Abstract

We report on a miniature spectrometer with a volume of 0.135 cm3 and dimensions of 3×3×11 mm, mounted directly on the surface of a CCD sensor. The spectrometer is formed by two flat diffraction gratings that are designed to perform both the dispersion and imaging functions, eliminating the need for any spherical optics. Two separate parts of the device were fabricated with the single-mask 1 μm lithography on a single glass wafer. The wafer was diced and the device was assembled and directly mounted onto a CCD sensor. The resolution of 3 nm, spectral range of 450 to 750 nm and the optical throughput of ∼9% were measured to be in a complete agreement with the model used for the development of the device.

©2007 Optical Society of America

1. Introduction

Spectroscopic devices have applications in practically all industries [1]. The cheaper and the smaller the spectrometer is, the more applications it can have. Small and cheap spectrometers are in demand.

Different approaches to the design of an inexpensive compact spectrometer have been presented during the last two decades [2] – [8]. One of the early works describes a Rowland spectrometer implemented in a planar waveguide [2]. The device was capable to separate two channels with the 10 nm wavelength difference. The recent progress in integrated optics and MEMS technology brought new possibilities for design of the compact spectrometers using very small integrated optical and opto-mechanical components. For example in [4] a planar waveguide spectrometer with the largest dimension of about 2 cm and spectral resolution of 9 nm within the 300 nm operating range is reported. A small Fourier Transform (FT) spectrometer with moving micro parts based on a 5 mm x 5 mm lamellar diffraction grating with variable depth was presented in [5] and demonstrated 1.6 nm resolution at a wavelength of 400 nm and 5.5 nm resolution at 800 nm. Static FT spectrometer with the 10 nm resolution at a wavelength of 570 nm based on a polarization interferometer realized with liquid-crystal materials was described in [6]. MEMS technology was used in [7] for the fabrication of an imaging diffraction grating used in a spectrometer as small as 1 cm3. The reported spectroscopic device demonstrated 5 nm resolution within 100 nm range. A promising result was obtained recently employing integrated optics in [8], describing a spectrometer having the largest dimension of the optical part of about 1.2 cm and having resolution varying from 0.3 to 4.6 nm in the 150 nm wavelength range.

In this article we report on an imaging spectrometer that combines wide operating spectral bandwidth of 300 nm with a good spectral resolution (3 nm) in a very small overall dimensions (3x3x11 mm3). The fabrication technology of the device is extremely simple – all parts of the device can be fabricated with a single lithographic step on a planar surface with a consequent batch assembling including wafer-to-wafer bonding through a spacer structure. Thus our design avoids difficulties with the alignment of different parts of the device common to classical spectrometers and complicated processing specific to integrated optics.

2. Theoretical considerations for a two-grating spectrometer

The concept of a general spectroscopic device is presented on Fig. 1. It includes an entrance slit, a dispersive element, a focusing element and an image sensor or – depending on the function – the exit slit [10]. Our decision to design a spectrometer with flat optics only, leaded to a grating that must provide both focusing and dispersion functions. Such a combination does not allow for high resolution in a wide spectral range, therefore we decided to use two gratings. The second grating provides additional compensation of the aberrations introduced by the first grating thus increasing the operating bandwidth considerably. A similar configuration was described in [9] for separate three-color imaging.

 figure: Fig. 1.

Fig. 1. General spectroscopic device

Download Full Size | PPT Slide | PDF

The theory of imaging gratings is well developed [11, 12] and various techniques including mechanical ruling, holography and lithography have been successfully used for their fabrication. Figure 2(a) presents the geometry of a diffraction grating which images point A into point B. Plane yoz of the coordinate system oxyz lies in the grating plane and the origin of the coordinate system o coincides with the point O.

 figure: Fig. 2.

Fig. 2. The grating imaging geometry. Point source A is imaged to an image point B. Single-grating (a) and double-grating (b) systems.

