Abstract

Modern miniaturized scanning grating spectrometers (SGSs) are often based on microelectromechanical system devices. In contrast to classical spectrometers, such systems exhibit additional design constraints, like a symmetrical motion of the grating with a limited deflection. A detailed mathematical analysis of typical SGS configurations based on the grating equation considering these constraints is presented. Equations that relate the basic angles on a scanning grating to the grating properties and the attainable wavelength range of a spectrometer are derived, and the solution set is examined. Furthermore, the analytical description can be used to optimize SGSs with symmetrically moving gratings. The attainable spectral range for a given deflection amplitude of the grating can be calculated. Alternatively, the required grating properties can be determined for a given spectral range.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. G. Loewen, Diffraction Gratings and Applications (Marcel Dekker1997).
  2. M. Born, E. Wolf, and A. B. Bhatia, Principal of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (CUP Archive, 2000).
    [PubMed]
  3. C. Palmer, Diffraction Grating Handbook, 6th ed.(Newport, 2005).
  4. G. R. Roaendahl, “Contributions to the optics of mirror systems and gratings with oblique incidence. I. Ray tracing formulas for the meridional plane,” J. Opt. Soc. Am. 51, 1–3 (1961).
    [CrossRef]
  5. A. B. Shafer, L. R. Megill, and L. Droppleman, “Optimization of the Czerny-Turner spectrometer,” J. Opt. Soc. Am. 54, 879–886 (1964).
    [CrossRef]
  6. K. Kudo, “Optical properties of plane-grating monochromator,” J. Opt. Soc. Am. 55, 150–161 (1965).
    [CrossRef]
  7. J. K. Pribram and C. M. Penchina, “Stray light in Czerny-Turner and Ebert spectrometers,” Appl. Opt. 7, 2005–2014 (1968).
    [CrossRef] [PubMed]
  8. R. A. Hill, “A new plane grating monochromator with off-axis paraboloids and curved slits,” Appl. Opt. 8, 575–581 (1969).
    [CrossRef] [PubMed]
  9. M. A. Gil and J. M. Simon, “New plane grating monochromator with off-axis parabolical mirrors,” Appl. Opt. 22, 152–158(1983).
    [CrossRef] [PubMed]
  10. H. Grüger, A. Wolter, T. Schuster, H. Schenk, and H. Lakner, “Realization of a spectrometer with micromachined scanning grating,” Proc. SPIE 4945, 46–53 (2003).
    [CrossRef]
  11. H. Ebert, “Zwei Formen von Spectrographen,” Ann. Phys. 274, 489–493 (1889).
    [CrossRef]
  12. M. Czerny and A. F. Turner, “Ueber den Astigmatimus bei Spiegelspektrometern,” Z. Phys. 61, 792–797 (1930).
    [CrossRef]
  13. M. Czerny and V. Plettig, “Ueber den Astigmatimus bei Spiegelspektrometern II,” Z. Phys. 63, 590–595(1930).
    [CrossRef]
  14. W. G. Fastie, “Image forming properties of the Ebert monochromator,” J. Opt. Soc. Am. 42, 647–650 (1952).
    [CrossRef]
  15. V. L. Chupp and P. C. Grantz, “Coma canceling monochromator with no slit mismatch,” Appl. Opt. 8, 925–929(1969).
    [CrossRef] [PubMed]
  16. L. Beiser, Unified Optical Scanning Technology (Wiley, 2003).
    [CrossRef]
  17. F. Zimmer, A. Heberer, T. Sandner, H. Grueger, H. Schenk, H. Lakner, A. Kenda, and W. Scherf, “Investigation and characterization of high-efficient NIR-scanning gratings used in NIR micro-spectrometer,” Proc. SPIE 6466, 646605 (2007).
    [CrossRef]
  18. S. Kurth, C. Kaufmann, R. Hahn, J. Mehner, W. Dotzel, and T. Gessner, MEMS Scanner for Laser Projection (Chemnitz University of Technology, Center for Microtechnologies, 2004).
  19. L. O. S. Ferreira and S. Moehlecke, “A silicon micromechanical galvanometric scanner,” Sens. Actuators A Phys. 73, 252–260 (1999).
    [CrossRef]

