In , you expended considerable effort learning how to use the calculus to draw graphs of a wide
of functions in two dimensions. Now that we have discussed lines and planes in
3, we continue our graphical development by drawing more complicated objects in three dimensions. Don't expect a general theory like we developed for two-dimensional graphs. Drawing curves and surfaces in three dimensions by hand or correctly interpreting computer-generated graphics is something of an art. After all, you must draw a two-dimensional image that somehow represents an object in three dimensions. Our goal here is not to produce artists, but rather to leave you with the ability to deal with a small group of surfaces in three dimensions. For our presentation over the next several chapters, you will want to have at your disposal a small number of familiar surfaces. You will need to recognize these when you see them and have a reasonable facility for drawing a picture by hand. We also urge you to learn to produce and interpret computer-generated graphs. Follow our hints carefully and work lots of problems. In numerous exercises in the chapters that follow, taking a few extra minutes to draw a better graph will often result in a huge savings of time and effort.
Cylindrical Surfaces
We begin with a simple type of three-dimensional surface. When you see the word cylinder, you probably think of a right circular . For instance, consider the graph of the equation
in three dimensions. Your first reaction might be to say that this is the equation for a circle, but you'd only be partly correct. The graph of
in two dimensions is the circle of radius 3, centered at the , but what about its graph in three dimensions? Consider the intersection of the surface with the plane
Since the equation has no
's in it, the intersection with every such plane (called the trace of the surface in the plane
) is the same: a circle of radius 3, centered at the . Think about it: whatever this three-dimensional surface is, its intersection with every plane
- plane is a circle of radius
3 , centered at the . This describes a right circular , in this case one of radius 3, whose axis is the
- axis (see ).
Figure 10.52
Right circular .
More generally, the term
is used to refer to any surface whose traces in every plane
to a given plane are the same. With this definition, many surfaces qualify as cylinders.
Sketching a Surface
&Draw a graph of the surface
&Notice that since there are no
's in the equation, the trace of the graph in the plane
is the same for every
This is then a
whose trace in every plane
- plane is the parabola
To draw this, we first draw the trace in the
- plane (see ) and then make several copies of the trace, locating the vertices at various points along the
- axis and finally, connect the traces with lines
- axis to give the drawing its three-dimensional look (see ). A computer-generated wireframe graph of the same surface is seen in . Notice that the wireframe consists of numerous traces for fixed values of
Figure 10.53a
Trace in the yz - plane.
Figure 10.53b
Figure 10.53c
Wireframe of
Sketching an Unusual Cylinder
&Draw a graph of the surface
&Notice that once again, one of the variables is missing. In this case, there are no
's and so, traces of the surface in any plane
- plane are the same. They all look like the two-dimensional graph of
z = sin x.
We draw one of these in the
- plane and then make copies in planes
- plane, finally connecting the endpoints with lines
- axis (see ). In , we show a computer-generated wireframe plot of the same surface. In this case, the
looks like a plane with ripples in it.
Figure 10.54a
The surface
z = sin x.
Figure 10.54b
Wireframe:
z = sin x.
Quadric Surfaces
The graph of the equation
ax2+by2+cz2+dxy+eyz+fxz+gx+hy+jz+k = 0
in three-dimensional space (where
a, b, c, d, e, f, g, h, j
are all constants and at least one of
a, b, c, d, e
is nonzero) is referred to as a .
The most familiar quadric surface is the :
(x-a)2+(y-b)2+(z-c)2 = r2
centered at the point
(a, b, c).
To draw the
centered at
(0, 0, 0),
first draw a circle of radius
centered at the
- plane. Then, to give the surface its three-dimensional look, draw circles of radius
centered at the , in both the
- planes, as in . Note that due to the perspective, these last two circles will look like ellipses and will only be partially visible (we indicate the hidden parts of the circles with dashed lines).
Figure 10.55
A generalization of the
(Notice that when
d = e = f,
the surface is a .)