Download Full Size | PPT Slide | PDF

Consider the grating central point O and a point P somewhere in the grating plain. Then the optical pass difference between light beams passing from A to B via the point P and the point O is:

ψ(λ,y,z)=APBAOB+N(y,z),

where N(y, z) is a function equal to the groove number at the point (y, z), m is a diffraction order and λ is a wavelength. Equation (1) could be decomposed into power series and take the form:

ψ(λ,y,z)=Σi=0Σj=0Fijyizj=Σi=0Σj=0(Mij+Nij)yizj

The coefficients Mij in Eq. (2) depend on the system configuration and the coefficients Nij are determined by the grating groove pattern [12]. The path difference equal to zero ψ = 0 corresponds to a spherical wavefront convergent to an image point B. Since the value of the coefficient Fij determines the deviation of the diffracted wavefront from the spherical one proportional to yizj the coefficients Fij in Eq. (2) can be interpreted as the aberration coefficients [12]. From the formal mathematical point of view Eq. (1)could also be decomposed in to McLoren series keeping λ as a parameter:

ψ(λ,y,z)=Σn=01n![y(y)0,0+z(z)0,0]nψ(λ,y,z)

We can impose a requirement for the aberration coefficients Fij in equation 2 to be zero, thus obtaining an aberration reduced image of a point A for the wavelength λ. From 3 we can obtain the formulation for the coefficients Nij of an abberation reduced grating:

Nij=Mij=1i!j![i+j(APBAOB)yizj](0,0)

Equation (4) allows calculation of the grating coefficients from the system geometry. Exact formulas for Mij are listed for example in [12]. Suppose we eliminated aberrations for one wavelength λ0 having calculated grating coefficients according to Eq. (4). The image of the point A produced by this grating for some another wavelength λ1 would be different from the point B, the image of the point A for the wavelength λ0 (see Fig. 2(a)). It means that the coefficients Mij in Eq. (2) change with the wavelength. However the product λNij change much faster than Mij. That is why aberrations eliminated for one wavelength appear again for another one. It could be shown that in case of two diffraction gratings as Fig. 2(b) presents, the Eq. (2) takes the form:

ψ(λ,y,z)=Σi=0Σj=0(Mij+mλNij+m˜λN˜ij)yizj

In Eq. (5) the coefficients Mij are defined by the system geometry, Nij and Ñij are dependent on the groove pattern of the first and second gratings respectively.

Consider two light beams having the wavelength λ1 and λ2 emitted from a source point A and diffracted from the grating 1. Light beam having the wavelength λ1 is imaged into point B1 and second beam - into point B2. The grating 1 can compensate aberrations for one of these beams, for example the beam 1. However, according to Eq. (2), it is impossible to compensate aberrations for the beam 2 with the same grating. Since the light beams under consideration become spatially separated after being diffracted, we can use grating 2 to compensate aberrations for the second beam. It can be achieved by an appropriate choice of the coefficients Ñij in Eq. (5). This approach has been demonstrated in [9] for a system that operates with three wavelengths and uses a three segments grating to compensate aberrations. However in case when the wavelengths λ1 and λ2 are close to each other the diffracted beams could run a long distance in space before they become separated. We used a single grating to reduce aberrations of the diffracted beams partially overlapped in space and tried to optimize it to provide sufficient compensation of aberrations for all wavelengths within the specified operating bandwidth. Using this approach, we were able to keep compact dimensions of the optical design. We also tried to avoid the problem of the alignment of different optical components: we proposed a planar configuration where all optical components are made on a single wafer using lithography technique.

3. Spectrometer design

Figure 3 presents schematics of our design.