2007 (1)

F. Zimmer, A. Heberer, T. Sandner, H. Grueger, H. Schenk, H. Lakner, A. Kenda, and W. Scherf, “Investigation and characterization of high-efficient NIR-scanning gratings used in NIR micro-spectrometer,” Proc. SPIE 6466, 646605 (2007).
[CrossRef]

2003 (1)

H. Grüger, A. Wolter, T. Schuster, H. Schenk, and H. Lakner, “Realization of a spectrometer with micromachined scanning grating,” Proc. SPIE 4945, 46–53 (2003).
[CrossRef]

1999 (1)

L. O. S. Ferreira and S. Moehlecke, “A silicon micromechanical galvanometric scanner,” Sens. Actuators A Phys. 73, 252–260 (1999).
[CrossRef]

1983 (1)

1969 (2)

1968 (1)

1965 (1)

1964 (1)

1961 (1)

1952 (1)

1930 (2)

M. Czerny and A. F. Turner, “Ueber den Astigmatimus bei Spiegelspektrometern,” Z. Phys. 61, 792–797 (1930).
[CrossRef]

M. Czerny and V. Plettig, “Ueber den Astigmatimus bei Spiegelspektrometern II,” Z. Phys. 63, 590–595(1930).
[CrossRef]

1889 (1)

H. Ebert, “Zwei Formen von Spectrographen,” Ann. Phys. 274, 489–493 (1889).
[CrossRef]

Beiser, L.

L. Beiser, Unified Optical Scanning Technology (Wiley, 2003).
[CrossRef]

Bhatia, A. B.

M. Born, E. Wolf, and A. B. Bhatia, Principal of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (CUP Archive, 2000).
[PubMed]

Born, M.

M. Born, E. Wolf, and A. B. Bhatia, Principal of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (CUP Archive, 2000).
[PubMed]

Chupp, V. L.

Czerny, M.

M. Czerny and A. F. Turner, “Ueber den Astigmatimus bei Spiegelspektrometern,” Z. Phys. 61, 792–797 (1930).
[CrossRef]

M. Czerny and V. Plettig, “Ueber den Astigmatimus bei Spiegelspektrometern II,” Z. Phys. 63, 590–595(1930).
[CrossRef]

Dotzel, W.

S. Kurth, C. Kaufmann, R. Hahn, J. Mehner, W. Dotzel, and T. Gessner, MEMS Scanner for Laser Projection (Chemnitz University of Technology, Center for Microtechnologies, 2004).

Droppleman, L.

Ebert, H.

H. Ebert, “Zwei Formen von Spectrographen,” Ann. Phys. 274, 489–493 (1889).
[CrossRef]

Fastie, W. G.

Ferreira, L. O. S.

L. O. S. Ferreira and S. Moehlecke, “A silicon micromechanical galvanometric scanner,” Sens. Actuators A Phys. 73, 252–260 (1999).
[CrossRef]

Gessner, T.

S. Kurth, C. Kaufmann, R. Hahn, J. Mehner, W. Dotzel, and T. Gessner, MEMS Scanner for Laser Projection (Chemnitz University of Technology, Center for Microtechnologies, 2004).

Gil, M. A.

Grantz, P. C.

Grueger, H.

F. Zimmer, A. Heberer, T. Sandner, H. Grueger, H. Schenk, H. Lakner, A. Kenda, and W. Scherf, “Investigation and characterization of high-efficient NIR-scanning gratings used in NIR micro-spectrometer,” Proc. SPIE 6466, 646605 (2007).
[CrossRef]

Grüger, H.

H. Grüger, A. Wolter, T. Schuster, H. Schenk, and H. Lakner, “Realization of a spectrometer with micromachined scanning grating,” Proc. SPIE 4945, 46–53 (2003).
[CrossRef]

Hahn, R.