Sketching an Ellipsoid
&Graph the ellipsoid
&To get an idea of what the graph looks like, first draw its traces in the three coordinate planes. (In general, you may need to look at the traces in planes
to the three coordinate planes, but the traces in the three coordinate planes will suffice, here.) In the
so we have the ellipse
which we graph in . Next, add to
the traces in the
- planes. These are
respectively, and are both ellipses (see ).
CASs have the capability of plotting functions of several variables in three dimensions. Many graphing calculators with three-dimensional plotting capabilities only
Figure 10.56a
Ellipse in yz - plane.
Figure 10.56b
Ellipsoid.
produce three-dimensional plots when given
For the problem
at hand, notice that we can solve for
and plot the two functions
to obtain the graph of the surface. Observe that the wireframe graph in
is not particularly smooth and appears to have some gaps. To correctly interpret such a graph, you must mentally fill in the gaps. This requires an understanding of how the graph should look, which we obtained drawing .
As an alternative, many CASs enable you to graph the equation
mode. In this mode, the CAS numerically solves the equation for the value of
corresponding to each one of a large number of sample values of
and plots the resulting points. The graph obtained in
is an improvement over , but doesn't show the traces that we used to construct .
The best option, when available, is often a . In three dimensions, this involves writing each of the three variables
in terms of two parameters, with the resulting surface plotted by plotting points corresponding to a sample of values of the two parameters. (A more extensive discussion of the mathematics of parametric surfaces is given in .) As we develop in the exercises, parametric equations for the ellipsoid are
x = sin s cos t, y = 2sin s sin t&
z = 3cos s,
with the parameters taken to be in the intervals
Notice how
shows a nice smooth plot and clearly shows the elliptical traces.
Figure 10.56c
Wireframe ellipsoid.
Figure 10.56d
Implicit wireframe plot.
Figure 10.56e
Parametric plot.&
Sketching a Paraboloid
&Draw a graph of the quadric surface
x2+y2 = z.
&To get an idea of what the graph looks like, first draw its traces in the three coordinate planes. In the
- plane, we have
(a parabola). In the
- plane, we have
(a parabola). In the
- plane, we have
(a pointthe origin). We sketch the traces in . Finally, since the trace in the
- plane is just a point, we consider the traces in the planes
). Notice that these are the circles
x2+y2 = k , where for larger values of
(i.e., larger values of
), we get circles of larger radius. We sketch the surface in
. Such surfaces are called
and since the traces in planes
- plane are circles, this is called a .
Graphing utilities with three-dimensional capabilities generally produce a graph like
z = x2 + y2.
Notice that the parabolic traces are visible, but not the circular cross-sections we drew in . The four peaks visible in
are due to the rectangular
used for the plot (in this case,
An improvement to this can be made by restricting the
of z - values. With
you can clearly see the circular cross-section in the plane
As in , a parametric surface plot is even better. Here, we have
x = scos t,
y = ssin t&
clearly shows the circular cross sections in the planes
Figure 10.57a
Figure 10.57b
Paraboloid.
Figure 10.57c
Wireframe paraboloid.
Figure 10.57d
Wireframe paraboloid for
Figure 10.57e
Parametric plot paraboloid. &
Notice that in each of the last several examples, we have had to work hard to produce computer-generated graphs that adequately show the important features of the given quadric surface. We want to encourage you to use your graphing calculator or CAS for drawing three-dimensional plots, because computer graphics are powerful tools for visualization and problem solving. However, be aware that you will need a basic understanding of the geometry of quadric surfaces to effectively produce and interpret computer-generated graphs.
Sketching an Elliptic Cone
&Draw a graph of the quadric surface
&Be careful not to jump to conclusions. While this equation may look a lot like that of an ellipsoid, there is a significant difference. (Look where the
term is!) Again, we start by looking at the traces in the coordinate planes. For the
- plane, we have
That is, the trace is a pair of
We show these in . Likewise, the trace in the
- plane is a pair of lines:
The trace in the
- plane is simply the . (Why?) Finally, the traces in the planes
- plane are the ellipses:
Adding these to the drawing gives us the double-cone seen in .