The spectrometer consists of two pieces of glass aligned parallel to each other. All spectrometer optical components, including the input slit and the diffraction gratings, work in reflection. The light from a source is reflected from a stripe mirror, acting as a slit and then diffracts on the grating 1, redirected by the second mirror to the grating 2 and finally goes to the photodetector through the glass plate. Both gratings operate in the -1 diffraction order. The size of the grating 1 defines the input aperture of the spectrometer since the non-diffracted light escapes the system through the glass plate as designated by the dashed arrows on the Fig. 3. The trajectory of light in the spectrometer is determined by the geometry and the position of the input slit fabricated as a mirror stripe, and the geometries of the two gratings. Higher incident angles and higher grating spatial frequencies allow to obtain a better performance. In practice, there are limits to these parameters. The available lithographic technology limited the maximum grating frequency to ∼1000 lines per millimeter. As the incident angle is being increased the amount of light reflected from the glass surfaces due to Fresnel reflections is also increasing. Such parasitic reflections contribute to the stray light as will be discussed below. Finally we had chosen the incident angle of the input light to be 60 degrees. The distance between glass plates was chosen to be 3 mm and the size of each peace was 3×11 mm2. Grating 1 and grating 2 had elliptical shape with axes of ellipses equal to 1 mm by 0.6 mm and 2.5 mm by 1 mm, respectively, defining the input numerical aperture to be equal to ∼0.05. We used Zemax [13] to calculate the parameters of the gratings via numerical optimization, since analytical calculations in case of two gratings require a huge amount of algebraic work. ZEMAX uses a polynomial representation of a diffractive surface similar to Eq. (2). A plane grating can be described by the function N(y, z) (see Eq. (1)) which equals to the number of the groove passing through the point (y, z). It follows from Eq. (1) that the number of the groove passing through the point (0,0) is zero. Suppose we can decompose function N into series as was done in Eq. (2). Then we can write:

 figure: Fig.3.

Fig.3. The design of a compact planar spectrometer.

Download Full Size | PPT Slide | PDF

N(y,z)=Σn=0cijyizj

Our numerical experiment has shown that 20 coefficients of the Eq. (6) are enough to describe our gratings and a larger number of coefficients does not improve the performance. We can choose orientation of the coordinate system so that the grating is symmetrical relative to z axis. Then all members of Eq. (6) with j odd are equal to zero and we have only 11 coefficients to optimize. So, for two gratings, our design task was an optimization problem with 22 parameters. With ZEMAX optimization engine we obtained the optimal values of grating coefficients which were then used in conjunction with Eq. (6) for mask design. Using Eq. (6) for each of the grating grooves we determined a set of points (y, z) belonging to the groove and describing its position in the grating plane. This procedure enabled drawing of the grating structure on the mask. The period of each of the gratings varied from 1 μ to 4 μ which was within the limits of the available lithography resolution. The lithographic technique did not allow fabrication of the blazed gratings, the only physical parameter we were able to optimize was the depth of the grating grooves which was chosen to be equal to 230 nm for Al-coated grating. The maximum diffraction efficiency for a single grating did not exceed 30%. An improvement of the diffraction efficiency would involve more complicated fabrication technology (for example holography) which is out of scope of our work.

4. Fabrication and testing of the spectrometer

All elements of the spectrometer including the slit and the gratings were fabricated on a single Al-coated glass wafer. The waver was covered with 650 nm of aluminum and then several process steps followed. First, the grating patterns were etched by plasma, to form 230 nm deep grooves. Then aluminum was patterned to define the input slit, the gratings and the intermediate mirror. Finally the wafer was diced and the two parts of the spectrometer were aligned in as presented in the Fig. 4 with the Videology 21K137 CCD board camera [14] that was used as a photo-detector.

 figure: Fig. 4.

Fig. 4. The experimental setup. The spectrometer is visible as two transparent rectangles mounted over the image sensor. The light is fed to the spectrograph from the fiber tip visible in the left top corner.

Download Full Size | PPT Slide | PDF

5. Spectrometer characterization

A multi-mode optical fiber was used to feed the spectrometer. One of the cleaved fiber tips was exposed to a light source under test and the second fiber tip illuminated the input slit of the spectrometer. As an example of the device performance, Fig. 5 represents the spectrum of a Ne lamp captured with the device CCD in the range 550 to 750 nm.