S. Kurth, C. Kaufmann, R. Hahn, J. Mehner, W. Dotzel, and T. Gessner, MEMS Scanner for Laser Projection (Chemnitz University of Technology, Center for Microtechnologies, 2004).

Heberer, A.

F. Zimmer, A. Heberer, T. Sandner, H. Grueger, H. Schenk, H. Lakner, A. Kenda, and W. Scherf, “Investigation and characterization of high-efficient NIR-scanning gratings used in NIR micro-spectrometer,” Proc. SPIE 6466, 646605 (2007).
[CrossRef]

Hill, R. A.

Kaufmann, C.

S. Kurth, C. Kaufmann, R. Hahn, J. Mehner, W. Dotzel, and T. Gessner, MEMS Scanner for Laser Projection (Chemnitz University of Technology, Center for Microtechnologies, 2004).

Kenda, A.

F. Zimmer, A. Heberer, T. Sandner, H. Grueger, H. Schenk, H. Lakner, A. Kenda, and W. Scherf, “Investigation and characterization of high-efficient NIR-scanning gratings used in NIR micro-spectrometer,” Proc. SPIE 6466, 646605 (2007).
[CrossRef]

Kudo, K.

Kurth, S.

S. Kurth, C. Kaufmann, R. Hahn, J. Mehner, W. Dotzel, and T. Gessner, MEMS Scanner for Laser Projection (Chemnitz University of Technology, Center for Microtechnologies, 2004).

Lakner, H.

F. Zimmer, A. Heberer, T. Sandner, H. Grueger, H. Schenk, H. Lakner, A. Kenda, and W. Scherf, “Investigation and characterization of high-efficient NIR-scanning gratings used in NIR micro-spectrometer,” Proc. SPIE 6466, 646605 (2007).
[CrossRef]

H. Grüger, A. Wolter, T. Schuster, H. Schenk, and H. Lakner, “Realization of a spectrometer with micromachined scanning grating,” Proc. SPIE 4945, 46–53 (2003).
[CrossRef]

Loewen, E. G.

E. G. Loewen, Diffraction Gratings and Applications (Marcel Dekker1997).

Megill, L. R.

Mehner, J.

S. Kurth, C. Kaufmann, R. Hahn, J. Mehner, W. Dotzel, and T. Gessner, MEMS Scanner for Laser Projection (Chemnitz University of Technology, Center for Microtechnologies, 2004).

Moehlecke, S.

L. O. S. Ferreira and S. Moehlecke, “A silicon micromechanical galvanometric scanner,” Sens. Actuators A Phys. 73, 252–260 (1999).
[CrossRef]

Palmer, C.

C. Palmer, Diffraction Grating Handbook, 6th ed.(Newport, 2005).

Penchina, C. M.

Plettig, V.

M. Czerny and V. Plettig, “Ueber den Astigmatimus bei Spiegelspektrometern II,” Z. Phys. 63, 590–595(1930).
[CrossRef]

Pribram, J. K.

Roaendahl, G. R.

Sandner, T.

F. Zimmer, A. Heberer, T. Sandner, H. Grueger, H. Schenk, H. Lakner, A. Kenda, and W. Scherf, “Investigation and characterization of high-efficient NIR-scanning gratings used in NIR micro-spectrometer,” Proc. SPIE 6466, 646605 (2007).
[CrossRef]

Schenk, H.

F. Zimmer, A. Heberer, T. Sandner, H. Grueger, H. Schenk, H. Lakner, A. Kenda, and W. Scherf, “Investigation and characterization of high-efficient NIR-scanning gratings used in NIR micro-spectrometer,” Proc. SPIE 6466, 646605 (2007).
[CrossRef]

H. Grüger, A. Wolter, T. Schuster, H. Schenk, and H. Lakner, “Realization of a spectrometer with micromachined scanning grating,” Proc. SPIE 4945, 46–53 (2003).
[CrossRef]

Scherf, W.