Since the traces in planes
- plane are ellipses, we refer to this as an .
Notice that one way to plot this with a CAS is to graph the two functions
In , we restrict the z -
to show the circular cross sections). Notice that this plot shows a large gap between the two halves of the cone. If you have drawn
yourself, this plotting deficiency won't fool you. Alternatively, the parametric plot shown in , with
shows the full cone with its circular and
Figure 10.58a
Trace in yz - plane.
Figure 10.58b
Elliptic cone.
Figure 10.58c
Wireframe cone.
Figure 10.58d
Parametric plot. &
Sketching a Hyperboloid of One Sheet
&Draw a graph of the quadric surface
&The traces in the coordinate planes are as follows:
Further, notice that the trace of the surface in each plane
(parallel to the
- plane) is also an ellipse:
Finally, observe that the larger
is, the larger the axes of the ellipses are. Adding this information to , we draw the surface seen in . We call this surface a .
To plot this with a CAS, you could graph the two functions
(See , where we have restricted the z -
to show the circular cross sections.) Notice that this plot shows a small gap between the two halves of the hyperboloid. If you have drawn
yourself, this plotting problem won't fool you.
Alternatively, the parametric plot seen in , with
x = 2cos s cosh t,
y = sin s cosh t&
shows the full hyperboloid with its circular and hyperbolic traces.
Figure 10.59a
Trace in yz - plane.
Figure 10.59b
Hyperboloid of one sheet.
Figure 10.59c
Wireframe hyperboloid.
Figure 10.59d
Parametric plot. &
Sketching a Hyperboloid of Two Sheets
&Draw a graph of the quadric surface
&First, notice that this is the same equation as in , except for the sign of the
- term. As we have done before, we first look at the traces in the three coordinate planes. The trace in the
) is defined by
Since it is clearly impossible for two negative numbers to add up to something positive, this is a contradiction and there is no trace in the
- plane. That is, the surface does not
intersect the
- plane. The traces in the other two coordinate planes are as follows:
We show these traces in . Finally, notice that for
we have that
so that the traces in the plane
are ellipses for
k2 > 4 . It is important to notice here
k2 < 4 , the equation
has no solution. (Why is that?) So, if
-2 < k < 2 , the surface has no trace at all in the plane
x = k , leaving a gap which separates the hyperbola into two sheets. Putting this all together, we have the surface seen in . We call this surface a .
Figure 10.60a
Figure 10.60b
Hyperboloid of two sheets.
Figure 10.60c
Wireframe hyperboloid.
Figure 10.60d
Parametric plot.
We can plot this on a CAS by graphing the two functions
. (See , where we have restricted the
to show the circular cross sections.) Notice that this plot shows large gaps between the two halves of the hyperboloid. If you have drawn
yourself, this plotting deficiency won't fool you.
Alternatively, the parametric plot with
x = 2 cosh s, y = sinh s cos t&
sinh ssin t,
produces the right half of the hyper-boloid with its circular and hyperbolic traces. The left half of the hyperboloid has parametric equations
We show both halves in .
As our final example, we offer one of the more interesting quadric surfaces. It is also one of the more difficult surfaces to sketch.
Sketching a Hyperbolic Paraboloid
&Sketch the graph of the quadric surface defined by the equation
z = 2y2-x2.
&We first consider the traces in planes
to each of the coordinate planes:
parallel to xy-plane (z = k): 2y2 -
x2 = k (hyperbola, for k
parallel to xz - plane (y = k) :z =
- x2 + 2k2
opening down)
parallel to yz-plane (x = k) :z = 2y2-k2 (parabola opening up).