The presence of stray light is clearly visible in Fig. 5. Stray light represents a serious problem because of the two features of the device. First, glass surface around the input slit reflects about 9% of the incoming light to the grating 1 (see Fig. 3) due to Fresnel reflection. Thus spectral image presented in Fig. 5 is actually a superposition of the dispersed images of the entrance slit and the fiber tip that was used for illumination of the slit. Such elliptical images of the fiber tip are also visible on the Fig. 5. The minor contribution to the stray light is due to low total diffraction efficiency of less than 10%, which means that approximately 90% of all input light is scattered. Most of this light leaves the device through transparent glass surfaces, but some light returns after parasitic reflections. Thus the elimination of Fresnel reflections would substantially decrease the level of stray light. It could be achieved by depositing an appropriate coating on the unused surfaces. The coating process would involve additional lithography step: the deposition of a light absorbing material on top of a wafer followed by the patterning of this material to open aluminum gratings and mirrors and that part of the glass plate which the light goes through to reach the photosensor. Implementation of such a coating is the scope of our future work.

 figure: Fig. 5.

Fig. 5. The spectrum of a Neon lamp produced with the planar spectrograph and registered with a CCD chip.

Download Full Size | PPT Slide | PDF

Figure 6 presents the spectrum of Ne, obtained with our spectrometer. The major lines are marked to identify them with the Ne spectral lines. Judging from the Fig. 6, the resolution of our spectrometer is approximately 3 nm, while the relative positions of spectral lines can be determined with a precision of better than 1 nm.

 figure: Fig. 6.

Fig. 6. Graphic representation of the registered spectrum of a Neon lamp.

Download Full Size | PPT Slide | PDF

In spite of its low diffraction efficiency, the device appeared to be sensitive enough to register spectra of different sources, such as lamps, LEDs, and groups of pixels on a computer screen. For example, Fig. 7 demonstrates the spectra of red, yellow and white pixels of the LCD computer screen. In this experiment the fiber tip was placed in front of the computer screen displaying a three-color image. These spectra were recorded at the lowest shutter speed (1/50 sec.) and automatic electronic gain control of the CCD camera. That is why they exhibit a periodic modulation which is a fixed pattern camera noise in a high sensitivity regime. The overall sensitivity of the spectrometer depends on the throughput of the optical part of the device and the sensitivity of the photo-detector. An optical throughput is:

 figure: Fig. 7.

Fig. 7. Spectra of different pixel groups of a color LCD screen.

Download Full Size | PPT Slide | PDF

L=SNAP

In the Eq. (7) S is the input slit area, NA is the input aperture and P is the efficiency of the optical scheme. In our design we have NA = 0.05 and P = 0.065. The relatively small value of the NA is a compromise between the throughput and the spectral resolution. A small value of P is the consequence of low diffraction grating efficiency (30%). The second grating decrease the device throughput further, so that less than 9% of the incoming light goes to the photo-detector. Relatively low throughput of our design is a disadvantage which is, however, not critical for applications requiring very low-cost high-resolution spectrometer coupled to a bright source.

6. Conclusion

We designed, fabricated and tested a new planar miniature spectrometer. Using a design with two planar gratings, we were able to obtain a 3 nm resolution within ∼300 nm bandwidth with the small 3×3×11 mm3 dimensions. All spectrometer components were fabricated lithographically on a single wafer. Planar miniature design of the spectrometer allows its integration with various types of photodetectors including low-cost CMOS sensors.

Acknowledgment

The authors acknowledge the financial support from STW The Dutch Technical Foundation, grant DET.6667. We appreciate the help from Dr. Lina Sarro, Jan Groeneweg, Jan Cornelis Wolff and Charles de Boer with the mask design and wafer processing and useful discussions with Dr. A. Simonov.