F. Zimmer, A. Heberer, T. Sandner, H. Grueger, H. Schenk, H. Lakner, A. Kenda, and W. Scherf, “Investigation and characterization of high-efficient NIR-scanning gratings used in NIR micro-spectrometer,” Proc. SPIE 6466, 646605 (2007).
[CrossRef]

Schuster, T.

H. Grüger, A. Wolter, T. Schuster, H. Schenk, and H. Lakner, “Realization of a spectrometer with micromachined scanning grating,” Proc. SPIE 4945, 46–53 (2003).
[CrossRef]

Shafer, A. B.

Simon, J. M.

Turner, A. F.

M. Czerny and A. F. Turner, “Ueber den Astigmatimus bei Spiegelspektrometern,” Z. Phys. 61, 792–797 (1930).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, and A. B. Bhatia, Principal of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (CUP Archive, 2000).
[PubMed]

Wolter, A.

H. Grüger, A. Wolter, T. Schuster, H. Schenk, and H. Lakner, “Realization of a spectrometer with micromachined scanning grating,” Proc. SPIE 4945, 46–53 (2003).
[CrossRef]

Zimmer, F.

F. Zimmer, A. Heberer, T. Sandner, H. Grueger, H. Schenk, H. Lakner, A. Kenda, and W. Scherf, “Investigation and characterization of high-efficient NIR-scanning gratings used in NIR micro-spectrometer,” Proc. SPIE 6466, 646605 (2007).
[CrossRef]

Ann. Phys. (1)

H. Ebert, “Zwei Formen von Spectrographen,” Ann. Phys. 274, 489–493 (1889).
[CrossRef]

Appl. Opt. (4)

J. Opt. Soc. Am. (4)

Proc. SPIE (2)

F. Zimmer, A. Heberer, T. Sandner, H. Grueger, H. Schenk, H. Lakner, A. Kenda, and W. Scherf, “Investigation and characterization of high-efficient NIR-scanning gratings used in NIR micro-spectrometer,” Proc. SPIE 6466, 646605 (2007).
[CrossRef]

H. Grüger, A. Wolter, T. Schuster, H. Schenk, and H. Lakner, “Realization of a spectrometer with micromachined scanning grating,” Proc. SPIE 4945, 46–53 (2003).
[CrossRef]

Sens. Actuators A Phys. (1)

L. O. S. Ferreira and S. Moehlecke, “A silicon micromechanical galvanometric scanner,” Sens. Actuators A Phys. 73, 252–260 (1999).
[CrossRef]

Z. Phys. (2)

M. Czerny and A. F. Turner, “Ueber den Astigmatimus bei Spiegelspektrometern,” Z. Phys. 61, 792–797 (1930).
[CrossRef]

M. Czerny and V. Plettig, “Ueber den Astigmatimus bei Spiegelspektrometern II,” Z. Phys. 63, 590–595(1930).
[CrossRef]

Other (5)

L. Beiser, Unified Optical Scanning Technology (Wiley, 2003).
[CrossRef]

S. Kurth, C. Kaufmann, R. Hahn, J. Mehner, W. Dotzel, and T. Gessner, MEMS Scanner for Laser Projection (Chemnitz University of Technology, Center for Microtechnologies, 2004).

E. G. Loewen, Diffraction Gratings and Applications (Marcel Dekker1997).

M. Born, E. Wolf, and A. B. Bhatia, Principal of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (CUP Archive, 2000).
[PubMed]

C. Palmer, Diffraction Grating Handbook, 6th ed.(Newport, 2005).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

SGS with its basic design parameters illustrated for the rest position of a symmetrically oscillating grating.

Fig. 2
Fig. 2

SGS with its basic design parameters illustrated for the rest position of a symmetrically oscillating grating in an arrangement with a common orientation of the slits and the grating in rest position.