We begin by drawing the traces in the
- planes, as seen in . Since the trace in the
- plane is the degenerate hyperbola
(two lines:
), we instead draw the trace in several of the planes
Notice that for
these are hyperbolas opening toward the positive and negative
- direction and for
these are hyperbolas opening toward the positive and negative
- direction. We indicate one of these for
and one for
in , where we show a sketch of the surface. We refer to this surface as a . More than anything else, the surface resembles a saddle. In fact, we refer to the
for this graph. (We'll discuss the significance of saddle points in .)
Figure 10.61a
Traces in the
Figure 10.61b
The surface
z = 2y2 - x2.
Figure 10.61c
Wireframe plot of
z = 2y2-x2.
A wireframe graph of
z = 2y2 - x2
is shown in
and where we limited the
Note that only the parabolic cross sections are drawn, but the graph shows all the features of . Plotting this surface parametrically is fairly tedious (requiring four different sets of equations) and doesn't improve the graph noticeably.
An Application
You may have noticed the large number of paraboloids around you. For instance, radiotelescopes and even home satellite television dishes have the shape of a portion of a paraboloid. Reflecting telescopes have parabolic mirrors which again, are a portion of a
paraboloid. There is a very good reason for this. It turns out that in all of these cases, light waves and radio waves striking
point on the parabolic dish or mirror are reflected toward
point, the focus of each parabolic cross section through the vertex of the paraboloid. This remarkable fact means that all light waves and radio waves end up being concentrated at just one point. In the case of a radiotelescope, placing a small receiver just in front of the focus can take a very faint signal and increase its effective strength immensely (see ). The same principle is used in optical telescopes to concentrate the light from a faint source (e.g., a distant star). In this case, a small mirror is mounted in a line from the parabolic mirror to the focus. The small mirror then reflects the concentrated light to an eyepiece for viewing (see ).
Figure 10.62
Radiotelescope.
Figure 10.63
Reflecting telescope.
& 2002 McGraw-Hill Companies, Inc.Wenhao Li,&Mikhail Balabas,&Xiang Peng,&Szymon
Pustelny,&Arne Wickenbrock,&Hong Guo,& and Dmitry
Budker, Characterization of high-temperature performance of cesium
vapor cells with anti-relaxation coating, . 121, 17); doi: 10.6017,
O. Yu. Tretiak, J. W. Blanchard, D. Budker, P. K. Olshin, S. N.
Smirnov, and M. V. Balabas, Raman and nuclear magnetic resonance
investigation of alkali metal vapor interaction with alkene-based
anti-relaxation coating,
144, 094707
A. M. Hibberd, S. J. Seltzer, M. V. Balabas, M. Morse, D. Budker,
and S. L. Bernasek, Light-induced changes in an alkali metal atomic
vapor cell coating studied by X-ray photoelectron spectroscopy,&114,
Eric P. Corsini, Todor Karaulanov, Mikhail
Balabas, and Dmitry Budker, Hyperfine frequency shift and Zeeman
relaxation in alkali-metal-vapor cells with antirelaxation alkene
87, 022901
Mikhail V. Balabas, Todor Karaulanov, Micah P. Ledbetter, and Dmitry
Budker, Polarized alkali vapor with minute-long transverse
spin-relaxation time,& 105, 070801
V. M. Acosta, A. Jarmola, D. Windes, E. Corsini, M. P. Ledbetter,
T.&Karaulanov, M. Auzinsh, S. A. Rangwala,
D. F. Jackson Kimball, and D. Budker, Rubidium dimers in
paraffin-coated cells,
12 ( (2010);
S. J. Seltzer, D. J. Michalak, M. H. Donaldson, M. V. Balabas, S. K.
Barber, S. L. Bernasek, M.-A. Bouchiat, A. Hexemer, A. M. Hibberd, D.
F. Jackson Kimball, C. Jaye, T. Karaulanov, F. A. Narducci, S. A.
Rangwala, H. G. Robinson, A. K. Shmakov, D. L. Voronov, V. V.
