From $\beta = \alpha - \Delta$, $\sin\beta = \sin\alpha \cos\Delta - \cos\alpha \sin\Delta$.When we define $x \equiv \sin\alpha$ and $p \equiv \frac{m\lambda}{d}$,

$$\sin \beta = \sin[\alpha - \Delta]$$ (67)

$$= \sin\alpha\cos\Delta - \cos\alpha\sin\Delta$$ (68)

$$= x\cos\Delta - \sqrt{1-x^2}\sin\Delta$$ (69)

because $-\frac{\pi}{2} < \alpha < \frac{\pi}{2}$. Therefore, the second condition becomes

$$p = x + x\cos\Delta - \sqrt{1-x^2}\sin\Delta$$ (70)

and

$$\sqrt{1-x^2}\sin\Delta = x + x\cos\Delta - p$$ (71)

Taking square of the both sides,

$$(1-x^2)\sin^2\Delta = x^2(1+\cos\Delta)^2 - 2xp(1+\cos\Delta) + p^2$$ (72)

or

$$0 = x^2[(1+\cos\Delta)^2+\sin^2\Delta] - 2p(1+\cos\Delta)x + p^2-\sin^2\Delta$$ (73)

When we define $x' \equiv \sin\beta$, an equation of the same shape comes out. This means that the two solutions $x_{\pm}$ of the second order equation corresponds to $\sin\alpha$ and $\sin\beta$.

Solutions of Equation 74 are

$$x_\pm = \frac{p(1+\cos\Delta)\pm\sin\Delta\times\sqrt{2(1+\cos\Delta)-p^2}} {2(1+\cos\Delta)}$$ (74)

Thus, $\alpha$ and $\beta$ can be derived as $\arcsin x_\pm$.

Following is a perl script to calculate $\alpha$ and $\beta$.

#!/usr/bin/perl -w

# subroutines
# arcsin from man perlfunc
sub asin { atan2($_[0], sqrt(1 - $_[0] * $_[0])) }

# usage
$usage = << '_USAGE';
usage: alphabeta.pl delta(deg) m d lambda
  with delta=|alpha-beta|, m=diffraction order, lambda=wavelength
  prints out solutions for alpha(deg) and beta(deg)
_USAGE

# Main routine
# usage
if (@ARGV != 4) {
  print $usage;
  exit -1;
}

# get command line options
$delta = shift(@ARGV);
$m = shift(@ARGV);
$d = shift(@ARGV);
$lambda = shift(@ARGV);

# calculation
$pi = atan2(1,1)*4;
$cosd = cos($delta*$pi/180);		# cos(delta)
$sind = sin($delta*$pi/180);		# sin(delta)
$p = $m*$lambda/$d;			# m*lambda/d
$p1 = (-$p*(1+$cosd)+$sind*sqrt(2*(1+$cosd)-$p*$p))/2/(1+$cosd);
$p2 = (-$p*(1+$cosd)-$sind*sqrt(2*(1+$cosd)-$p*$p))/2/(1+$cosd);
$a1 = asin($p1)*180/$pi;		# solution in degrees
$a2 = asin($p2)*180/$pi;		# solution in degrees

# results
print "$a1 $a2\n";

File locations

Original source of this document is at siroan:~mos/design/010521_Grating/ and copied to gin-an:~tomono/Presen01/010521_Grating/. Revised version is edited at gin-an:~tomono/Presen01/011228_Grating/. Some of the plots are generated from siroan:/win98/My Documents/tomono/Design01/Fiber/010514_OptTrain/SlitFunctions.xls. HTML version of this document is posted at http://www.rzg.mpg.de/~tomono/MOS/auth/Reports/011228_Grating/.

Revision history

2001.5.23.

-

• First release

2001.5.23.

-

• Errata: parenthesis in Equation 4
• Another grating shape added (Figure 2(b), Equations 61, and 62)

2001.5.24.

-

• Derivation of $\alpha$ and $\beta$ from $\Delta$
• Errata: $\frac{d'}{d}$ coefficient in Equation 65

2001.5.28.

-

• J-band grating
• Plots are updated according to change of Equation
• Second release
• calculation of $f_{cam}$

2001.12.21.

-

• Source file copied to tomono@gin-an
• Link to the postscript file added to the html version
• Spell checked

2001.12.28.

- Revised version

• Consideration of wide entrance slit

2002.1.10.

-

• Equations 12 and 75 are corrected.

2002.1.11.

-

• Perl script added for calculation of $\alpha$ and $\beta$.

2002.1.17.

-

• Section 3.4 added.

2002.2.4.

-

• Section 4 added.

About this document ...

Interferometry on a blazed grating ¹

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Footnotes

... grating¹

Revised on December 2001.