Fig. 3
Fig. 3

The grating in a SGS in three different positions. In (a), λ max hits the exit slit, in (b), λ d hits the exit slit, and in (c), λ min hits the exit slit. A rotation about φ ^ φ + φ ^ results in a spectral range Δ λ = λ max λ min .

Fig. 4
Fig. 4

Plot of the angles ω d , δ / 2 , α d , and β d = f ( φ ^ ) in an SGS for m = 1 , g = 700 nm , λ min = 380 nm , and λ max = 780 nm . The limitations for the deflection amplitude φ ^ have been omitted.

Fig. 5
Fig. 5

Plot of the angles ω d , δ / 2 , α d , and β d = f ( φ ^ ) in an SGS for m = 1 , g = 1600 nm , λ min = 950 nm , and λ max = 1900 nm . The limitations for the deflection amplitude φ ^ have been omitted.

Fig. 6
Fig. 6

Solutions for the angles ω d , δ / 2 , α d , and β d = f ( φ ^ ) in an SGS for m = 1 , g = 700 nm , λ min = 380 nm , and λ max = 780 nm for possible SGS configurations.

Fig. 7
Fig. 7

Solutions for the angles ω d , δ / 2 , α d , and β d = f ( φ ^ ) in an SGS for m = 1 , g = 1600 nm , λ min = 950 nm , and λ max = 1900 nm for possible SGS configurations.

Fig. 8
Fig. 8

Plots of the angles ω d , δ / 2 , α d , and β d = f ( φ ^ ) in an SGS for m = 1 , g = 700 nm , λ min = 380 nm , and λ max = 780 nm within the solution space 9.06 ° φ ^ 13.48 ° .

Fig. 9
Fig. 9

Plots of the angles ω d , δ / 2 , α d , and β d = f ( φ ^ ) in an SGS for m = 1 , g = 1600 nm , λ min = 950 nm , and λ max = 1900 nm within the solution space 9.58 ° φ ^ 13.87 ° .

Fig. 10
Fig. 10

Plot of the attainable wavelength range for an SGS with m = 1 , g = { 417 , 556 , 700 , 833 } nm , and λ m = 580 nm for a MEMS device with a maximum scanning amplitude up to 15 ° .

Fig. 11
Fig. 11

Plot of the attainable wavelength range for an SGS with m = 1 , g = { 1052 , 1428 , 1600 , 1818 } nm , and λ m = 1425 nm for a MEMS device with a maximum scanning amplitude up to 15 ° .

Tables (4)

Tables Icon

Table 1 Oscillatory Optical Scanning Technologies Applied in MEMS Devices and the Associated Maximum Scanning Amplitudes

Tables Icon

Table 2 Solution Set of the Angles φ ^ , δ / 2 , ω d , α d , and β d for the Exemplary SGS Shown in Figs. 6, 7

Tables Icon

Table 3 Values of the Angles δ / 2 , ω d , α d , and β d for the Exemplary SGS Shown in Figs. 8, 9 for a MEMS Grating with φ ^ = 10 °

Tables Icon

Table 4 Maximum Attainable Wavelength Ranges Δ λ max for the Exemplary SGS in Figs. 10, 11 with Different Values of g and m = 1 , λ m = 580 nm , and φ ^ 15 ° , or m = 1 , λ m = 1425 nm , and φ ^ 15 ° , Respectively

Equations (36)