Yashchuk, A. Pines, and D. Budker, Investigation of Anti-Relaxation
Coatings for Alkali-Metal Vapor Cells Using Surface Science
Techniques,
133, 144703
D. F. Jackson Kimball, Khoa Nguyen, K. Ravi, Arijit Sharma, Vaibhav
S. Prabhudesai, S. A. Rangwala, V. V. Yashchuk, M. V. Balabas, and D.
Budker, Electric-field-induced change of alkali-metal vapor density in
paraffin-coated cells, &79, 09);
T. Karaulanov, M. T. Graf, D. English, S. M. Rochester, Y. Rosen, K.
Tsigutkin, D. Budker, M. V. Balabas, D. F. Jackson Kimball, F. A.
Narducci, S. Pustelny, V. V. Yashchuk, Controlling atomic vapor
density in paraffin-coated cells using light-induced atomic
desorption,& 79, 09);
J. S. Guzman, A. Wojciechowski, J. E. Stalnaker, K. Tsigutkin, V. V.
Yashchuk, D. Budker, Nonlinear magneto-optical rotation, Zeeman and
hyperfine relaxation of potassium atoms in a paraffin-coated
cell,& 74(5) &053415
M. T. Graf, D. F. Kimball, S. M. Rochester, K. Kerner, C. Wong, D.
Budker, E. B. Alexandrov, M. V. Balabas, and V. V. Yashchuk,
Relaxation of atomic polarization in paraffin-coated cesium vapor
D. Budker, L. Hollberg, D. F. Kimball, J. Kitching, S. Pustelny, and
V. V. Yashchuk, Investigation of microwave transitions and nonlinear
magneto-optical rotation in anti-relaxation-coated cells,
E. B. Alexandrov, M. V. Balabas, D. Budker, D. S. English, D.F.
Kimball, C.-H. Li, and V.V. Yashchuk, ,
66, 02); erratum:
70, e049902
Nonlinear magneto-
and electro- atomic magnetometry
Guzhi Bao,&Arne Wickenbrock,&Simon Rochester,&and
Dmitry Budker, Suppression of nonlinear Zeeman effect and heading
error in earth-field-range alkali-vapor magnetometers,
E. Mariotti et al, Forty years after the first dark resonance
experiment: an overview of the COSMA project results,
10226, 19th International Conference and School on Quantum
Electronics: Laser Physics and Applications, 102260K (2017);
doi:10.4896
Arne Wickenbrock, Nathan Leefer, John W. Blanchard, and Dmitry
Budker, Eddy current imaging with an atomic radio-frequency
magnetometer,
108, 183507
Elena Zhivun, Arne Wickenbrock,
Julia Sudyka, Szymon Pustelny, Brian Patton, and Dmitry Budker, Light
shift averaging in paraffin-coate&
24(14), 1);