References and links

1. R. F. Wolffenbuttel, “MEMS-based optical mini- and microspectrometers for the visible and infrared spectral range,” J. Micromech. Microeng. 15,S145–S152 (2005) [CrossRef]  

2. H. W. Yen, H. R. Friedrich, R. J. Morrison, and G. L. Tangonan, “Planar Rowland spectrometer for fiber-optic wavelength demultiplexing,” Opt. Lett. 6,639–641 (1981). [CrossRef]   [PubMed]  

3. Don.S. Goldman, P.L. White, and N.C. Anheier, “Miniaturized spectrometer employing planar waveguidesand grating couplers for chemical analysis,” Appl. Opt. 29,4583–4589 (1990). [CrossRef]   [PubMed]  

4. Dietmar Sander and Jorg Muller, “Selffocusing phase transmission grating for an integrated optical microspectrometer,” Sens. Actuators A. 88,1–9 (2001). [CrossRef]  

5. O. Manzardo, R. Michaely, F. Schdelin, W. Noell, T. Overstolz, N. De Rooij, and H. P. Herzig, “Miniature lamellar grating interferometer based on silicon technology,” Opt. Lett. 29,1437–1439 (2004) [CrossRef]   [PubMed]  

6. G. Boer, P. Ruffieux, T. Scharf, P. Seitz, and R. Dndliker, “Compact Liquid-Crystal-Polymer Fourier-Transform Spectrometer,” Appl. Opt. 43,2201–2208 (2004) [CrossRef]   [PubMed]  

7. S. Ura, F. Okayama, K. Shiroshita, K. Nishio, T. Sasaki, H. Nishihara, T. Yotsuya, M. Okano, and K. Satoh, “Planar Reflection Grating Lens for Compact Spectroscopic Imaging System,” Appl. Opt. 42,175–180 (2003). [CrossRef]   [PubMed]  

8. K. Chaganti, I. Salakhutdinov, I. Avrutsky, and G. W. Auner, “A simple miniature optical spectrometer with a planar waveguide grating coupler in combination with a plano-convex lens,” Opt. Express 14,4064–4072 (2006). [CrossRef]   [PubMed]  

9. S. Ura, T. Sasaki, and H. Nishihara, “Combination of Grating Lenses for Color Splitting and Imaging,” Appl. Opt. 40,5819–5824 (2001). [CrossRef]  

10. J. F. James and R. S. Sternberg, “The Design Of Optical Spectrometers,”, Chapman and Hall, London1969

11. H. Noda, T. Namioka, and M. Seya ,“Geometric theory of the grating,” J. Opt. Soc. Am. 64,103–1036 (1974). [CrossRef]  

12. Cristopher Palmer and Wayne R. McKinney ,“Imaging theory of plane-symmetric varied line-space grating systems,” Opt. Eng. ,33,820–829 (1994). [CrossRef]  

13. ZEMAX Optical Design Program, User’s Guide, Version 9.0 (Focus Software, Inc., Tucson, Ariz., 2000).

14. http://www.videologyinc.com

References

  • View by:

  1. R. F. Wolffenbuttel, “MEMS-based optical mini- and microspectrometers for the visible and infrared spectral range,” J. Micromech. Microeng. 15,S145–S152 (2005)
    [Crossref]
  2. H. W. Yen, H. R. Friedrich, R. J. Morrison, and G. L. Tangonan, “Planar Rowland spectrometer for fiber-optic wavelength demultiplexing,” Opt. Lett. 6,639–641 (1981).
    [Crossref] [PubMed]
  3. Don.S. Goldman, P.L. White, and N.C. Anheier, “Miniaturized spectrometer employing planar waveguidesand grating couplers for chemical analysis,” Appl. Opt. 29,4583–4589 (1990).
    [Crossref] [PubMed]
  4. Dietmar Sander and Jorg Muller, “Selffocusing phase transmission grating for an integrated optical microspectrometer,” Sens. Actuators A. 88,1–9 (2001).
    [Crossref]
  5. O. Manzardo, R. Michaely, F. Schdelin, W. Noell, T. Overstolz, N. De Rooij, and H. P. Herzig, “Miniature lamellar grating interferometer based on silicon technology,” Opt. Lett. 29,1437–1439 (2004)
    [Crossref] [PubMed]
  6. G. Boer, P. Ruffieux, T. Scharf, P. Seitz, and R. Dndliker, “Compact Liquid-Crystal-Polymer Fourier-Transform Spectrometer,” Appl. Opt. 43,2201–2208 (2004)
    [Crossref] [PubMed]
  7. S. Ura, F. Okayama, K. Shiroshita, K. Nishio, T. Sasaki, H. Nishihara, T. Yotsuya, M. Okano, and K. Satoh, “Planar Reflection Grating Lens for Compact Spectroscopic Imaging System,” Appl. Opt. 42,175–180 (2003).
    [Crossref] [PubMed]
  8. K. Chaganti, I. Salakhutdinov, I. Avrutsky, and G. W. Auner, “A simple miniature optical spectrometer with a planar waveguide grating coupler in combination with a plano-convex lens,” Opt. Express 14,4064–4072 (2006).
    [Crossref] [PubMed]
  9. S. Ura, T. Sasaki, and H. Nishihara, “Combination of Grating Lenses for Color Splitting and Imaging,” Appl. Opt. 40,5819–5824 (2001).
    [Crossref]
  10. J. F. James and R. S. Sternberg, “The Design Of Optical Spectrometers,”, Chapman and Hall, London1969
  11. H. Noda, T. Namioka, and M. Seya ,“Geometric theory of the grating,” J. Opt. Soc. Am. 64,103–1036 (1974).
    [Crossref]
  12. Cristopher Palmer and Wayne R. McKinney ,“Imaging theory of plane-symmetric varied line-space grating systems,” Opt. Eng., 33,820–829 (1994).
    [Crossref]
  13. ZEMAX Optical Design Program, User’s Guide, Version 9.0 (Focus Software, Inc., Tucson, Ariz., 2000).
  14. http://www.videologyinc.com