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

m λ g = sin ( α ) + sin ( β ) .
α = α d + φ , β = β d + φ , ω = ω d + φ .
m λ g = sin ( sign ( m ) ( α d + φ ) ) + sin ( sign ( m ) ( β d + φ ) ) .
| m | λ g = sin ( α d + φ ) + sin ( β d + φ ) .
| m | λ min g = sin ( α d φ ^ ) + sin ( β d φ ^ ) ,
| m | λ max g = sin ( α d + φ ^ ) + sin ( β d + φ ^ ) .
| m | λ d g = sin ( α d ) + sin ( β d ) .
| m | ( λ max + λ min ) g = sin ( α d + φ ^ ) + sin ( α d φ ^ ) + sin ( β d + φ ^ ) + sin ( β d φ ^ ) ,
| m | ( λ max λ min ) g = sin ( α d + φ ^ ) sin ( α d φ ^ ) + sin ( β d + φ ^ ) sin ( β d φ ^ ) .
| m | ( λ max + λ min ) g = 2 cos ( φ ^ ) ( sin ( α d ) + sin ( β d ) ) ,
| m | ( λ max λ min ) g = 2 sin ( φ ^ ) ( cos ( α d ) + cos ( β d ) ) .
| m | ( λ max + λ min ) 2 g cos ( φ ^ ) = 2 cos ( α d β d 2 ) sin ( α d + β d 2 ) ,
| m | ( λ max λ min ) 2 g sin ( φ ^ ) = 2 cos ( α d β d 2 ) cos ( α d + β d 2 ) .
δ = β α = const.
δ 2 = β d α d 2 .
α d = ω d δ 2 , β d = ω d + δ 2 .
ω d = β d + α d 2 .
| m | ( λ max + λ min ) 2 g cos ( φ ^ ) = 2 sin ( ω d ) cos ( δ 2 ) ,
| m | ( λ max λ min ) 2 g sin ( φ ^ ) = 2 cos ( ω d ) cos ( δ 2 ) .
( | m | ( λ max + λ min ) 2 g cos ( φ ^ ) ) 2 + ( | m | ( λ max λ min ) 2 g sin ( φ ^ ) ) 2 = 4 cos 2 ( δ 2 ) .
δ 2 = arccos ( | m | 2 g ( λ max + λ min 2 cos ( φ ^ ) ) 2 + ( λ max λ min 2 sin ( φ ^ ) ) 2 ) .
ω d = arcsin ( | m | 2 g λ max + λ min 2 cos ( φ ^ ) cos ( δ 2 ) ) .
ω d = arcsin ( λ max + λ min 2 cos ( φ ^ ) ( λ max + λ min 2 cos ( φ ^ ) ) 2 + ( λ max λ min 2 sin ( φ ^ ) ) 2 ) .
α d = arcsin ( λ max + λ min 2 cos ( φ ^ ) ( λ max + λ min 2 cos ( φ ^ ) ) 2 + ( λ max λ min 2 sin ( φ ^ ) ) 2 ) arccos ( | m | 2 g ( λ max + λ min 2 cos ( φ ^ ) ) 2 + ( λ max λ min 2 sin ( φ ^ ) ) 2 ) ,
β d = arcsin ( λ max + λ min 2 cos ( φ ^ ) ( λ max + λ min 2 cos ( φ ^ ) ) 2 + ( λ max λ min 2 sin ( φ ^ ) ) 2 ) + arccos ( | m | 2 g ( λ max + λ min 2 cos ( φ ^ ) ) 2 + ( λ max λ min 2 sin ( φ ^ ) ) 2 ) .
R = λ d λ = m N = m W g g .
δ 2 = sign ( m ) δ 2 , ω d = sign ( m ) ω d , α d = sign ( m ) α d , β d = sign ( m ) β d .
if     δ 2 = 0 { α d = β d = ω d | d α d d φ ^ | = | d β d d φ ^ | = | d δ d φ ^ | = .
| α , β | π 2 .
| α ^ , β ^ | π 2 .
| α d , β d | π 2 φ ^ .
φ ^ min = 1 2 arccos ( λ max λ min ( | m | 2 g ) 2 + ( 1 ( λ max | m | 2 g ) 2 ) ( 1 ( λ min | m | 2 g ) 2 ) ) ,
φ ^ max = 1 2 arccos ( λ min | m | 2 g + ( 1 λ max | m | 2 g ) ( 1 λ min 2 λ max | m | 2 g ) ) .
λ min = λ m Δ λ 2 , λ max = λ m + Δ λ 2 ,
φ ^ min = f ( m , g , λ m , Δ λ ) .
Δ λ max = f ( m , g , λ m , φ ^ ) .

Metrics