S. Pustelny, L. Busaite, A.
Akulshin, M. Auzinsh, N. Leefer, and D. Budker, Nonlinear
Magneto-Optical Rotation in Rubidium Vapor Excited with Blue
92, 053410
I. Mateos, B. Patton, E. Zhivun,
D. Budker, D. Wurm, J. Ramos-Castro, Noise characterization of an
atomic magnetometer at sub-millihertz frequencies,
Elena Zhivun, Arne Wickenbrock,
Brian Patton, and Dmitry Budker, Alkali-vapor magnetic resonance
driven by fictitious ra &105,
B. Patton, E. Zhivun, D. C.
Hovde, and D. Budker, All-Optical Vector Atomic Magnetometer,
113, 013001
Dmitry Budker, Magnetometry: Techniques,
Developments, Applications, at &: The A to Z
of Quantum Science (May 2011); &&
Brian Patton,&Oscar
Versolato,&D. Chris Hovde,&Eric Corsini,&James Higbie,
and Dmitry Budker, &A Remotely& Interrogated
All-Optical&87Rb Magnetometer, & 101, 083502
Chris Hovde, Brian Patton, Oscar
Versolato, Eric Corsini, Simon Rochester, and Dmitry Budker, Heading
error in an alignment-based magnetometer,&DOI:&10.953
T. Zigdon, A. D. Wilson-Gordon,
S. Guttikonda, E. J. Bahr, O. Neitzke,&S. M. Rochester, and D.
Budker,&Nonlinear magneto-optical rotation in the presence of
radio-frequency field,
Adam Wojciechowski,&Eric
Corsini,&Jerzy Zachorowski, and&Wojciech
Gawlik,&Nonlinear Faraday rotation and detection of
superposition states in cold atoms, &Phys.
Rev. A 81, 053420
Chris Hovde, Brian Patton, Eric Corsini, James Higbie, and Dmitry
Budker, Sensitive optical atomic magnetometer based on nonlinear
magneto-optical rotation,& 7693, 10); (
M. Auzinsh, D. Budker, and S. M. Rochester, Light-induced
polarization effects in atoms with partially resolved hyperfine
structure and applications to absorption, fluorescence, and nonlinear
magneto-optical rotation, 80(5), 09). ()
D. F. Jackson Kimball, L. R. Jacome, Srikanth Guttikonda, Eric J.
Bahr, and Lok Fai Chan, Magnetometric sensitivity optimization for
nonlinear optical rotation with frequency-modulated light: Rubidium D2
line,. 106, 09). ()
K. Jensen, V. M. Acosta, J. M. Higbie, M. P. Ledbetter, S. M.
Rochester, and D. Budker, Cancellation of nonlinear Zeeman shifts with
light shifts, 79(2), 09); &al
Chris Hovde, Victor M. Acosta, Eric Corsini, James M. Higbie, Micah
P. Ledbetter, and Dmitry Budker, Nonlinear magneto-optical rotation
for sensitive measurement of magnetic fields, in&Biomedical
Optics, OSA Technical Digest (CD) (Optical Society of
America, 2008), .&
V. M. Acosta, M. Auzinsh, W. Gawlik, P. Grisins, J. M.
Higbie,& Derek F. Jackson Kimball, L. Krzemien, M. P. Ledbetter,
S. Pustelny, S. M. Rochester, V. V. Yashchuk, and D. Budker,
Production and detection of atomic hexadecapole at Earth's magnetic
&& 16(15),& (2008) &
M. P. Ledbetter, I. M. Savukov, V. M. Acosta, D. Budker, and M. V.
Romalis, Spin-exchange relaxation free magnetometry with Cs vapor, Phys.
A77(3), 033408
Review article:&D. Budker and M. V. Romalis, Optical
Magnetometry,& 3, 227 - 234 (2007);&
S. Pustelny, A. Wojciechowski, M. Kotyrba, K. Sycz, J. Zachorowski,
W. Gawlik,&A. Cingoz, N. Leefer, J. M. Higbie, E. Corsini, M. P.
Ledbetter,&S. M. Rochester, A. O. Sushkov, and D. Budker,
All-optical atomic magnetometers based on nonlinear magneto-optical
rotation with amplitude modulated light,
6604 (2007);
M. P. Ledbetter, V. M. Acosta, S. M. Rochester, D. Budker, S.
Pustelny, and V. V. Yashchuk, Detection of radio frequency magnetic
fields using nonlinear magneto-optical rotation,&75(2),&023405
J. M. Higbie, E. Corsini, and D. Budker, Robust, High-Speed,
All-Optical Atomic Magnetometer, . 77(11),
S. Pustelny, D. F. Jackson Kimball, S. M. Rochester, V. V. Yashchuk,
D. Budker, Influence of magnetic-field inhomogeneity on nonlinear
magneto-optical resonances,& 74, 06); ;
this paper is also available from
S. Pustelny, S. M. Rochester, D. F. Jackson Kimball, V. V. Yashchuk,
D. Budker, and W. Gawlik, Nonlinear magneto-optical rotation with
modulated light in tilted magnetic fields,
74, 06); ;
this paper is also available from
V. Acosta, M. P. Ledbetter, S. M. Rochester, D. Budker, D. F.
Jackson-Kimball, D. C. Hovde, W. Gawlik, S. Pustelny, and J.