2006 (1)

2005 (1)

R. F. Wolffenbuttel, “MEMS-based optical mini- and microspectrometers for the visible and infrared spectral range,” J. Micromech. Microeng. 15,S145–S152 (2005)
[Crossref]

2004 (2)

2003 (1)

2001 (2)

S. Ura, T. Sasaki, and H. Nishihara, “Combination of Grating Lenses for Color Splitting and Imaging,” Appl. Opt. 40,5819–5824 (2001).
[Crossref]

Dietmar Sander and Jorg Muller, “Selffocusing phase transmission grating for an integrated optical microspectrometer,” Sens. Actuators A. 88,1–9 (2001).
[Crossref]

1994 (1)

Cristopher Palmer and Wayne R. McKinney ,“Imaging theory of plane-symmetric varied line-space grating systems,” Opt. Eng., 33,820–829 (1994).
[Crossref]

1990 (1)

1981 (1)

1974 (1)

H. Noda, T. Namioka, and M. Seya ,“Geometric theory of the grating,” J. Opt. Soc. Am. 64,103–1036 (1974).
[Crossref]

Anheier, N.C.

Auner, G. W.

Avrutsky, I.

Boer, G.

Chaganti, K.

Dndliker, R.

Friedrich, H. R.

Goldman, Don.S.

Herzig, H. P.

James, J. F.

J. F. James and R. S. Sternberg, “The Design Of Optical Spectrometers,”, Chapman and Hall, London1969

Manzardo, O.

McKinney, Wayne R.

Cristopher Palmer and Wayne R. McKinney ,“Imaging theory of plane-symmetric varied line-space grating systems,” Opt. Eng., 33,820–829 (1994).
[Crossref]

Michaely, R.

Morrison, R. J.

Muller, Jorg

Dietmar Sander and Jorg Muller, “Selffocusing phase transmission grating for an integrated optical microspectrometer,” Sens. Actuators A. 88,1–9 (2001).
[Crossref]

Namioka, T.

H. Noda, T. Namioka, and M. Seya ,“Geometric theory of the grating,” J. Opt. Soc. Am. 64,103–1036 (1974).
[Crossref]

Nishihara, H.

Nishio, K.

Noda, H.

H. Noda, T. Namioka, and M. Seya ,“Geometric theory of the grating,” J. Opt. Soc. Am. 64,103–1036 (1974).
[Crossref]

Noell, W.

Okano, M.

Okayama, F.

Overstolz, T.