Zachorowski, Nonlinear magneto-optical rotation with
frequency-modulated light in the geophysical field range,
S. Pustelny, D. F. Jackson Kimball, S. M. Rochester, V. V. Yashchuk,
W. Gawlik, and D. Budker, Pump-probe nonlinear magneto-optical
rotation with frequency modulated light,
M. V. Balabas, D. Budker, J. Kitching, P. D. D. Schwindt, and J. E.
Stalnaker, Magnetometry with millimeter-scale anti-relaxation-coated
W. Gawlik, L. Krzemien, S. Pustelny, D. Sangla, J. Zachorowski, M.
Graf, A.O. Sushkov, and D. Budker, Nonlinear Magneto-Optical Rotation
with Amplitude-Modulated L ,
Review article: E. B.
Alexandrov, M. Auzinsh, D. Budker, D. F. Kimball, S. M. Rochester, and
V. V. Yashchuk, Dynamic effects in nonlinear magneto-optics of atoms
in a Special Issue of
on Nonlinear and Integrated Magneto-O ;
Book chapter:&Yu. Malakyan, D. Budker, S. Rochester, D.
Kimball, V. Yashchuk and W. Gawlik,&, in&,&NATO
Science Series, v. 189,&V.M.&Akulin, A.&Sarfati,
G.&Kurizki and S.&Pellegrin, Eds.,
&pp.&91-104,&Springer Netherlands, 2005
Comment: D. Budker and S. M. Rochester,& A relation
between electromagnetically induced absorption resonances and
nonlinear magneto-optics in Lambda-systems,
Yu. P. Malakyan, S. M. Rochester, D. Budker, D. F. Kimball, and V.
V. Yashchuk, Nonlinear magneto-optical rotation of frequency-modulated
light resonant with a low-J transition,
V.V. Yashchuk, D. Budker, W. Gawlik, D.F. Kimball, Yu. P. Malakyan,
S.M. Rochester, Selective addressing of high-rank atomic polarization
90, 03) ; .
D. Budker,
(Nature News and Views), &.
Review article: D. Budker, W. Gawlik, D.F. Kimball, S.M.
Rochester, V.V. Yashchuk, A. Weis, ;
S.M. Rochester and D. Budker, Nonlinear magneto-optical rotation in
optically thick media, , 49(14-15),
D. Budker, D.F. Kimball, V.V. Yashchuk, and M. Zolotorev, ,
D. Budker, D.F. Kimball, S.M. Rochester, and V.V. Yashchuk, ,
S.M. Rochester, D. S. Hsiung, D. Budker, R. Y. Chiao, D.F. Kimball,
and V.V. Yashchuk, Self-rotation of resonant elliptically polarized
light in collision-free rubidium vapor,
D. Budker, D.F. Kimball, S.M. Rochester, and V.V. Yashchuk,
(ICAP-2000 abstract).
D. Budker, D.F. Kimball, S.M. Rochester, and V.V. Yashchuk, ,
D. Budker, D.F. Kimball, S.M. Rochester, V.V. Yashchuk, and M.
Zolotorev, ,
D. Budker, R.Y. Chiao, D. S. Hsiung, S.M. Rochester, and V.V.
(QELS-2000, paper QFC4).
V.V. Yashchuk, D. Budker, and J. Davis, Laser Frequency
Stabilization Using Linear Magneto-Optics,
71(2), 341, 2000.