Palmer, Cristopher

Cristopher Palmer and Wayne R. McKinney ,“Imaging theory of plane-symmetric varied line-space grating systems,” Opt. Eng., 33,820–829 (1994).
[Crossref]

Rooij, N. De

Ruffieux, P.

Salakhutdinov, I.

Sander, Dietmar

Dietmar Sander and Jorg Muller, “Selffocusing phase transmission grating for an integrated optical microspectrometer,” Sens. Actuators A. 88,1–9 (2001).
[Crossref]

Sasaki, T.

Satoh, K.

Scharf, T.

Schdelin, F.

Seitz, P.

Seya, M.

H. Noda, T. Namioka, and M. Seya ,“Geometric theory of the grating,” J. Opt. Soc. Am. 64,103–1036 (1974).
[Crossref]

Shiroshita, K.

Sternberg, R. S.

J. F. James and R. S. Sternberg, “The Design Of Optical Spectrometers,”, Chapman and Hall, London1969

Tangonan, G. L.

Ura, S.

White, P.L.

Wolffenbuttel, R. F.

R. F. Wolffenbuttel, “MEMS-based optical mini- and microspectrometers for the visible and infrared spectral range,” J. Micromech. Microeng. 15,S145–S152 (2005)
[Crossref]

Yen, H. W.

Yotsuya, T.

Appl. Opt. (4)

J. Micromech. Microeng. (1)

R. F. Wolffenbuttel, “MEMS-based optical mini- and microspectrometers for the visible and infrared spectral range,” J. Micromech. Microeng. 15,S145–S152 (2005)
[Crossref]

J. Opt. Soc. Am. (1)

H. Noda, T. Namioka, and M. Seya ,“Geometric theory of the grating,” J. Opt. Soc. Am. 64,103–1036 (1974).
[Crossref]

Opt. Eng. (1)

Cristopher Palmer and Wayne R. McKinney ,“Imaging theory of plane-symmetric varied line-space grating systems,” Opt. Eng., 33,820–829 (1994).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Sens. Actuators A. (1)

Dietmar Sander and Jorg Muller, “Selffocusing phase transmission grating for an integrated optical microspectrometer,” Sens. Actuators A. 88,1–9 (2001).
[Crossref]

Other (3)

J. F. James and R. S. Sternberg, “The Design Of Optical Spectrometers,”, Chapman and Hall, London1969

ZEMAX Optical Design Program, User’s Guide, Version 9.0 (Focus Software, Inc., Tucson, Ariz., 2000).

http://www.videologyinc.com

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1.
Fig. 1. General spectroscopic device
Fig. 2.
Fig. 2. The grating imaging geometry. Point source A is imaged to an image point B. Single-grating (a) and double-grating (b) systems.
Fig.3.
Fig.3. The design of a compact planar spectrometer.
Fig. 4.
Fig. 4. The experimental setup. The spectrometer is visible as two transparent rectangles mounted over the image sensor. The light is fed to the spectrograph from the fiber tip visible in the left top corner.
Fig. 5.
Fig. 5. The spectrum of a Neon lamp produced with the planar spectrograph and registered with a CCD chip.
Fig. 6.
Fig. 6. Graphic representation of the registered spectrum of a Neon lamp.
Fig. 7.
Fig. 7. Spectra of different pixel groups of a color LCD screen.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

ψ ( λ , y , z ) = APB AOB + N ( y , z ) ,
ψ ( λ , y , z ) = Σ i = 0 Σ j = 0 F ij y i z j = Σ i = 0 Σ j = 0 ( M ij + N ij ) y i z j
ψ ( λ , y , z ) = Σ n = 0 1 n ! [ y ( y ) 0,0 + z ( z ) 0,0 ] n ψ ( λ , y , z )
N ij = M ij = 1 i ! j ! [ i + j ( APB AOB ) y i z j ] ( 0,0 )
ψ ( λ , y , z ) = Σ i = 0 Σ j = 0 ( M ij + mλN ij + m ˜ λ N ˜ ij ) y i z j
N ( y , z ) = Σ n = 0 c ij y i z j
L = S NA P

Metrics