D. Budker, D.F. Kimball, S.M. Rochester, and V.V. Yashchuk, ,
LBNL PUB-5453, Berkeley, California, December, 1999
D. Budker, D.F. Kimball, S.M. Rochester, and V.V. Yashchuk, ,
D. Budker, V.V. Yashchuk, and M. Zolotorev, ,
81(26), ).
V.V. Yashchuk, D. Budker, and M. Zolotorev, , in:
Trapped Charged Particles and Fundamental Physics, D.H.E. Dubin and D.
Schneider, eds. AIP conference proceedings 457, pp. 177-181.
D. Budker, D.J. Orlando, and V.V. Yashchuk,
(Word-97);
67(7), 584 (1999).
D. Budker, V.V. Yashchuk, and M. Zolotorev,
(Word-97, PDF,
.rtf, or .PS); Sib. J. Phys.1 (1999); see also: .
Detection of
magnetic microparticles
D. Maser, S. Pandey, H. Ring, M.
P. Ledbetter, S. Knappe, J. Kitching, and D. Budker, Detection of a
single cobalt microparticle with a microfabricated atomic
magnetometer, . 82, 086112
S. Xu, M. H. Donaldson, A.
Pines, S. M. Rochester, D. Budker, and V. V. Yashchuk, Application of
atomic magnetometry in magnetic particle detection,&;& ;
this paper is also available from the &
Ion Traps and Antimatter
Nathan Leefer, Kai Krimmel, William Bertsche, Dmitry Budker, Joel
Fajans, Ron Folman, Hartmut H鋐fner, and Ferdinand Schmidt-Kaler,
Investigation of two-frequency Paul traps for antihydrogen production,
, 238(1), 1-18, DOI
10.-016-1388-0,
Cosmic Axion Spin-Precession Experiment (CASPEr)
Antoine Garcon, Deniz Aybas, John W.
Blanchard, Gary Centers, Nataniel L. Figueroa, PeterW. Graham, Derek
F. Jackson Kimball,Surjeet Rajendran, Marina G. Sendra, Alexander O.
Sushkov, Lutz Trahms, Tao Wang, Arne Wickenbrock, Teng Wu, and
Dmitry Budker,
Tao Wang, Derek F. Jackson Kimball, Alexander O. Sushkov, Deniz
Aybas, John W. Blanchard, Gary Centers, Sean R. O Kelley, Jiancheng
Fang, and Dmitry Budker, Application of Spin-Exchange Relaxation-Free
Magnetometry to the Cosmic Axion Spin Precession Experiment,
Dmitry Budker, Peter W. Graham, Micah Ledbetter, Surjeet Rajendran,
and Alex Sushkov, Cosmic Axion Spin Precession Experiment (CASPEr), PHYSICAL
REVIEW X 4, 021030
(2014); ()
Global Network of Optical Magnetometers for Exotic
physics searches (GNOME)
D. F. Jackson Kimball, D. Budker, J. Eby, M.
Pospelov, S.
Pustelny, T. Scholtes, Y. V. Stadnik, A. Weis, and A. Wickenbrock,
Searching for axion stars and Q-balls with a terrestrial
magnetometer network,
S. Pustelny, D. F. Jackson Kimball, C. Pankow, M. P. Ledbetter, P.
Wlodarczyk, P. Wcislo, M. Pospelov, J. Smith, J. Read, W. Gawlik, and
D. Budker, &Global Network of Optical Magnetometers for Exotic
Physics:& Novel scheme for exo Annalen der
Physik (2013);
M. Pospelov,&S. Pustelny,&M. P. Ledbetter,&D. F.
Jackson Kimball,&W. Gawlik, and D. Budker, &Detecting Domain
Walls of Axionlike Models Using Terrestrial Experiments,&Phys.
Rev. Lett. 110,
13) (How do you know if you ran through a
see also a
about this work
Przemyslaw Wlodarczyk, Szymon Pustelny, Dmitry Budker, and Marcin
Lipinski, Multi-Channel Data Acquisition System with Absolute Time
Synchronization,
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